Abstract.
The quantum discord of bipartite systems is one of the best-known measures of non-classical correlations and an important quantum resource. In the recent work appeared in [Phys. Rev. Lett 2020, 124:110401], the quantum discord has been generalized to multipartite systems. In this paper, we give analytic solutions of the quantum discord for tripartite states with fourteen parameters.
Key words and phrases:
Quantum discord, quantum correlations, tripartite quantum states, optimization on manifolds
*Corresponding author: Xiaoli Hu (xiaolihumath@jhun.edu.cn)
The quantum discord usually involves with quantum entanglement and umentangled quantum correlations in quantum systems. It measures the total non-classical correlation in a quantum system, and has attracted widespread attention since its appearance. Applications of the non-entanglement quantum correlations in quantum information processings have been extensively studied, including the quantum computing scheme of DQC1 [1] and Grover search algorithm [2] etc. This partly explains why quantum schemes surpass classical schemes. Meanwhile, the quantum discord as a non-classical correlation is one of the important quantum resources and is ubiquitous in many areas of modern physics ranging from condensed matter physics, quantum optics, high-energy physics to quantum chemistry, thus can be regarded as one of the fundamental non-classical correlations besides entanglement and EPR-steerable states [3, 4].
The quantum discord is defined as the maximal difference between the quantum mutual information without and with a von Neumann projective measurement applying to one part of the bipartite system. For tripartite and lager systems, some generalizations of the discord have been proposed [5, 6, 7, 8, 9, 10], and have been used in quantum information processings. It is well-known that quantum discord is extremely difficult to evaluate and most exact solutions are only for the X-type quantum states (cf. [11, 12, 13, 14]). This paper is devoted to quantification of the quantum correlation in tripartite and larger systems to derive some exact solutions for non-X-type states, and we hope it can contribute to better understanding and more effective use of quantum states in realizing quantum information processing schemes.
The paper is organized as follows. We first introduce the generalized discord for tripartite systems [10] based on that of bipartite systems [3]. We derive analytic solutions for tripartite states with fourteen parameters. Furthermore, the quantum discord of some well-known states (such as GHZ states) are computed.
2. Generalization of quantum discord to tripartite states
For a bipartite state Ļbcsuperscriptitalic-Ļšš\phi^bcitalic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT on system HBāHCtensor-productsubscriptš»šµsubscriptš»š¶H_B\otimes H_Citalic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ā italic_H start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, the quantum mutual information is I(Ļbc):=SB(Ļb)+SC(Ļc)-SBC(Ļbc)assignš¼superscriptitalic-Ļššsubscriptššµsuperscriptitalic-Ļšsubscriptšš¶superscriptitalic-Ļšsubscriptššµš¶superscriptitalic-ĻššI(\phi^bc):=S_B(\phi^b)+S_C(\phi^c)-S_BC(\phi^bc)italic_I ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) := italic_S start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) + italic_S start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ), where S(ĻX)=TrĻXlog2(ĻX)šsuperscriptitalic-ĻšTrsuperscriptitalic-Ļšsubscript2superscriptitalic-ĻšS(\phi^X)=\mathrmTr\phi^X\log_2(\phi^X)italic_S ( italic_Ļ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) = roman_Tr italic_Ļ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) is the von Neumann entropy of the quantum state on system X. Set Ī kBsubscriptsuperscriptĪ šµš\\Pi^B_k\ roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT to be an one-dimensional von Neumann projection operator on subsystem BšµBitalic_B which satisfies ākĪ kB=I,(Ī kB)2=Ī kB,Ī kBĪ kā²B=Ī“kkā²formulae-sequencesubscriptšsuperscriptsubscriptĪ ššµš¼formulae-sequencesuperscriptsuperscriptsubscriptĪ ššµ2superscriptsubscriptĪ ššµsuperscriptsubscriptĪ ššµsuperscriptsubscriptĪ superscriptšā²šµsubscriptšæšsuperscriptšā²\sum_k\Pi_k^B=I,(\Pi_k^B)^2=\Pi_k^B,\Pi_k^B\Pi_k^\prime% ^B=\delta_kk^^\primeā start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Ī start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = italic_I , ( roman_Ī start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = roman_Ī start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , roman_Ī start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT roman_Ī start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ā² end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = italic_Ī“ start_POSTSUBSCRIPT italic_k italic_k start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ā² end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Then the state Ļbcsuperscriptitalic-Ļšš\phi^bcitalic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT under the measurement Ī kBsubscriptsuperscriptĪ šµš\\Pi^B_k\ roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is changed into
Ļkc=1pkTrB(IāĪ kB)Ļbc(IāĪ kB)subscriptsuperscriptitalic-Ļšš1subscriptššsubscriptTršµtensor-productš¼subscriptsuperscriptĪ šµšsuperscriptitalic-Ļšštensor-productš¼subscriptsuperscriptĪ šµš\phi^c_k=\frac1p_k\mathrmTr_B(I\otimes\Pi^B_k)\phi^bc(I% \otimes\Pi^B_k)italic_Ļ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_I ā roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ( italic_I ā roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )
with the probability pk=Tr(IāĪ kB)Ļbc(IāĪ kB)subscriptššTrtensor-productš¼subscriptsuperscriptĪ šµšsuperscriptitalic-Ļšštensor-productš¼subscriptsuperscriptĪ šµšp_k=\mathrmTr(I\otimes\Pi^B_k)\phi^bc(I\otimes\Pi^B_k)italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = roman_Tr ( italic_I ā roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ( italic_I ā roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ). For simplicity, we denote by Ī XsuperscriptĪ š\Pi^Xroman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT the measurement Ī kXsuperscriptsubscriptĪ šš\\Pi_k^X\ roman_Ī start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT on system XšXitalic_X. The quantum conditional entropy is simply given by SC|Ī B(Ļbc)=ākpkS(Ļkc)subscriptšconditionalš¶superscriptĪ šµsuperscriptitalic-Ļššsubscriptšsubscriptšššsubscriptsuperscriptitalic-ĻššS_C(\phi^bc)=\sum_kp_kS(\phi^c_k)italic_S start_POSTSUBSCRIPT italic_C | roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = ā start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_S ( italic_Ļ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ). Then the measurement-induced quantum mutual information is given by
C(Ļbc)=SC(Ļc)-minSC|Ī B(Ļbc).š¶superscriptitalic-Ļššsubscriptšš¶superscriptitalic-Ļšsubscriptšconditionalš¶superscriptĪ šµsuperscriptitalic-ĻššC(\phi^bc)=S_C(\phi^c)-\min S_C(\phi^bc).italic_C ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) - roman_min italic_S start_POSTSUBSCRIPT italic_C | roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) .
By Olliver and Zurek [3], the original definition of the quantum discord Q(Ļ)ššQ(\rho)italic_Q ( italic_Ļ ) is the difference of the quantum mutual information I(Ļbc)š¼superscriptitalic-ĻššI(\phi^bc)italic_I ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) and the measurement-induced quantum mutual information C(Ļbc)š¶superscriptitalic-ĻššC(\phi^bc)italic_C ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ), i.e.
(2.1) Q(Ļbc)=I(Ļbc)-C(Ļbc)=minĪ BSC,šsuperscriptitalic-Ļššš¼superscriptitalic-Ļššš¶superscriptitalic-ĻššsubscriptsuperscriptĪ šµsubscriptšconditionalš¶superscriptĪ šµsuperscriptitalic-Ļššsubscriptšconditionalš¶šµsuperscriptitalic-ĻššQ(\phi^bc)=I(\phi^bc)-C(\phi^bc)=\min_\Pi^B\S_C(\phi^bc% )-S_B(\phi^bc)\,italic_Q ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = italic_I ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_C ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = roman_min start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_C ,
where SC|B(Ļbc)=SBC(Ļbc)-SB(Ļb)subscriptšconditionalš¶šµsuperscriptitalic-Ļššsubscriptššµš¶superscriptitalic-Ļššsubscriptššµsuperscriptitalic-ĻšS_B(\phi^bc)=S_BC(\phi^bc)-S_B(\phi^b)italic_S start_POSTSUBSCRIPT italic_C | italic_B end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) is the unmeasured conditional state on subsystem Cš¶Citalic_C.
For the tripartite system HAāHBāHCtensor-productsubscriptš»š“subscriptš»šµsubscriptš»š¶H_A\otimes H_B\otimes H_Citalic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ā italic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ā italic_H start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, we consider the BCšµš¶BCitalic_B italic_C composite system as the first subsystem and Aš“Aitalic_A-system as the second subsystem. The state Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT of system HAāHBāHCtensor-productsubscriptš»š“subscriptš»šµsubscriptš»š¶H_A\otimes H_B\otimes H_Citalic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ā italic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ā italic_H start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT gives arise to a state on BCšµš¶BCitalic_B italic_C-subsystem after the von Neumann measurement Ī jAsuperscriptsubscriptĪ šš“\\Pi_j^A\ roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT on Aš“Aitalic_A subsystem. Namely, it takes the following form:
(2.2) Ļjbc=1pjbcTrA(Ī jAāI)Ļabc(Ī jAāI)subscriptsuperscriptšššš1subscriptsuperscriptššššsubscriptTrš“tensor-productsuperscriptsubscriptĪ šš“š¼superscriptšššštensor-productsuperscriptsubscriptĪ šš“š¼\beginsplit\rho^bc_j=\frac1p^bc_j\mathrmTr_A(\Pi_j^A% \otimes I)\rho^abc(\Pi_j^A\otimes I)\endsplitstart_ROW start_CELL italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG roman_Tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā italic_I ) italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā italic_I ) end_CELL end_ROW
with probability pjbc=Tr(Ī jAāI)Ļabc(Ī jAāI)subscriptsuperscriptššššTrtensor-productsuperscriptsubscriptĪ šš“š¼superscriptšššštensor-productsuperscriptsubscriptĪ šš“š¼p^bc_j=\mathrmTr(\Pi_j^A\otimes I)\rho^abc(\Pi_j^A\otimes I)italic_p start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = roman_Tr ( roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā italic_I ) italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā italic_I ).The measured quantum mutual information of Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT is naturally given by
(2.3) š„(Ļabc|Ī A)=SBC(Ļbc)-SBC|Ī A(Ļabc).š„conditionalsuperscriptššššsuperscriptĪ š“subscriptššµš¶superscriptšššsubscriptšconditionalšµš¶superscriptĪ š“superscriptšššš\beginsplit\mathcalJ(\rho^abc|\Pi^A)=S_BC(\rho^bc)-S_\Pi^A(% \rho^abc).\endsplitstart_ROW start_CELL caligraphic_J ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) . end_CELL end_ROW
The quantity of classical correlation of the tripartite state Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT is
(2.4) š(Ļabc)=maxĪ Aš„(Ļabc|Ī A)=SBC(Ļbc)-minĪ ASBC|Ī A(Ļabc).šsuperscriptššššsubscriptsuperscriptĪ š“š„conditionalsuperscriptššššsuperscriptĪ š“subscriptššµš¶superscriptšššsubscriptsuperscriptĪ š“subscriptšconditionalšµš¶superscriptĪ š“superscriptšššš\beginsplit\mathcalC(\rho^abc)=\max_\Pi^A\mathcalJ(\rho^abc|\Pi^% A)=S_BC(\rho^bc)-\min_\Pi^AS_BC(\rho^abc).\endsplitstart_ROW start_CELL caligraphic_C ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = roman_max start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT caligraphic_J ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - roman_min start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) . end_CELL end_ROW
We know that the quantum mutual information I(Ļabc)=SA(Ļa)+SBC(Ļbc)-SABC(Ļabc)š¼superscriptššššsubscriptšš“superscriptššsubscriptššµš¶superscriptšššsubscriptšš“šµš¶superscriptššššI(\rho^abc)=S_A(\rho^a)+S_BC(\rho^bc)-S_ABC(\rho^abc)italic_I ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) + italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ). Similar to Eq.(2.1), the generalized quantum discord of the tripartite state Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT can be defined as
(2.5) š¬(Ļabc)=I(Ļabc)-C(Ļabc)=minĪ ASBC,š¬superscriptššššš¼superscriptššššš¶superscriptššššsubscriptsuperscriptĪ š“subscriptšconditionalšµš¶superscriptĪ š“superscriptššššsubscriptšconditionalšµš¶š“superscriptšššš\beginsplit\mathcalQ(\rho^abc)=I(\rho^abc)-C(\rho^abc)=\min_\Pi^A% \S_BC(\rho^abc)-S_BC(\rho^abc)\,\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = italic_I ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) - italic_C ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = roman_min start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) , end_CELL end_ROW
where SBC|A(Ļabc)=SABC(Ļabc)-SA(Ļa)subscriptšconditionalšµš¶š“superscriptššššsubscriptšš“šµš¶superscriptššššsubscriptšš“superscriptššS_BC(\rho^abc)=S_ABC(\rho^abc)-S_A(\rho^a)italic_S start_POSTSUBSCRIPT italic_B italic_C | italic_A end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) is the unmeasured conditional entropy on BCšµš¶BCitalic_B italic_C-bipartite subsystem.
In order to evaluate the quantity minĪ ASBC|Ī A(Ļabc)subscriptsuperscriptĪ š“subscriptšconditionalšµš¶superscriptĪ š“superscriptšššš\min_\Pi^AS_BC(\rho^abc)roman_min start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ), the multipartite measurement based on conditional operators can be constructed as follows: [15]
(2.6) Ī jkAB=Ī jAāĪ k|jBsuperscriptsubscriptĪ ššš“šµtensor-productsuperscriptsubscriptĪ šš“superscriptsubscriptĪ conditionalšššµ\beginsplit\Pi_jk^AB=\Pi_j^A\otimes\Pi_j^B\endsplitstart_ROW start_CELL roman_Ī start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā roman_Ī start_POSTSUBSCRIPT italic_k | italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_CELL end_ROW
with the measurement ordering from Aš“Aitalic_A to BšµBitalic_B. The projector Ī k|jBsuperscriptsubscriptĪ conditionalšššµ\Pi_k^Broman_Ī start_POSTSUBSCRIPT italic_k | italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT on subsystem BšµBitalic_B is conditional measurement outcome of Aš“Aitalic_A. These projectors satisfy ākĪ k|jB=IB,ājĪ jA=IAformulae-sequencesubscriptšsubscriptsuperscriptĪ šµconditionalššsuperscriptš¼šµsubscriptšsuperscriptsubscriptĪ šš“superscriptš¼š“\sum_k\Pi^B_k=I^B,\sum_j\Pi_j^A=I^Aā start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k | italic_j end_POSTSUBSCRIPT = italic_I start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , ā start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = italic_I start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT. Then after the measurement Ī jkABsuperscriptsubscriptĪ ššš“šµ\Pi_jk^ABroman_Ī start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, the state Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT is collapsed to a state on subsystem Cš¶Citalic_C, i.e.
(2.7) Ļjkc=1pjkcTrAB(Ī jkABāI)Ļabc(Ī jkABāI)subscriptsuperscriptšššš1subscriptsuperscriptššššsubscriptTrš“šµtensor-productsuperscriptsubscriptĪ ššš“šµš¼superscriptšššštensor-productsuperscriptsubscriptĪ ššš“šµš¼\beginsplit\rho^c_jk=\frac1p^c_jk\mathrmTr_AB(\Pi_jk^AB% \otimes I)\rho^abc(\Pi_jk^AB\otimes I)\endsplitstart_ROW start_CELL italic_Ļ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT end_ARG roman_Tr start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( roman_Ī start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ā italic_I ) italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Ī start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ā italic_I ) end_CELL end_ROW
with the probability pjkc=Tr(Ī jkABāI)Ļabc(Ī jkABāI)subscriptsuperscriptššššTrtensor-productsuperscriptsubscriptĪ ššš“šµš¼superscriptšššštensor-productsuperscriptsubscriptĪ ššš“šµš¼p^c_jk=\mathrmTr(\Pi_jk^AB\otimes I)\rho^abc(\Pi_jk^AB\otimes I)italic_p start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT = roman_Tr ( roman_Ī start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ā italic_I ) italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Ī start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ā italic_I ). The conditional entropy after the ABš“šµABitalic_A italic_B-bipartite measurement is
SC|Ī AB(Ļabc)=ājklpjkcĪ»l(jk)log2Ī»l(jk),subscriptšconditionalš¶superscriptĪ š“šµsuperscriptššššsubscriptšššsuperscriptsubscriptššššsuperscriptsubscriptššššsubscript2superscriptsubscriptššššS_\Pi^AB(\rho^abc)=\sum_jklp_jk^c\lambda_l^(jk)\log_2% \lambda_l^(jk),italic_S start_POSTSUBSCRIPT italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = ā start_POSTSUBSCRIPT italic_j italic_k italic_l end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j italic_k ) end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j italic_k ) end_POSTSUPERSCRIPT ,
where Ī»l(jk)superscriptsubscriptšššš\lambda_l^(jk)italic_Ī» start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j italic_k ) end_POSTSUPERSCRIPT are eigenvalues of state Ļjkcsubscriptsuperscriptšššš\rho^c_jkitalic_Ļ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT.
Let ĻĪ X=āĪ XĪ XĻĪ XsubscriptšsuperscriptĪ šsubscriptsuperscriptĪ šsuperscriptĪ ššsuperscriptĪ š\rho_\Pi^X=\sum_\Pi^X\Pi^X\rho\Pi^Xitalic_Ļ start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ā start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT italic_Ļ roman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT be the state after measurement Ī XsuperscriptĪ š\Pi^Xroman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT. Then for a bipartite state Ļabsuperscriptššš\rho^abitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT, the conditional entropy on subsystem BšµBitalic_B after the measurement on subsystem Aš“Aitalic_A is
(2.8) SB|Ī A(Ļab)=ājpjSB(Ļjb).subscriptšconditionalšµsuperscriptĪ š“superscriptšššsubscriptšsubscriptššsubscriptššµsubscriptsuperscriptššš\beginsplitS_B(\rho^ab)=\sum_jp_jS_B(\rho^b_j).\endsplitstart_ROW start_CELL italic_S start_POSTSUBSCRIPT italic_B | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ) = ā start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) . end_CELL end_ROW
By [10, Eq.(6)], the entropy of the measured system can always be decomposed as
(2.9) SAB(ĻĪ Aab)=SA(ĻĪ Aab)+SB|Ī A(Ļab).subscriptšš“šµsubscriptsuperscriptšššsuperscriptĪ š“subscriptšš“subscriptsuperscriptšššsuperscriptĪ š“subscriptšconditionalšµsuperscriptĪ š“superscriptšššS_AB(\rho^ab_\Pi^A)=S_A(\rho^ab_\Pi^A)+S_B(\rho^ab).italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) + italic_S start_POSTSUBSCRIPT italic_B | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ) .
For the tripartite system, using the measurement Ī ABsuperscriptĪ š“šµ\Pi^ABroman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, we have
(2.10) SABC(ĻĪ ABabc)-SAB(ĻĪ ABabc)=SC|Ī AB(Ļabc),subscriptšš“šµš¶subscriptsuperscriptššššsuperscriptĪ š“šµsubscriptšš“šµsubscriptsuperscriptššššsuperscriptĪ š“šµsubscriptšconditionalš¶superscriptĪ š“šµsuperscriptššššS_ABC(\rho^abc_\Pi^AB)-S_AB(\rho^abc_\Pi^AB)=S_C(% \rho^abc),italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) ,
when the measurement on Aš“Aitalic_A system is Ī AsuperscriptĪ š“\Pi^Aroman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT, then we have
(2.11) SABC(ĻĪ Aabc)-SA(ĻĪ Aabc)=SBC|Ī A(Ļabc).subscriptšš“šµš¶subscriptsuperscriptššššsuperscriptĪ š“subscriptšš“subscriptsuperscriptššššsuperscriptĪ š“subscriptšconditionalšµš¶superscriptĪ š“superscriptššššS_ABC(\rho^abc_\Pi^A)-S_A(\rho^abc_\Pi^A)=S_\Pi^A(\rho^% abc).italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) .
By Eq.(2.9), Eq.(2.10), Eq.(2.11), we have that
SBC|Ī A(Ļabc)=SB|Ī A(Ļab)+SC|Ī AB(Ļabc).subscriptšconditionalšµš¶superscriptĪ š“superscriptššššsubscriptšconditionalšµsuperscriptĪ š“superscriptšššsubscriptšconditionalš¶superscriptĪ š“šµsuperscriptššššS_\Pi^A(\rho^abc)=S_B(\rho^ab)+S_C(\rho^abc).italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ) + italic_S start_POSTSUBSCRIPT italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) .
Meanwhile, SA(ĻĪ ABabc)=SA(ĻĪ Aabc)subscriptšš“subscriptsuperscriptššššsuperscriptĪ š“šµsubscriptšš“subscriptsuperscriptššššsuperscriptĪ š“S_A(\rho^abc_\Pi^AB)=S_A(\rho^abc_\Pi^A)italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ), so the generalization discord of a tripartite state can be written as [10]
(2.12) š¬(Ļ):=minĪ AB[-SBC|A(Ļ)+SB|Ī A(Ļ)+SC|Ī AB(Ļ)].assignš¬šsubscriptsuperscriptĪ š“šµsubscriptšconditionalšµš¶š“šsubscriptšconditionalšµsuperscriptĪ š“šsubscriptšconditionalš¶superscriptĪ š“šµš\beginsplit\mathcalQ(\rho):=\min_\Pi^AB[-S_BC(\rho)+S_B(% \rho)+S_\Pi^AB(\rho)].\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ ) := roman_min start_POSTSUBSCRIPT roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ - italic_S start_POSTSUBSCRIPT italic_B italic_C | italic_A end_POSTSUBSCRIPT ( italic_Ļ ) + italic_S start_POSTSUBSCRIPT italic_B | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ ) + italic_S start_POSTSUBSCRIPT italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ ) ] . end_CELL end_ROW
3. Quantum Discord of non-X Qubit-Qutrit state
For the product states in the tripartite system, the discord has the special property that it reduces to the standard bipartite discord when only bipartite quantum correlations are present. This means š¬ABC(ĻxāĻy)=š¬X(Ļx)subscriptš¬š“šµš¶tensor-productsuperscriptšš„superscriptšš¦subscriptš¬šsuperscriptšš„\mathcalQ_ABC(\rho^x\otimes\rho^y)=\mathcalQ_X(\rho^x)caligraphic_Q start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ā italic_Ļ start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ) = caligraphic_Q start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_Ļ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ) for X=AB,BCšš“šµšµš¶X=AB,BCitalic_X = italic_A italic_B , italic_B italic_C and ACš“š¶ACitalic_A italic_C subsystem. We consider the following tripartite states
(3.1) Ļabc=18(I8+a3Ļ3āI4+I2āb3Ļ3āI2+I4āāi3ciĻi+āi3riĻiāĻiāI2+āi3siĻiāI2āĻi+āi3TiĻiāĻiāĻi),superscriptšššš18subscriptš¼8tensor-productsubscriptš3subscriptš3subscriptš¼4tensor-producttensor-productsubscriptš¼2subscriptš3subscriptš3subscriptš¼2tensor-productsubscriptš¼4superscriptsubscriptš3subscriptššsubscriptššsuperscriptsubscriptš3tensor-productsubscriptššsubscriptššsubscriptššsubscriptš¼2superscriptsubscriptš3tensor-productsubscriptš šsubscriptššsubscriptš¼2subscriptššsuperscriptsubscriptš3tensor-productsubscriptššsubscriptššsubscriptššsubscriptšš\beginsplit\rho^abc=&\frac18(I_8+a_3\sigma_3\otimes I_4+I_2% \otimes b_3\sigma_3\otimes I_2+I_4\otimes\sum_i^3c_i\sigma_i\\ +&\sum_i^3r_i\sigma_i\otimes\sigma_i\otimes I_2+\sum_i^3s_i% \sigma_i\otimes I_2\otimes\sigma_i+\sum_i^3T_i\sigma_i\otimes% \sigma_i\otimes\sigma_i),\endsplitstart_ROW start_CELL italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 8 end_ARG ( italic_I start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ā ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + end_CELL start_CELL ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , end_CELL end_ROW
where Idsubscriptš¼šI_ditalic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT represents the unit matrix of order dšditalic_d, and Ļi(i=1,2,3)subscriptššš123\sigma_i(i=1,2,3)italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i = 1 , 2 , 3 ) are Pauli matrices. The parameters a3,b3,ci,ri,si,Tiāāsubscriptš3subscriptš3subscriptššsubscriptššsubscriptš šsubscriptššāa_3,b_3,c_i,r_i,s_i,T_i\in\mathbbRitalic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā blackboard_R and they are confined within the internal [-1,1]11[-1,1][ - 1 , 1 ]. Its matrix has the following form:
(3.2) Ļ=(**000****00*0**00****0*00*****00*****000****00*0*00****000**).š000missing-subexpression000000000000missing-subexpression000missing-subexpression000missing-subexpression000\beginsplit\rho=\left(\beginarray[]cccccccc*&*&0&0&0&*&*&*\\ &*&0&0&*&0&*&*\\ 0&0&*&*&*&*&0&*\\ 0&0&*&*&*&*&*&0\\ 0&*&*&*&*&*&0&0\\ &0&*&*&*&*&0&0\\ &*&0&*&0&0&*&*\\ &*&*&0&0&0&*&*\\ \endarray\right).\endsplitstart_ROW start_CELL italic_Ļ = ( start_ARRAY start_ROW start_CELL * end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL * end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL * end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL * end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL * end_CELL start_CELL * end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL * end_CELL start_CELL * end_CELL end_ROW end_ARRAY ) . end_CELL end_ROW
Let ,j=0,1formulae-sequenceketšbrašš01\ italic_j ā© āØ italic_j be the computational base, then any von Neumann measurement on system XšXitalic_X can be written as ,j=0,1\V^\dagger , italic_j = 0 , 1 for some unitary matrix VāSU(2)šSU2V\in\mathrmSU(2)italic_V ā roman_SU ( 2 ). Any unitary matrix can be written as V=tI+-1ākykĻkšš”š¼1subscriptšsubscriptš¦šsubscriptššV=tI+\sqrt-1\sum_ky_k\sigma_kitalic_V = italic_t italic_I + square-root start_ARG - 1 end_ARG ā start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT with t,ykāāš”subscriptš¦šāt,y_k\in\mathbbRitalic_t , italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ā blackboard_R. When the measurement Ī jXsubscriptsuperscriptĪ šš\\Pi^X_j\ roman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is performed locally on one part of the composite system YāXtensor-productššY\otimes Xitalic_Y ā italic_X, the ensemble ĻjY,pjYsuperscriptsubscriptšššsuperscriptsubscriptššš\\rho_j^Y,p_j^Y\ italic_Ļ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT , italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT is given by ĻjY=1pjYTrX(IāĪ jX)ĻYX(IāĪ jX)superscriptsubscriptššš1superscriptsubscriptšššsubscriptTrštensor-productš¼subscriptsuperscriptĪ ššsuperscriptššštensor-productš¼subscriptsuperscriptĪ šš\rho_j^Y=\frac1p_j^Y\mathrmTr_X(I\otimes\Pi^X_j)\rho^YX(% I\otimes\Pi^X_j)italic_Ļ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT end_ARG roman_Tr start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_I ā roman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUPERSCRIPT italic_Y italic_X end_POSTSUPERSCRIPT ( italic_I ā roman_Ī start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) with the probability pjY=Tr[ĻYX(IāĪ jX)]superscriptsubscriptšššTrdelimited-[]superscriptššštensor-productš¼superscriptsubscriptĪ ššp_j^Y=\mathrmTr[\rho^YX(I\otimes\Pi_j^X)]italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Y end_POSTSUPERSCRIPT = roman_Tr [ italic_Ļ start_POSTSUPERSCRIPT italic_Y italic_X end_POSTSUPERSCRIPT ( italic_I ā roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) ].
It follows from symmetry that
(3.3) Vā Ļ1V=(t2+y12-y22-y32)Ļ1+2(ty3+y1y2)Ļ2+2(-ty2+y1y3)Ļ3,Vā Ļ2V=(t2+y22-y32-y12)Ļ1+2(ty1+y2y3)Ļ3+2(-ty3+y1y2)Ļ1,Vā Ļ3V=(t2+y32-y12-y22)Ļ1+2(ty2+y1y3)Ļ3+2(-ty1+y2y3)Ļ2.formulae-sequencesuperscriptšā subscriptš1šsuperscriptš”2superscriptsubscriptš¦12superscriptsubscriptš¦22superscriptsubscriptš¦32subscriptš12š”subscriptš¦3subscriptš¦1subscriptš¦2subscriptš22š”subscriptš¦2subscriptš¦1subscriptš¦3subscriptš3formulae-sequencesuperscriptšā subscriptš2šsuperscriptš”2superscriptsubscriptš¦22superscriptsubscriptš¦32superscriptsubscriptš¦12subscriptš12š”subscriptš¦1subscriptš¦2subscriptš¦3subscriptš32š”subscriptš¦3subscriptš¦1subscriptš¦2subscriptš1superscriptšā subscriptš3šsuperscriptš”2superscriptsubscriptš¦32superscriptsubscriptš¦12superscriptsubscriptš¦22subscriptš12š”subscriptš¦2subscriptš¦1subscriptš¦3subscriptš32š”subscriptš¦1subscriptš¦2subscriptš¦3subscriptš2\beginsplitV^\dagger\sigma_1V&=(t^2+y_1^2-y_2^2-y_3^2)% \sigma_1+2(ty_3+y_1y_2)\sigma_2+2(-ty_2+y_1y_3)\sigma_3,\\ V^\dagger\sigma_2V&=(t^2+y_2^2-y_3^2-y_1^2)\sigma_1+2(ty_% 1+y_2y_3)\sigma_3+2(-ty_3+y_1y_2)\sigma_1,\\ V^\dagger\sigma_3V&=(t^2+y_3^2-y_1^2-y_2^2)\sigma_1+2(ty_% 2+y_1y_3)\sigma_3+2(-ty_1+y_2y_3)\sigma_2.\\ \endsplitstart_ROW start_CELL italic_V start_POSTSUPERSCRIPT ā end_POSTSUPERSCRIPT italic_Ļ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_V end_CELL start_CELL = ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2 ( italic_t italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 2 ( - italic_t italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_V start_POSTSUPERSCRIPT ā end_POSTSUPERSCRIPT italic_Ļ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_V end_CELL start_CELL = ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2 ( italic_t italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 2 ( - italic_t italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_V start_POSTSUPERSCRIPT ā end_POSTSUPERSCRIPT italic_Ļ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_V end_CELL start_CELL = ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2 ( italic_t italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 2 ( - italic_t italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT . end_CELL end_ROW
Introduce new variables z1X=2(-ty2+y1y3),z2X=2(ty1+y2y3),z3X=(t2+y32-y12-y22)formulae-sequencesuperscriptsubscriptš§1š2š”subscriptš¦2subscriptš¦1subscriptš¦3formulae-sequencesuperscriptsubscriptš§2š2š”subscriptš¦1subscriptš¦2subscriptš¦3superscriptsubscriptš§3šsuperscriptš”2superscriptsubscriptš¦32superscriptsubscriptš¦12superscriptsubscriptš¦22z_1^X=2(-ty_2+y_1y_3),z_2^X=2(ty_1+y_2y_3),z_3^X=(t^2% +y_3^2-y_1^2-y_2^2)italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT = 2 ( - italic_t italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT = 2 ( italic_t italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT = ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ), then (z1X)2+(z2X)2+(z3X)2=1superscriptsuperscriptsubscriptš§1š2superscriptsuperscriptsubscriptš§2š2superscriptsuperscriptsubscriptš§3š21(z_1^X)^2+(z_2^X)^2+(z_3^X)^2=1( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1. Therefore Ī jXĻkĪ jX=(-1)jzkXĪ jXsuperscriptsubscriptĪ šš Discord servers ššsuperscriptsubscriptĪ ššsuperscript1šsuperscriptsubscriptš§ššsuperscriptsubscriptĪ šš\Pi_j^X\sigma_k\Pi_j^X=(-1)^jz_k^X\Pi_j^Xroman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT = ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT for j=0,1š01j=0,1italic_j = 0 , 1 and k=1,2,3š123k=1,2,3italic_k = 1 , 2 , 3.
For the tripartite state Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT, the conditional state on BCšµš¶BCitalic_B italic_C subsystem after measurement Ī jA(j=0,1)superscriptsubscriptĪ šš“š01\\Pi_j^A(j=0,1)\ roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_j = 0 , 1 ) on subsystem Aš“Aitalic_A is
(3.4) Ļjbc=1pjbc(Ī 1AāI2āI2)Ļabc(Ī 1AāI2āI2)=1pjbc[(1+(-1)ja3z3A)I2āI2+b3Ļ3āI2+(-1)jāi3riziAĻiāI2+āi3(ci+(-1)jsiziA)I2āĻi+(-1)jāi3TiziAĻiāĻi],superscriptsubscriptšššš1superscriptsubscriptšššštensor-productsuperscriptsubscriptĪ 1š“subscriptš¼2subscriptš¼2superscriptšššštensor-productsuperscriptsubscriptĪ 1š“subscriptš¼2subscriptš¼21superscriptsubscriptššššdelimited-[]tensor-product1superscript1šsubscriptš3superscriptsubscriptš§3š“subscriptš¼2subscriptš¼2tensor-productsubscriptš3subscriptš3subscriptš¼2superscript1šsuperscriptsubscriptš3tensor-productsubscriptššsuperscriptsubscriptš§šš“subscriptššsubscriptš¼2superscriptsubscriptš3tensor-productsubscriptššsuperscript1šsubscriptš šsuperscriptsubscriptš§šš“subscriptš¼2subscriptššsuperscript1šsuperscriptsubscriptš3tensor-productsubscriptššsuperscriptsubscriptš§šš“subscriptššsubscriptšš\beginsplit\rho_j^bc&=\frac1p_j^bc(\Pi_1^A\otimes I_2% \otimes I_2)\rho^abc(\Pi_1^A\otimes I_2\otimes I_2)\\ &=\frac1p_j^bc[(1+(-1)^ja_3z_3^A)I_2\otimes I_2+b_3% \sigma_3\otimes I_2+(-1)^j\sum_i^3r_iz_i^A\sigma_i\otimes I_% 2\\ &+\sum_i^3(c_i+(-1)^js_iz_i^A)I_2\otimes\sigma_i+(-1)^j% \sum_i^3T_iz_i^A\sigma_i\otimes\sigma_i],\endsplitstart_ROW start_CELL italic_Ļ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT end_ARG ( roman_Ī start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Ī start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT end_ARG [ ( 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] , end_CELL end_ROW
where the probabilities are
pjbc=Tr((Ī jAāI2āI2)Ļabc(Ī jAāI2āI2))=12[1+(-1)ja3z3A],superscriptsubscriptššššTrtensor-productsuperscriptsubscriptĪ šš“subscriptš¼2subscriptš¼2superscriptšššštensor-productsuperscriptsubscriptĪ šš“subscriptš¼2subscriptš¼212delimited-[]1superscript1šsubscriptš3superscriptsubscriptš§3š“p_j^bc=\mathrmTr((\Pi_j^A\otimes I_2\otimes I_2)\rho^abc(\Pi_% j^A\otimes I_2\otimes I_2))=\frac12[1+(-1)^ja_3z_3^A],italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT = roman_Tr ( ( roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Ī start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ] ,
and āi3(ziA)2=1superscriptsubscriptš3superscriptsubscriptsuperscriptš§š“š21\sum_i^3(z^A_i)^2=1ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_z start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1. Therefore the reduced state of Ļjbcsuperscriptsubscriptšššš\rho_j^bcitalic_Ļ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT is
Ļjb=TrCĻjbc=12(1+(-1)ja3z3A)[(1+(-1)ja3z3A)I2+b3Ļ3+(-1)jāi3riziAĻi]superscriptsubscriptšššsubscriptTrš¶superscriptsubscriptšššš121superscript1šsubscriptš3superscriptsubscriptš§3š“delimited-[]1superscript1šsubscriptš3superscriptsubscriptš§3š“subscriptš¼2subscriptš3subscriptš3superscript1šsuperscriptsubscriptš3subscriptššsuperscriptsubscriptš§šš“subscriptšš\rho_j^b=\mathrmTr_C\rho_j^bc=\frac12(1+(-1)^ja_3z_3^A)% [(1+(-1)^ja_3z_3^A)I_2+b_3\sigma_3+(-1)^j\sum_i^3r_iz_% i^A\sigma_i]italic_Ļ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_ARG [ ( 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ]
with the probability pjb=pjbc=12[1+(-1)ja3z3A]superscriptsubscriptšššsuperscriptsubscriptšššš12delimited-[]1superscript1šsubscriptš3superscriptsubscriptš§3š“p_j^b=p_j^bc=\frac12[1+(-1)^ja_3z_3^A]italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT = italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ]. The eigenvalues of Ļjbsuperscriptsubscriptššš\rho_j^bitalic_Ļ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT are
Ī»jĀ±=12(1+(-1)ja3z3A)[1+(-1)ja3z3AĀ±(b3+(-1)jr3z3A)2+āi2(riziA)2].subscriptsuperscriptšplus-or-minusš121superscript1šsubscriptš3superscriptsubscriptš§3š“delimited-[]plus-or-minus1superscript1šsubscriptš3superscriptsubscriptš§3š“superscriptsubscriptš3superscript1šsubscriptš3superscriptsubscriptš§3š“2superscriptsubscriptš2superscriptsubscriptššsuperscriptsubscriptš§šš“2\lambda^\pm_j=\frac12(1+(-1)^ja_3z_3^A)[1+(-1)^ja_3z_3^% A\pm\sqrt(b_3+(-1)^jr_3z_3^A)^2+\sum_i^2(r_iz_i^A)^2% ].italic_Ī» start_POSTSUPERSCRIPT Ā± end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_ARG [ 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT Ā± square-root start_ARG ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] .
We define the following entropy function
(3.5) HĪµ(x)=12[(1+Īµ+x)log2(1+Īµ+x)+(1+Īµ-x)log2(1+Īµ-x)].subscriptš»šš„12delimited-[]1šš„subscript21šš„1šš„subscript21šš„\beginsplitH_\varepsilon(x)=\frac12[(1+\varepsilon+x)\log_2(1+% \varepsilon+x)+(1+\varepsilon-x)\log_2(1+\varepsilon-x)].\endsplitstart_ROW start_CELL italic_H start_POSTSUBSCRIPT italic_Īµ end_POSTSUBSCRIPT ( italic_x ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ( 1 + italic_Īµ + italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_Īµ + italic_x ) + ( 1 + italic_Īµ - italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_Īµ - italic_x ) ] . end_CELL end_ROW
Then measured conditional entropy of BšµBitalic_B subsystem can be obtained as [3, 4, 14, 16, 17, 18]
(3.6) SB|Ī A(Ļ)=-ājpjb(Ī»j+log2Ī»j++Ī»j-log2Ī»j-)=-12[Ha3z3A(A+)+H-a3z3A(A-)-2H(a3z3A)-2],subscriptšconditionalšµsuperscriptĪ š“šsubscriptšsuperscriptsubscriptšššsubscriptsuperscriptššsubscript2subscriptsuperscriptššsubscriptsuperscriptššsubscript2subscriptsuperscriptšš12delimited-[]subscriptš»subscriptš3superscriptsubscriptš§3š“subscriptš“subscriptš»subscriptš3superscriptsubscriptš§3š“subscriptš“2š»subscriptš3superscriptsubscriptš§3š“2\beginsplitS_\Pi^A(\rho)&=-\sum_jp_j^b(\lambda^+_j\log_2% \lambda^+_j+\lambda^-_j\log_2\lambda^-_j)\\ &=-\frac12[H_a_3z_3^A(A_+)+H_-a_3z_3^A(A_-)-2H(a_3z_% 3^A)-2],\endsplitstart_ROW start_CELL italic_S start_POSTSUBSCRIPT italic_B | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ ) end_CELL start_CELL = - ā start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_Ī» start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_Ī» start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = - divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) - 2 italic_H ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - 2 ] , end_CELL end_ROW
where AĀ±=(b3Ā±r3z3A)2+āi2(riziA)2subscriptš“plus-or-minussuperscriptplus-or-minussubscriptš3subscriptš3superscriptsubscriptš§3š“2superscriptsubscriptš2superscriptsubscriptššsuperscriptsubscriptš§šš“2A_\pm=\sqrt(b_3\pm r_3z_3^A)^2+\sum_i^2(r_iz_i^A)^2italic_A start_POSTSUBSCRIPT Ā± end_POSTSUBSCRIPT = square-root start_ARG ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Ā± italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG.
After measurement Ī k|jBsuperscriptsubscriptĪ conditionalšššµ\Pi_j^Broman_Ī start_POSTSUBSCRIPT italic_k | italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT on BCšµš¶BCitalic_B italic_C system, the state Ļjbcsuperscriptsubscriptšššš\rho_j^bcitalic_Ļ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT is changed to
(3.7) Ļjkc=1pjkc[(1+(-1)ja3z3A+(-1)kb3z3B+(-1)j+kāi3riziAziB)I2+āi3(ci+(-1)jsiziA+(-1)k+jTiziAziB)Ļi],(j,k=0,1)fragmentssuperscriptsubscriptšššš1superscriptsubscriptššššfragments[fragments(1superscriptfragments(1)šsubscriptš3superscriptsubscriptš§3š“superscriptfragments(1)šsubscriptš3superscriptsubscriptš§3šµsuperscriptfragments(1)ššsuperscriptsubscriptš3subscriptššsuperscriptsubscriptš§šš“superscriptsubscriptš§ššµ)subscriptš¼2superscriptsubscriptš3fragments(subscriptššsuperscriptfragments(1)šsubscriptš šsuperscriptsubscriptš§šš“superscriptfragments(1)ššsubscriptššsuperscriptsubscriptš§šš“superscriptsubscriptš§ššµ)subscriptšš],fragments(š,š0,1)\beginsplit\rho_jk^c=&\frac1p_jk^c[(1+(-1)^ja_3z_3^A+(-1% )^kb_3z_3^B+(-1)^j+k\sum_i^3r_iz_i^Az_i^B)I_2\\ +&\sum_i^3(c_i+(-1)^js_iz_i^A+(-1)^k+jT_iz_i^Az_i^B)% \sigma_i],(j,k=0,1)\endsplitstart_ROW start_CELL italic_Ļ start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_ARG [ ( 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j + italic_k end_POSTSUPERSCRIPT ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + end_CELL start_CELL ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k + italic_j end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] , ( italic_j , italic_k = 0 , 1 ) end_CELL end_ROW
with the probability (k=0,1š01k=0,1italic_k = 0 , 1)
(3.8) p0kc=12(1+a3z3A)(1+Ī±k),p1kc=12(1-a3z3A)(1+Ī²k),formulae-sequencesuperscriptsubscriptš0šš121subscriptš3superscriptsubscriptš§3š“1subscriptš¼šsuperscriptsubscriptš1šš121subscriptš3superscriptsubscriptš§3š“1subscriptš½šp_0k^c=\frac12(1+a_3z_3^A)(1+\alpha_k),\ \ p_1k^c=\frac1% 2(1-a_3z_3^A)(1+\beta_k),italic_p start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_ARG ( 1 + italic_Ī± start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , italic_p start_POSTSUBSCRIPT 1 italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_ARG ( 1 + italic_Ī² start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ,
where Ī±k=a3z3A+(-1)k(b3z3B+āi3riziAziB),Ī²k=-a3z3A+(-1)k(b3z3B-āi3riziAziB)formulae-sequencesubscriptš¼šsubscriptš3superscriptsubscriptš§3š“superscript1šsubscriptš3superscriptsubscriptš§3šµsuperscriptsubscriptš3subscriptššsuperscriptsubscriptš§šš“superscriptsubscriptš§ššµsubscriptš½šsubscriptš3superscriptsubscriptš§3š“superscript1šsubscriptš3superscriptsubscriptš§3šµsuperscriptsubscriptš3subscriptššsuperscriptsubscriptš§šš“superscriptsubscriptš§ššµ\alpha_k=a_3z_3^A+(-1)^k(b_3z_3^B+\sum_i^3r_iz_i^Az_% i^B),\beta_k=-a_3z_3^A+(-1)^k(b_3z_3^B-\sum_i^3r_iz_% i^Az_i^B)italic_Ī± start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) , italic_Ī² start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT - ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ). The non-zero eigenvalues of Ļjkcsubscriptsuperscriptšššš\rho^c_jkitalic_Ļ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT are given by
(3.9) Ī»0kĀ±=12(1+Ī±k)(1+Ī±kĀ±Ī³k),Ī»1kĀ±=12(1+Ī²k)(1+Ī²kĀ±Ī“k),k=0,1,formulae-sequencesuperscriptsubscriptš0šplus-or-minus121subscriptš¼šplus-or-minus1subscriptš¼šsubscriptš¾šformulae-sequencesuperscriptsubscriptš1šplus-or-minus121subscriptš½šplus-or-minus1subscriptš½šsubscriptšæšš01\lambda_0k^\pm=\frac12(1+\alpha_k)(1+\alpha_k\pm\gamma_k),\ \ % \lambda_1k^\pm=\frac12(1+\beta_k)(1+\beta_k\pm\delta_k),k=0,1,italic_Ī» start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Ā± end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_Ī± start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG ( 1 + italic_Ī± start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT Ā± italic_Ī³ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , italic_Ī» start_POSTSUBSCRIPT 1 italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Ā± end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_Ī² start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG ( 1 + italic_Ī² start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT Ā± italic_Ī“ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , italic_k = 0 , 1 ,
where
Ī³k=[āi3(ci+siziA+(-1)kTiziAziB)2]12,Ī“k=[āi3(-ci+siziA+(-1)kTiziAziB)2]12.formulae-sequencesubscriptš¾šsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptššsubscriptš šsuperscriptsubscriptš§šš“superscript1šsubscriptššsuperscriptsubscriptš§šš“subscriptsuperscriptš§šµš212subscriptšæšsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptššsubscriptš šsuperscriptsubscriptš§šš“superscript1šsubscriptššsuperscriptsubscriptš§šš“subscriptsuperscriptš§šµš212\beginsplit\gamma_k&=[\sum_i^3(c_i+s_iz_i^A+(-1)^kT_iz_i% ^Az^B_i)^2]^\frac12,\\ \delta_k&=[\sum_i^3(-c_i+s_iz_i^A+(-1)^kT_iz_i^Az^B_i% )^2]^\frac12.\endsplitstart_ROW start_CELL italic_Ī³ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_Ī“ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( - italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT . end_CELL end_ROW
According to the fact that the eigenvalues in Eq.(3.9) are nonnegative, we have āi3ai2+āi3bi2+āi3ri2ā¤1superscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš3superscriptsubscriptšš21\sqrt\sum_i^3a_i^2+\sqrt\sum_i^3b_i^2+\sqrt\sum_i^3r_% i^2\leq 1square-root start_ARG ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + square-root start_ARG ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + square-root start_ARG ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ā¤ 1.
The entropy of Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT under the measurement Ī ABsuperscriptĪ š“šµ\Pi^ABroman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is given by
(3.10) SC|Ī AB(Ļ)=-āj,kpjkc(Ī»jk+log2Ī»jk++Ī»jk-log2Ī»jk-)=-12(1+a3z3A)[HĪ±0(Ī³0)+HĪ±1(Ī³1)-2Ha3z3A(Ī±0-Ī±12)]-12(1-a3z3A)[HĪ²0(Ī“0)+HĪ²1(Ī“1)-2H-a3z3A(Ī²0-Ī²12)]+2.subscriptšconditionalš¶superscriptĪ š“šµšsubscriptššsubscriptsuperscriptššššsuperscriptsubscriptšššsubscript2superscriptsubscriptšššsuperscriptsubscriptšššsubscript2superscriptsubscriptššš121subscriptš3superscriptsubscriptš§3š“delimited-[]subscriptš»subscriptš¼0subscriptš¾0subscriptš»subscriptš¼1subscriptš¾12subscriptš»subscriptš3superscriptsubscriptš§3š“subscriptš¼0subscriptš¼12121subscriptš3superscriptsubscriptš§3š“delimited-[]subscriptš»subscriptš½0subscriptšæ0subscriptš»subscriptš½1subscriptšæ12subscriptš»subscriptš3superscriptsubscriptš§3š“subscriptš½0subscriptš½122\beginsplitS_\Pi^AB(\rho)=&-\sum_j,kp^c_jk(\lambda_jk^+\log_% 2\lambda_jk^++\lambda_jk^-\log_2\lambda_jk^-)\\ =&-\frac12(1+a_3z_3^A)[H_\alpha_0(\gamma_0)+H_\alpha_1(% \gamma_1)-2H_a_3z_3^A(\frac\alpha_0-\alpha_12)]\\ &-\frac12(1-a_3z_3^A)[H_\beta_0(\delta_0)+H_\beta_1(\delta% _1)-2H_-a_3z_3^A(\frac\beta_0-\beta_12)]+2.\endsplitstart_ROW start_CELL italic_S start_POSTSUBSCRIPT italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ ) = end_CELL start_CELL - ā start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT ( italic_Ī» start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT + italic_Ī» start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 2 italic_H start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( divide start_ARG italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 2 italic_H start_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( divide start_ARG italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) ] + 2 . end_CELL end_ROW
In particularly, a3z3A=Ī±0+Ī±12subscriptš3superscriptsubscriptš§3š“subscriptš¼0subscriptš¼12a_3z_3^A=\frac\alpha_0+\alpha_12italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = divide start_ARG italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG.
Let G(z1A,z2A,z3A)=1-SB|Ī A(Ļ)šŗsuperscriptsubscriptš§1š“superscriptsubscriptš§2š“superscriptsubscriptš§3š“1subscriptšconditionalšµsuperscriptĪ š“šG(z_1^A,z_2^A,z_3^A)=1-S_\Pi^A(\rho)italic_G ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = 1 - italic_S start_POSTSUBSCRIPT italic_B | roman_Ī start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ ) and F(z1A,z2A,z3A,z1B,z2B,z3B)=2-SC|Ī AB(Ļ)š¹superscriptsubscriptš§1š“superscriptsubscriptš§2š“superscriptsubscriptš§3š“subscriptsuperscriptš§šµ1subscriptsuperscriptš§šµ2subscriptsuperscriptš§šµ32subscriptšconditionalš¶superscriptĪ š“šµšF(z_1^A,z_2^A,z_3^A,z^B_1,z^B_2,z^B_3)=2-S_\Pi^AB% (\rho)italic_F ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = 2 - italic_S start_POSTSUBSCRIPT italic_C | roman_Ī start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Ļ ), then we have the following result.
For the non-X-states Ļš\rhoitalic_Ļ in Eq(3.1) with 14 parameters, the quantum discord is given by
(3.11) š¬(Ļ)=-SABC(Ļ)+SA(Ļ)+minĪ A(Ļ)+SC=3+āi=18Ī»ilog2Ī»i-āk=12Ī»kalog2Ī»ka-maxziXā[0,1],āi(ziX)2=1G+F,š¬šsubscriptšš“šµš¶šsubscriptšš“šsubscriptšconditionalšµsuperscriptĪ š“šsubscriptšconditionalš¶superscriptĪ š“šµš3superscriptsubscriptš18subscriptššsubscript2subscriptššsuperscriptsubscriptš12superscriptsubscriptšššsubscript2superscriptsubscriptšššsubscriptformulae-sequencesubscriptsuperscriptš§šš01subscriptšsuperscriptsuperscriptsubscriptš§šš21šŗš¹\beginsplit\mathcalQ(\rho)=&-S_ABC(\rho)+S_A(\rho)+\min\S_B% (\rho)+S_\Pi^AB(\rho)\\\ =&3+\sum_i=1^8\lambda_i\log_2\lambda_i-\sum_k=1^2\lambda_k^% a\log_2\lambda_k^a-\max_z^X_i\in[0,1],\sum_i(z_i^X)^2=1\% G+F\,\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ ) = end_CELL start_CELL - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_Ļ ) + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_Ļ ) + roman_min italic_S start_POSTSUBSCRIPT italic_B end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL 3 + ā start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - ā start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT - roman_max start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā [ 0 , 1 ] , ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 end_POSTSUBSCRIPT italic_G + italic_F , end_CELL end_ROW
where Ī»i(i=1,āÆ,8)subscriptššš1normal-āÆ8\lambda_i(i=1,\cdots,8)italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i = 1 , āÆ , 8 ) are the eigenvalues of Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT, Ī»ka=12[1+(-1)ka3],(k=0,1)superscriptsubscriptššš12delimited-[]1superscript1šsubscriptš3š01\lambda_k^a=\frac12[1+(-1)^ka_3],(k=0,1)italic_Ī» start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] , ( italic_k = 0 , 1 ) are eigenvalues of Ļabcsuperscriptšššš\rho^abcitalic_Ļ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT on subsystem Aš“Aitalic_A and XšXitalic_X represents subsystem A,Bš“šµA,Bitalic_A , italic_B.
Theorem 3.2.
Let r=maxr2šsubscriptš1subscriptš2r=\max\r_1italic_r = roman_max italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , then maxziXā[0,1],āi(ziX)2=1G+Fsubscriptformulae-sequencesubscriptsuperscriptš§šš01subscriptšsuperscriptsuperscriptsubscriptš§šš21šŗš¹\max_z^X_i\in[0,1],\sum_i(z_i^X)^2=1\G+F\roman_max start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā [ 0 , 1 ] , ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 end_POSTSUBSCRIPT italic_G + italic_F can be explicitly computed as follows.
Case1: when a3b3r3ā¤0,r32-r2ā„a3b3r3formulae-sequencesubscriptš3subscriptš3subscriptš30superscriptsubscriptš32superscriptš2subscriptš3subscriptš3subscriptš3a_3b_3r_3\leq 0,r_3^2-r^2\geq a_3b_3r_3italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā¤ 0 , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ā„ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, and (b3+r3)(c3+s3)ā¤0subscriptš3subscriptš3subscriptš3subscriptš 30(b_3+r_3)(c_3+s_3)\leq 0( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ( italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ā¤ 0, we have
(3.12) maxziXā[0,1],āi(ziX)2=1G+F=G(0,0,1)+F(0,0,1,0,0,1),subscriptformulae-sequencesubscriptsuperscriptš§šš01subscriptšsuperscriptsuperscriptsubscriptš§šš21šŗš¹šŗ001š¹001001\max_z^X_i\in[0,1],\sum_i(z_i^X)^2=1\G+F\=G(0,0,1)+F(0,0,1,0,0% ,1),roman_max start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā [ 0 , 1 ] , ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 end_POSTSUBSCRIPT italic_G + italic_F = italic_G ( 0 , 0 , 1 ) + italic_F ( 0 , 0 , 1 , 0 , 0 , 1 ) ,
where
(3.13) G(0,0,1)=12[Ha3(|b3+r3|)+H-a3(|b3-r3|)-2H(a3)]šŗ00112delimited-[]subscriptš»subscriptš3subscriptš3subscriptš3subscriptš»subscriptš3subscriptš3subscriptš32š»subscriptš3\beginsplitG(0,0,1)=\frac12[H_a_3(|b_3+r_3|)+H_-a_3(|b_3-r% _3|)-2H(a_3)]\endsplitstart_ROW start_CELL italic_G ( 0 , 0 , 1 ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | ) + italic_H start_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | ) - 2 italic_H ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] end_CELL end_ROW
and
(3.14) F(0,0,1,0,0,1)=12(1+a3)[HĪ±0(Ī³0)+HĪ±1(Ī³1)-2Ha3(b3+r3)]+12(1-a3)[HĪ²0(Ī“0)+HĪ²1(Ī“1)-2H-a3(b3-r3)].š¹001001121subscriptš3delimited-[]subscriptš»subscriptš¼0subscriptš¾0subscriptš»subscriptš¼1subscriptš¾12subscriptš»subscriptš3subscriptš3subscriptš3121subscriptš3delimited-[]subscriptš»subscriptš½0subscriptšæ0subscriptš»subscriptš½1subscriptšæ12subscriptš»subscriptš3subscriptš3subscriptš3\beginsplitF(0,0,1,0,0,1)=&\frac12(1+a_3)[H_\alpha_0(\gamma_0)+H% _\alpha_1(\gamma_1)-2H_a_3(b_3+r_3)]\\ +&\frac12(1-a_3)[H_\beta_0(\delta_0)+H_\beta_1(\delta_1)-2H_% -a_3(b_3-r_3)].\endsplitstart_ROW start_CELL italic_F ( 0 , 0 , 1 , 0 , 0 , 1 ) = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 2 italic_H start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] end_CELL end_ROW start_ROW start_CELL + end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 2 italic_H start_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] . end_CELL end_ROW
In this case, the parameters are degenerated into (k=0,1)š01(k=0,1)( italic_k = 0 , 1 )
Ī±k=a3+(-1)k(b3+r3),Ī³k=[āi3ci2+s32+T32+2(c3s3+(-1)k(c3T3+s3T3))]12,formulae-sequencesubscriptš¼šsubscriptš3superscript1šsubscriptš3subscriptš3subscriptš¾šsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš 32superscriptsubscriptš322subscriptš3subscriptš 3superscript1šsubscriptš3subscriptš3subscriptš 3subscriptš312\alpha_k=a_3+(-1)^k(b_3+r_3),\gamma_k=[\sum_i^3c_i^2+s_3% ^2+T_3^2+2(c_3s_3+(-1)^k(c_3T_3+s_3T_3))]^\frac12,italic_Ī± start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , italic_Ī³ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 ( italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ) ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ,
Ī²k=-a3+(-1)k(b3-r3),Ī“k=[āi3ci2+s32+T32+2(-c3s3+(-1)k(s3T3-c3T3))]12.formulae-sequencesubscriptš½šsubscriptš3superscript1šsubscriptš3subscriptš3subscriptšæšsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš 32superscriptsubscriptš322subscriptš3subscriptš 3superscript1šsubscriptš 3subscriptš3subscriptš3subscriptš312\beta_k=-a_3+(-1)^k(b_3-r_3),\delta_k=[\sum_i^3c_i^2+s_3% ^2+T_3^2+2(-c_3s_3+(-1)^k(s_3T_3-c_3T_3))]^\frac12.italic_Ī² start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , italic_Ī“ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 ( - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ) ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT .
Case 2: (1) When b3=0,c1s1ā¤0,s1ā¤|c1|formulae-sequencesubscriptš30formulae-sequencesubscriptš1subscriptš 10subscriptš 1subscriptš1b_3=0,c_1s_1\leq 0,s_1\leq|c_1|italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ā¤ 0 , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ā¤ | italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | and max=|r1|subscriptš1subscriptš2subscriptš3subscriptš1\max\r_3=|r_1|roman_max italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = | italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |, we have
(3.15) maxziXā[0,1],āi(ziX)2=1G+F=G(1,0,0)+F(1,0,0,1,0,0),subscriptformulae-sequencesubscriptsuperscriptš§šš01subscriptšsuperscriptsuperscriptsubscriptš§šš21šŗš¹šŗ100š¹100100\max_z^X_i\in[0,1],\sum_i(z_i^X)^2=1\G+F\=G(1,0,0)+F(1,0,0,1,0% ,0),roman_max start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā [ 0 , 1 ] , ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 end_POSTSUBSCRIPT italic_G + italic_F = italic_G ( 1 , 0 , 0 ) + italic_F ( 1 , 0 , 0 , 1 , 0 , 0 ) ,
where
(3.16) G(1,0,0)=12[Ha3(r1)+H-a3(r1)-2H(a3)]šŗ10012delimited-[]subscriptš»subscriptš3subscriptš1subscriptš»subscriptš3subscriptš12š»subscriptš3G(1,0,0)=\frac12[H_a_3(r_1)+H_-a_3(r_1)-2H(a_3)]italic_G ( 1 , 0 , 0 ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 2 italic_H ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ]
and
(3.17) F(1,0,0,1,0,0)=12[Hr1(Ī³0)+H-r1(Ī³1)+Hr1(Ī“0)+H-r1(Ī“1)-4H(r1)].š¹10010012delimited-[]subscriptš»subscriptš1subscriptš¾0subscriptš»subscriptš1subscriptš¾1subscriptš»subscriptš1subscriptšæ0subscriptš»subscriptš1subscriptšæ14š»subscriptš1\beginsplitF(1,0,0,1,0,0)=\frac12[H_r_1(\gamma_0)+H_-r_1(% \gamma_1)+H_r_1(\delta_0)+H_-r_1(\delta_1)-4H(r_1)].\endsplitstart_ROW start_CELL italic_F ( 1 , 0 , 0 , 1 , 0 , 0 ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 4 italic_H ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] . end_CELL end_ROW
In this case, the parameters are degenerated into (k=0,1)š01(k=0,1)( italic_k = 0 , 1 )
Ī³k=[āi3ci2+s12+T12+2(c1s1+(-1)k(c1T1+s1T1))]12,Ī“k=[āi3ci2+s12+T12+2(-c1s1+(-1)k(c1T1-s1T1))]12.formulae-sequencesubscriptš¾šsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš 12superscriptsubscriptš122subscriptš1subscriptš 1superscript1šsubscriptš1subscriptš1subscriptš 1subscriptš112subscriptšæšsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš 12superscriptsubscriptš122subscriptš1subscriptš 1superscript1šsubscriptš1subscriptš1subscriptš 1subscriptš112\beginsplit\gamma_k&=[\sum_i^3c_i^2+s_1^2+T_1^2+2(c_1s_% 1+(-1)^k(c_1T_1+s_1T_1))]^\frac12,\\ \delta_k&=[\sum_i^3c_i^2+s_1^2+T_1^2+2(-c_1s_1+(-1)^k(% c_1T_1-s_1T_1))]^\frac12.\endsplitstart_ROW start_CELL italic_Ī³ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_Ī“ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 ( - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT . end_CELL end_ROW
(2) When b3=0,c1s1ā¤0,s1ā¤|c1|formulae-sequencesubscriptš30formulae-sequencesubscriptš1subscriptš 10subscriptš 1subscriptš1b_3=0,c_1s_1\leq 0,s_1\leq|c_1|italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ā¤ 0 , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ā¤ | italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | and maxr2=|r2|subscriptš1subscriptš2subscriptš3subscriptš2\max\=|r_2|roman_max italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = | italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT |, we have
(3.18) maxziXā[0,1],āi(ziX)2=1G+F=G(0,1,0)+F(0,1,0,0,1,0),subscriptformulae-sequencesubscriptsuperscriptš§šš01subscriptšsuperscriptsuperscriptsubscriptš§šš21šŗš¹šŗ010š¹010010\max_z^X_i\in[0,1],\sum_i(z_i^X)^2=1\G+F\=G(0,1,0)+F(0,1,0,0,1% ,0),roman_max start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā [ 0 , 1 ] , ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 end_POSTSUBSCRIPT italic_G + italic_F = italic_G ( 0 , 1 , 0 ) + italic_F ( 0 , 1 , 0 , 0 , 1 , 0 ) ,
where
(3.19) G(0,1,0)=12[Ha3(r2)+H-a3(r2)-2H(a3)]šŗ01012delimited-[]subscriptš»subscriptš3subscriptš2subscriptš»subscriptš3subscriptš22š»subscriptš3G(0,1,0)=\frac12[H_a_3(r_2)+H_-a_3(r_2)-2H(a_3)]italic_G ( 0 , 1 , 0 ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - 2 italic_H ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ]
and
(3.20) F(0,1,0,0,1,0)=12[Hr2(Ī³0)+H-r2(Ī³1)+Hr2(Ī“0)+H-r2(Ī“1)-4H(r2)].š¹01001012delimited-[]subscriptš»subscriptš2subscriptš¾0subscriptš»subscriptš2subscriptš¾1subscriptš»subscriptš2subscriptšæ0subscriptš»subscriptš2subscriptšæ14š»subscriptš2\beginsplitF(0,1,0,0,1,0)=&\frac12[H_r_2(\gamma_0)+H_-r_2(% \gamma_1)+H_r_2(\delta_0)+H_-r_2(\delta_1)-4H(r_2)].\endsplitstart_ROW start_CELL italic_F ( 0 , 1 , 0 , 0 , 1 , 0 ) = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 4 italic_H ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ] . end_CELL end_ROW
In this case, the parameters are degenerated into (k=0,1)š01(k=0,1)( italic_k = 0 , 1 )
Ī³k=[āi3ci2+s22+T22+2(c2s2+(-1)kc2T2+s2T2)]12;Ī“k=[āi3ci2+s22+T22+2(-c2s2+(-1)kc2T2-s2T2)]12.formulae-sequencesubscriptš¾šsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš 22superscriptsubscriptš222subscriptš2subscriptš 2superscript1šsubscriptš2subscriptš2subscriptš 2subscriptš212subscriptšæšsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš 22superscriptsubscriptš222subscriptš2subscriptš 2superscript1šsubscriptš2subscriptš2subscriptš 2subscriptš212\beginsplit\gamma_k&=[\sum_i^3c_i^2+s_2^2+T_2^2+2(c_2s_% 2+(-1)^kc_2T_2+s_2T_2)]^\frac12;\\ \delta_k&=[\sum_i^3c_i^2+s_2^2+T_2^2+2(-c_2s_2+(-1)^kc% _2T_2-s_2T_2)]^\frac12.\endsplitstart_ROW start_CELL italic_Ī³ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 ( italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ; end_CELL end_ROW start_ROW start_CELL italic_Ī“ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 ( - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT . end_CELL end_ROW
By definition, we have
(3.21) G+F=Ha3z3A(B+)+H-a3z3A(B-)-2H(a3z3A)+12(1+a3z3A)[HĪ±0(Ī³0)+HĪ±1(Ī³1)-2Ha3z3A(Ī±0-Ī±12)]+12(1-a3z3A)[HĪ²0(Ī“0)+HĪ²1(Ī“1)-2H-a3z3A(Ī²0-Ī²12)].šŗš¹subscriptš»subscriptš3superscriptsubscriptš§3š“subscriptšµsubscriptš»subscriptš3superscriptsubscriptš§3š“subscriptšµ2š»subscriptš3superscriptsubscriptš§3š“121subscriptš3superscriptsubscriptš§3š“delimited-[]subscriptš»subscriptš¼0subscriptš¾0subscriptš»subscriptš¼1subscriptš¾12subscriptš»subscriptš3superscriptsubscriptš§3š“subscriptš¼0subscriptš¼12121subscriptš3superscriptsubscriptš§3š“delimited-[]subscriptš»subscriptš½0subscriptšæ0subscriptš»subscriptš½1subscriptšæ12subscriptš»subscriptš3superscriptsubscriptš§3š“subscriptš½0subscriptš½12\beginsplitG+F&=H_a_3z_3^A(B_+)+H_-a_3z_3^A(B_-)-2H(a_3% z_3^A)\\ &+\frac12(1+a_3z_3^A)[H_\alpha_0(\gamma_0)+H_\alpha_1(% \gamma_1)-2H_a_3z_3^A(\frac\alpha_0-\alpha_12)]\\ &+\frac12(1-a_3z_3^A)[H_\beta_0(\delta_0)+H_\beta_1(\delta% _1)-2H_-a_3z_3^A(\frac\beta_0-\beta_12)].\endsplitstart_ROW start_CELL italic_G + italic_F end_CELL start_CELL = italic_H start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_B start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_B start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) - 2 italic_H ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 2 italic_H start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( divide start_ARG italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - 2 italic_H start_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( divide start_ARG italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) ] . end_CELL end_ROW
Note that Fš¹Fitalic_F is a function of six variables and the first three are exactly the variables of GšŗGitalic_G. Our strategy of locating the extremal points of G+Fšŗš¹G+Fitalic_G + italic_F is first finding the critical points z1A,z2A,z3Asuperscriptsubscriptš§1š“superscriptsubscriptš§2š“superscriptsubscriptš§3š“z_1^A,z_2^A,z_3^Aitalic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT of GšŗGitalic_G and verify that at those points the critical points of GšŗGitalic_G are attainable, then we can find the maximal points of F+Gš¹šŗF+Gitalic_F + italic_G.
For case 1: a3b3r3ā¤0subscriptš3subscriptš3subscriptš30a_3b_3r_3\leq 0italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā¤ 0 and r32-r2ā„a3b3r3superscriptsubscriptš32superscriptš2subscriptš3subscriptš3subscriptš3r_3^2-r^2\geq a_3b_3r_3italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ā„ italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, by [14] we know that maxG(z1A,z2A,z3A)=G(0,0,1)šŗsuperscriptsubscriptš§1š“superscriptsubscriptš§2š“superscriptsubscriptš§3š“šŗ001\maxG(z_1^A,z_2^A,z_3^A)=G(0,0,1)roman_max italic_G ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_G ( 0 , 0 , 1 ), then the parameters in function Fš¹Fitalic_F are degenerated into (k=0,1š01k=0,1italic_k = 0 , 1)
Ī±k=a3+(-1)k(b3z3B+r3z3B),Ī²k=-a3+(-1)k(b3z3B-r3z3B),Ī³k=āi3ci2+s32+T32(z3B)2+2[c3s3+(-1)k(s3T3z3B+c3T3z3B)]12,Ī“k=āi3ci2+s32+T32(z3B)2+2[-c3s3+(-1)k(s3T3z3B-c3T3z3B)]12.formulae-sequencesubscriptš¼šsubscriptš3superscript1šsubscriptš3superscriptsubscriptš§3šµsubscriptš3superscriptsubscriptš§3šµformulae-sequencesubscriptš½šsubscriptš3superscript1šsubscriptš3superscriptsubscriptš§3šµsubscriptš3superscriptsubscriptš§3šµformulae-sequencesubscriptš¾šsuperscriptsuperscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš 32superscriptsubscriptš32superscriptsubscriptsuperscriptš§šµ322delimited-[]subscriptš3subscriptš 3superscript1šsubscriptš 3subscriptš3subscriptsuperscriptš§šµ3subscriptš3subscriptš3subscriptsuperscriptš§šµ312subscriptšæšsuperscriptsuperscriptsubscriptš3superscriptsubscriptšš2superscriptsubscriptš 32superscriptsubscriptš32superscriptsubscriptsuperscriptš§šµ322delimited-[]subscriptš3subscriptš 3superscript1šsubscriptš 3subscriptš3subscriptsuperscriptš§šµ3subscriptš3subscriptš3subscriptsuperscriptš§šµ312\beginsplit\alpha_k&=a_3+(-1)^k(b_3z_3^B+r_3z_3^B),\\ \beta_k&=-a_3+(-1)^k(b_3z_3^B-r_3z_3^B),\\ \gamma_k&=\\sum_i^3c_i^2+s_3^2+T_3^2(z^B_3)^2+2[c_3% s_3+(-1)^k(s_3T_3z^B_3+c_3T_3z^B_3)]\^\frac12,\\ \delta_k&=\\sum_i^3c_i^2+s_3^2+T_3^2(z^B_3)^2+2[-c_3% s_3+(-1)^k(s_3T_3z^B_3-c_3T_3z^B_3)]\^\frac12.\end% splitstart_ROW start_CELL italic_Ī± start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) , end_CELL end_ROW start_ROW start_CELL italic_Ī² start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT - italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) , end_CELL end_ROW start_ROW start_CELL italic_Ī³ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 [ italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_Ī“ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 [ - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT . end_CELL end_ROW
Therefore, we have
(3.22) F(z1A,z2A,z3A,z1B,z2B,z3B)=F(0,0,1,z3B)=12(1+a3)[HĪ±0(Ī³0)+HĪ±1(Ī³1)]+12(1-a3)[HĪ²0(Ī³0)+HĪ²1(Ī³1)].š¹subscriptsuperscriptš§š“1subscriptsuperscriptš§š“2subscriptsuperscriptš§š“3subscriptsuperscriptš§šµ1subscriptsuperscriptš§šµ2subscriptsuperscriptš§šµ3š¹001subscriptsuperscriptš§šµ3121subscriptš3delimited-[]subscriptš»subscriptš¼0subscriptš¾0subscriptš»subscriptš¼1subscriptš¾1121subscriptš3delimited-[]subscriptš»subscriptš½0subscriptš¾0subscriptš»subscriptš½1subscriptš¾1\beginsplit&F(z^A_1,z^A_2,z^A_3,z^B_1,z^B_2,z^B_3)=F% (0,0,1,z^B_3)\\ &=\frac12(1+a_3)[H_\alpha_0(\gamma_0)+H_\alpha_1(\gamma_1)]+% \frac12(1-a_3)[H_\beta_0(\gamma_0)+H_\beta_1(\gamma_1)].\end% splitstart_ROW start_CELL end_CELL start_CELL italic_F ( italic_z start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_F ( 0 , 0 , 1 , italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] + divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] . end_CELL end_ROW
When (b3+r3)(c3+s3)ā¤0subscriptš3subscriptš3subscriptš3subscriptš 30(b_3+r_3)(c_3+s_3)\leq 0( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ( italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ā¤ 0, it can be observed that Fš¹Fitalic_F is an even function for z3Bā[-1,1]superscriptsubscriptš§3šµ11z_3^B\in[-1,1]italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ā [ - 1 , 1 ], so we just need to consider z3Bā[0,1]superscriptsubscriptš§3šµ01z_3^B\in[0,1]italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ā [ 0 , 1 ]. The derivative of Fš¹Fitalic_F on z3Bsuperscriptsubscriptš§3šµz_3^Bitalic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT is given by
(3.23) āFāz3B=14(1+a3)(b3+r3)log2(1+Ī±1)2[(1+Ī±0)2-Ī³02](1+Ī±0)2[(1+Ī±1)2-Ī³12]+-c3T3-s3T3+T32z3BĪ³1log21+Ī±1+Ī³11+Ī±1-Ī³1+c3T3+s3T3+T32z3BĪ³0log21+Ī±0+Ī³01+Ī±0-Ī³0+14(1-a3)(b3-r3)log2(1+Ī²1)2[(1+Ī²0)2-Ī“02](1+Ī²0)2[(1+Ī²1)2-Ī“12]+c3T3-s3T3+T32z3BĪ“1log21+Ī²1+Ī“11+Ī²1-Ī“1+-c3T3+s3T3+T32z3BĪ“0log21+Ī²0+Ī“01+Ī²0-Ī“0;š¹superscriptsubscriptš§3šµ141subscriptš3subscriptš3subscriptš3subscript2superscript1subscriptš¼12delimited-[]superscript1subscriptš¼02superscriptsubscriptš¾02superscript1subscriptš¼02delimited-[]superscript1subscriptš¼12superscriptsubscriptš¾12subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptš¾1subscript21subscriptš¼1subscriptš¾11subscriptš¼1subscriptš¾1subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptš¾0subscript21subscriptš¼0subscriptš¾01subscriptš¼0subscriptš¾0141subscriptš3subscriptš3subscriptš3subscript2superscript1subscriptš½12delimited-[]superscript1subscriptš½02superscriptsubscriptšæ02superscript1subscriptš½02delimited-[]superscript1subscriptš½12superscriptsubscriptšæ12subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptšæ1subscript21subscriptš½1subscriptšæ11subscriptš½1subscriptšæ1subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptšæ0subscript21subscriptš½0subscriptšæ01subscriptš½0subscriptšæ0\beginsplit&\frac\partialF\partialz_3^B=\frac14(1+a_3)\(b% _3+r_3)\log_2\frac(1+\alpha_1)^2[(1+\alpha_0)^2-\gamma_0^2]% (1+\alpha_0)^2[(1+\alpha_1)^2-\gamma_1^2]\\ &+\frac-c_3T_3-s_3T_3+T_3^2z_3^B\gamma_1\log_2\frac1+% \alpha_1+\gamma_11+\alpha_1-\gamma_1+\fracc_3T_3+s_3T_3+T_% 3^2z_3^B\gamma_0\log_2\frac1+\alpha_0+\gamma_01+\alpha_% 0-\gamma_0\\\ &+\frac14(1-a_3)\(b_3-r_3)\log_2\frac(1+\beta_1)^2[(1+\beta_% 0)^2-\delta_0^2](1+\beta_0)^2[(1+\beta_1)^2-\delta_1^2]% \\ &+\fracc_3T_3-s_3T_3+T_3^2z_3^B\delta_1\log_2\frac1+% \beta_1+\delta_11+\beta_1-\delta_1\ +\frac-c_3T_3+s_3T_3+T% _3^2z_3^B\delta_0\log_2\frac1+\beta_0+\delta_01+\beta_0% -\delta_0\;\endsplitstart_ROW start_CELL end_CELL start_CELL divide start_ARG ā italic_F end_ARG start_ARG ā italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG 4 ( 1 + italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_ARG ( 1 + italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG 1 end_ARG start_ARG 4 ( 1 - italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_ARG ( 1 + italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ; end_CELL end_ROW
If b3+r3ā¤0subscriptš3subscriptš30b_3+r_3\leq 0italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā¤ 0 and c3+s3ā„0subscriptš3subscriptš 30c_3+s_3\geq 0italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā„ 0, we have Ī³0ā„Ī³1subscriptš¾0subscriptš¾1\gamma_0\geq\gamma_1italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ā„ italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, Ī±1ā„Ī±0subscriptš¼1subscriptš¼0\alpha_1\geq\alpha_0italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ā„ italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, Ī“0ā„Ī“1subscriptšæ0subscriptšæ1\delta_0\geq\delta_1italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ā„ italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and Ī²1ā„Ī²0subscriptš½1subscriptš½0\beta_1\geq\beta_0italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ā„ italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, then
(3.24) (b3+r3)log2(1+Ī±1)2[(1+Ī±0)2-Ī³02](1+Ī±0)2[(1+Ī±1)2-Ī³12]ā„0;(b3-r3)log2(1+Ī²1)2[(1+Ī²0)2-Ī“02](1+Ī²0)2[(1+Ī²1)2-Ī“12]ā„0;formulae-sequencesubscriptš3subscriptš3subscript2superscript1subscriptš¼12delimited-[]superscript1subscriptš¼02superscriptsubscriptš¾02superscript1subscriptš¼02delimited-[]superscript1subscriptš¼12superscriptsubscriptš¾120subscriptš3subscriptš3subscript2superscript1subscriptš½12delimited-[]superscript1subscriptš½02superscriptsubscriptšæ02superscript1subscriptš½02delimited-[]superscript1subscriptš½12superscriptsubscriptšæ120(b_3+r_3)\log_2\frac(1+\alpha_1)^2[(1+\alpha_0)^2-\gamma_0^2% ](1+\alpha_0)^2[(1+\alpha_1)^2-\gamma_1^2]\geq 0;(b_3-r_3)% \log_2\frac(1+\beta_1)^2[(1+\beta_0)^2-\delta_0^2](1+\beta_0% )^2[(1+\beta_1)^2-\delta_1^2]\geq 0;( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_ARG ( 1 + italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG ā„ 0 ; ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_ARG ( 1 + italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG ā„ 0 ;
(3.25) -c3T3-s3T3+T32z3BĪ³1log21+Ī±1+Ī³11+Ī±1-Ī³1+c3T3+s3T3+T32z3BĪ³0log21+Ī±0+Ī³01+Ī±0-Ī³0ā„-c3T3-s3T3+T32z3BĪ³0log21+Ī±1+Ī³11+Ī±1-Ī³1+c3T3+s3T3+T32z3BĪ³0log21+Ī±1+Ī³11+Ī±1-Ī³1=2T32z3BĪ³0log21+Ī±1+Ī³11+Ī±1-Ī³1ā„0;subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptš¾1subscript21subscriptš¼1subscriptš¾11subscriptš¼1subscriptš¾1subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptš¾0subscript21subscriptš¼0subscriptš¾01subscriptš¼0subscriptš¾0subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptš¾0subscript21subscriptš¼1subscriptš¾11subscriptš¼1subscriptš¾1subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptš¾0subscript21subscriptš¼1subscriptš¾11subscriptš¼1subscriptš¾12superscriptsubscriptš32subscriptsuperscriptš§šµ3subscriptš¾0subscript21subscriptš¼1subscriptš¾11subscriptš¼1subscriptš¾10\beginsplit&\frac-c_3T_3-s_3T_3+T_3^2z_3^B\gamma_1\log% _2\frac1+\alpha_1+\gamma_11+\alpha_1-\gamma_1+\fracc_3T_3+s% _3T_3+T_3^2z_3^B\gamma_0\log_2\frac1+\alpha_0+\gamma_0% 1+\alpha_0-\gamma_0\\ \geq&\frac-c_3T_3-s_3T_3+T_3^2z_3^B\gamma_0\log_2\frac% 1+\alpha_1+\gamma_11+\alpha_1-\gamma_1+\fracc_3T_3+s_3T_3% +T_3^2z_3^B\gamma_0\log_2\frac1+\alpha_1+\gamma_11+% \alpha_1-\gamma_1\\ =&\frac2T_3^2z^B_3\gamma_0\log_2\frac1+\alpha_1+\gamma_1% 1+\alpha_1-\gamma_1\geq 0;\endsplitstart_ROW start_CELL end_CELL start_CELL divide start_ARG - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL ā„ end_CELL start_CELL divide start_ARG - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL divide start_ARG 2 italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_Ī³ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī± start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī³ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ā„ 0 ; end_CELL end_ROW
(3.26) c3T3-s3T3+T32z3BĪ“1log21+Ī²1+Ī“11+Ī²1-Ī“1+-c3T3+s3T3+T32z3BĪ“0log21+Ī²0+Ī“01+Ī²0-Ī“0ā„c3T3-s3T3+T32z3BĪ“0log21+Ī²1+Ī“11+Ī²1-Ī“1+-c3T3+s3T3+T32z3BĪ“0log21+Ī²1+Ī“11+Ī²1-Ī“1=2T32z3BĪ“0log21+Ī²1+Ī“11+Ī²1-Ī“1ā„0.subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptšæ1subscript21subscriptš½1subscriptšæ11subscriptš½1subscriptšæ1subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptšæ0subscript21subscriptš½0subscriptšæ01subscriptš½0subscriptšæ0subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptšæ0subscript21subscriptš½1subscriptšæ11subscriptš½1subscriptšæ1subscriptš3subscriptš3subscriptš 3subscriptš3superscriptsubscriptš32superscriptsubscriptš§3šµsubscriptšæ0subscript21subscriptš½1subscriptšæ11subscriptš½1subscriptšæ12superscriptsubscriptš32subscriptsuperscriptš§šµ3subscriptšæ0subscript21subscriptš½1subscriptšæ11subscriptš½1subscriptšæ10\beginsplit&\fracc_3T_3-s_3T_3+T_3^2z_3^B\delta_1\log_% 2\frac1+\beta_1+\delta_11+\beta_1-\delta_1+\frac-c_3T_3+s_% 3T_3+T_3^2z_3^B\delta_0\log_2\frac1+\beta_0+\delta_01% +\beta_0-\delta_0\\ \geq&\fracc_3T_3-s_3T_3+T_3^2z_3^B\delta_0\log_2\frac% 1+\beta_1+\delta_11+\beta_1-\delta_1+\frac-c_3T_3+s_3T_3+T% _3^2z_3^B\delta_0\log_2\frac1+\beta_1+\delta_11+\beta_1% -\delta_1\\ =&\frac2T_3^2z^B_3\delta_0\log_2\frac1+\beta_1+\delta_1% 1+\beta_1-\delta_1\geq 0.\endsplitstart_ROW start_CELL end_CELL start_CELL divide start_ARG italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL ā„ end_CELL start_CELL divide start_ARG italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL divide start_ARG 2 italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_Ī“ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_Ī² start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Ī“ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ā„ 0 . end_CELL end_ROW
Hence in this case we get āFāz3Bā„0š¹subscriptsuperscriptš§šµ30\frac\partialF\partialz^B_3\geq 0divide start_ARG ā italic_F end_ARG start_ARG ā italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ā„ 0 when z3Bā[0,1]superscriptsubscriptš§3šµ01z_3^B\in[0,1]italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ā [ 0 , 1 ].
If b3+r3ā„0subscriptš3subscriptš30b_3+r_3\geq 0italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā„ 0 and c3+s3ā¤0subscriptš3subscriptš 30c_3+s_3\leq 0italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā¤ 0, we also can show that āFāz3Bā„0š¹superscriptsubscriptš§3šµ0\frac\partial F\partial z_3^B\geq 0divide start_ARG ā italic_F end_ARG start_ARG ā italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG ā„ 0 similarly. So Fš¹Fitalic_F is a strictly monotonically increasing function with z3Bā[0,1]subscriptsuperscriptš§šµ301z^B_3\in[0,1]italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ā [ 0 , 1 ]. Similarly we can check that Fš¹Fitalic_F is a strictly monotonically increasing function with respect to z1Bā[0,1]subscriptsuperscriptš§šµ101z^B_1\in[0,1]italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ā [ 0 , 1 ] or z2Bā[0,1]subscriptsuperscriptš§šµ201z^B_2\in[0,1]italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā [ 0 , 1 ] in case 2. ā
Theorem 3.3.
For the Werner-GHZ state Ļw=c|Ļā©āØĻ|+(1-c)I8subscriptšš¤šketšbraš1šš¼8\rho_w=c|\psi\rangle\langle\psi|+(1-c)\fracI8italic_Ļ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = italic_c | italic_Ļ ā© āØ italic_Ļ | + ( 1 - italic_c ) divide start_ARG italic_I end_ARG start_ARG 8 end_ARG, where |Ļā©=|000ā©+|111ā©2ketšket000ket1112|\psi\rangle=\frac2| italic_Ļ ā© = divide start_ARG | 000 ā© + | 111 ā© end_ARG start_ARG 2 end_ARG, the quantum discord is
(3.27) š¬=18(1-c)log2(1-c)+1+7c8log2(1+7c)-14(1+3c)log2(1+3c).š¬181šššsubscriptš21š17š8ššsubscriptš217š1413šššsubscriptš213š\beginsplit\mathcalQ=\frac18(1-c)log_2(1-c)+\frac1+7c8log_2(1+% 7c)-\frac14(1+3c)log_2(1+3c).\endsplitstart_ROW start_CELL caligraphic_Q = divide start_ARG 1 end_ARG start_ARG 8 end_ARG ( 1 - italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c ) + divide start_ARG 1 + 7 italic_c end_ARG start_ARG 8 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + 7 italic_c ) - divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 + 3 italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + 3 italic_c ) . end_CELL end_ROW
Obviously, maxG(z1A,z2A,z3A)=H(c).šŗsuperscriptsubscriptš§1š“superscriptsubscriptš§2š“superscriptsubscriptš§3š“š»š\max\G(z_1^A,z_2^A,z_3^A)\=H(c).roman_max italic_G ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_H ( italic_c ) . Let Īø=cz3Bššsuperscriptsubscriptš§3šµ\theta=cz_3^Bitalic_Īø = italic_c italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, then
(3.28) F(z1A,z2A,z3A,z1B,z2B,z3B)=F(Īø)=12[HĪø(|c+Īø|)+HĪø(|c-Īø|)+H-Īø(|c+Īø|)+H-Īø(|c-Īø|)]-2H(Īø).š¹superscriptsubscriptš§1š“superscriptsubscriptš§2š“superscriptsubscriptš§3š“superscriptsubscriptš§1šµsuperscriptsubscriptš§2šµsuperscriptsubscriptš§3šµš¹š12delimited-[]subscriptš»šššsubscriptš»šššsubscriptš»šššsubscriptš»ššš2š»š\beginsplit&F(z_1^A,z_2^A,z_3^A,z_1^B,z_2^B,z_3^B)=F% (\theta)\\ =&\frac12[H_\theta(|c+\theta|)+H_\theta(|c-\theta|)+H_-\theta(|c+% \theta|)+H_-\theta(|c-\theta|)]-2H(\theta).\endsplitstart_ROW start_CELL end_CELL start_CELL italic_F ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) = italic_F ( italic_Īø ) end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_Īø end_POSTSUBSCRIPT ( | italic_c + italic_Īø | ) + italic_H start_POSTSUBSCRIPT italic_Īø end_POSTSUBSCRIPT ( | italic_c - italic_Īø | ) + italic_H start_POSTSUBSCRIPT - italic_Īø end_POSTSUBSCRIPT ( | italic_c + italic_Īø | ) + italic_H start_POSTSUBSCRIPT - italic_Īø end_POSTSUBSCRIPT ( | italic_c - italic_Īø | ) ] - 2 italic_H ( italic_Īø ) . end_CELL end_ROW
It is easy to see that F(Īø)š¹šF(\theta)italic_F ( italic_Īø ) is monotonically increasing with respect to Īøā[0,1]š01\theta\in[0,1]italic_Īø ā [ 0 , 1 ]. So maxF(Īø)=F(maxĪø)=F(c)š¹šš¹šš¹š\max\F(\theta)\=F(\max\\theta\)=F(c)roman_max italic_F ( italic_Īø ) = italic_F ( roman_max italic_Īø ) = italic_F ( italic_c ). Fig.1 shows the behavior of the function š¬š¬\mathcalQcaligraphic_Q.
Next, we consider the following general tripartite state
(3.29) Ļ=18(I8+āi3aiĻiāI4+I2āāi3biĻiāI2+I4āāi3ciĻi+āi3riĻiāĻiāI2+āi3siĻiāI2āĻi+āi3viI2āĻiāĻi+āi3TiĻiāĻiāĻi).š18subscriptš¼8superscriptsubscriptš3tensor-productsubscriptššsubscriptššsubscriptš¼4tensor-productsubscriptš¼2superscriptsubscriptš3tensor-productsubscriptššsubscriptššsubscriptš¼2tensor-productsubscriptš¼4superscriptsubscriptš3subscriptššsubscriptššsuperscriptsubscriptš3tensor-productsubscriptššsubscriptššsubscriptššsubscriptš¼2superscriptsubscriptš3tensor-productsubscriptš šsubscriptššsubscriptš¼2subscriptššsuperscriptsubscriptš3tensor-productsubscriptš£šsubscriptš¼2subscriptššsubscriptššsuperscriptsubscriptš3tensor-productsubscriptššsubscriptššsubscriptššsubscriptšš\beginsplit\rho&=\frac18(I_8+\sum_i^3a_i\sigma_i\otimes I_4+% I_2\otimes\sum_i^3b_i\sigma_i\otimes I_2+I_4\otimes\sum_i^3c% _i\sigma_i\\ &+\sum_i^3r_i\sigma_i\otimes\sigma_i\otimes I_2+\sum_i^3s_i% \sigma_i\otimes I_2\otimes\sigma_i\\ &+\sum_i^3v_iI_2\otimes\sigma_i\otimes\sigma_i+\sum_i^3T_i% \sigma_i\otimes\sigma_i\otimes\sigma_i).\endsplitstart_ROW start_CELL italic_Ļ end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 8 end_ARG ( italic_I start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ā ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ā italic_Ļ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) . end_CELL end_ROW
Let a=āi3ai2šsuperscriptsubscriptš3superscriptsubscriptšš2a=\sqrt\sum_i^3a_i^2italic_a = square-root start_ARG ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG and b=āi3bi2šsuperscriptsubscriptš3superscriptsubscriptšš2b=\sqrt\sum_i^3b_i^2italic_b = square-root start_ARG ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG, then we can get the quantum discord for some special cases.
Theorem 3.4.
For the general tripartite state Ļš\rhoitalic_Ļ in Eq.(3.29), we have the following results:
Case 1: when ai=vi=Ti=0,r1=r2=r3=rformulae-sequencesubscriptššsubscriptš£šsubscriptšš0subscriptš1subscriptš2subscriptš3ša_i=v_i=T_i=0,r_1=r_2=r_3=ritalic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_r, we have that
(3.30) š¬(Ļ)=āi8Ī»ilog2Ī»i+4+Hb(r)+H-b(r)-H(|b+r|)-12[Hb+B(r)+Hb-B(r)+H-b+B(r)+H-b-B(r)],š¬šsuperscriptsubscriptš8subscriptššsubscript2subscriptšš4subscriptš»ššsubscriptš»ššš»šš12delimited-[]subscriptš»šBšsubscriptš»šBšsubscriptš»šBšsubscriptš»šBš\beginsplit\mathcalQ(\rho)=&\sum_i^8\lambda_i\log_2\lambda_i+4% +H_b(r)+H_-b(r)-H(|b+r|)\\ -&\frac12[H_b+\mathrmB(r)+H_b-\mathrmB(r)+H_-b+\mathrmB(r)+H_% -b-\mathrmB(r)],\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ ) = end_CELL start_CELL ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 4 + italic_H start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_b end_POSTSUBSCRIPT ( italic_r ) - italic_H ( | italic_b + italic_r | ) end_CELL end_ROW start_ROW start_CELL - end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_b + roman_B end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT italic_b - roman_B end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_b + roman_B end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_b - roman_B end_POSTSUBSCRIPT ( italic_r ) ] , end_CELL end_ROW
where B=[āi3(sibib+ci)2]12normal-Bsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptš šsubscriptšššsubscriptšš212\mathrmB=[\sum_i^3(s_i\fracb_ib+c_i)^2]^\frac12roman_B = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG + italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT.
Case 2: when bi=vi=Ti=0,r1=r2=r3=rformulae-sequencesubscriptššsubscriptš£šsubscriptšš0subscriptš1subscriptš2subscriptš3šb_i=v_i=T_i=0,r_1=r_2=r_3=ritalic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_r, we have that
(3.31) š¬(Ļ)=āi8Ī»ilog2Ī»i+3-H(a2)-12[Ha(r)+H-a(r)-2H(a)]-12(1+a)[Ha+A(r)+Ha-A(r)-2Ha(r)]-12(1-a)[H-a+A(r)+H-a-A(r)-2H-a(r)],š¬šsuperscriptsubscriptš8subscriptššsubscript2subscriptšš3š»superscriptš212delimited-[]subscriptš»ššsubscriptš»šš2š»š121šdelimited-[]subscriptš»šAšsubscriptš»šAš2subscriptš»šš121šdelimited-[]subscriptš»šAšsubscriptš»šAš2subscriptš»šš\beginsplit\mathcalQ(\rho)&=\sum_i^8\lambda_i\log_2\lambda_i+3% -H(a^2)-\frac12[H_a(r)+H_-a(r)-2H(a)]\\ &-\frac12(1+a)[H_a+\mathrmA(r)+H_a-\mathrmA(r)-2H_a(r)]\\ &-\frac12(1-a)[H_-a+\mathrmA(r)+H_-a-\mathrmA(r)-2H_-a(r)],\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ ) end_CELL start_CELL = ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 3 - italic_H ( italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_a end_POSTSUBSCRIPT ( italic_r ) - 2 italic_H ( italic_a ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_a + roman_A end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT italic_a - roman_A end_POSTSUBSCRIPT ( italic_r ) - 2 italic_H start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_r ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a ) end_ARG [ italic_H start_POSTSUBSCRIPT - italic_a + roman_A end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_a - roman_A end_POSTSUBSCRIPT ( italic_r ) - 2 italic_H start_POSTSUBSCRIPT - italic_a end_POSTSUBSCRIPT ( italic_r ) ] , end_CELL end_ROW
where A=[āi3(siaia+ci)2]12normal-Asuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptš šsubscriptšššsubscriptšš212\mathrmA=[\sum_i^3(s_i\fraca_ia+c_i)^2]^\frac12roman_A = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_a end_ARG + italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT.
Case 3: when ri=Ti=vi=0subscriptššsubscriptššsubscriptš£š0r_i=T_i=v_i=0italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0, we have that
(3.32) š¬(Ļ)=āi8Ī»ilog2Ī»i+3-H(a2)-12[Hb(a)+H-b(a)-2H(a)]-12(1+a)[Ha+A(b)+Ha-A(b)-2Ha(b)]-12(1-a)[Hb+A(b)+H-a-A(b)-2H-a(b)],š¬šsuperscriptsubscriptš8subscriptššsubscript2subscriptšš3š»superscriptš212delimited-[]subscriptš»ššsubscriptš»šš2š»š121šdelimited-[]subscriptš»šAšsubscriptš»šAš2subscriptš»šš121šdelimited-[]subscriptš»šAšsubscriptš»šAš2subscriptš»šš\beginsplit\mathcalQ(\rho)&=\sum_i^8\lambda_i\log_2\lambda_i+3% -H(a^2)-\frac12[H_b(a)+H_-b(a)-2H(a)]\\ &-\frac12(1+a)[H_a+\mathrmA(b)+H_a-\mathrmA(b)-2H_a(b)]\\ &-\frac12(1-a)[H_b+\mathrmA(b)+H_-a-\mathrmA(b)-2H_-a(b)],\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ ) end_CELL start_CELL = ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 3 - italic_H ( italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_a ) + italic_H start_POSTSUBSCRIPT - italic_b end_POSTSUBSCRIPT ( italic_a ) - 2 italic_H ( italic_a ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_a + roman_A end_POSTSUBSCRIPT ( italic_b ) + italic_H start_POSTSUBSCRIPT italic_a - roman_A end_POSTSUBSCRIPT ( italic_b ) - 2 italic_H start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_b ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_b + roman_A end_POSTSUBSCRIPT ( italic_b ) + italic_H start_POSTSUBSCRIPT - italic_a - roman_A end_POSTSUBSCRIPT ( italic_b ) - 2 italic_H start_POSTSUBSCRIPT - italic_a end_POSTSUBSCRIPT ( italic_b ) ] , end_CELL end_ROW
where A=[āi3(siaia+ci)2]12normal-Asuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptš šsubscriptšššsubscriptšš212\mathrmA=[\sum_i^3(s_i\fraca_ia+c_i)^2]^\frac12roman_A = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_a end_ARG + italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT.
Case 4: when ai=ci=si=Ti=0,r1=r2=r3=r,v1=v2=v3=vformulae-sequencesubscriptššsubscriptššsubscriptš šsubscriptšš0subscriptš1subscriptš2subscriptš3šsubscriptš£1subscriptš£2subscriptš£3š£a_i=c_i=s_i=T_i=0,r_1=r_2=r_3=r,v_1=v_2=v_3=vitalic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_r , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_v, we have that
(3.33) š¬(Ļ)=āi8Ī»ilog2Ī»i+Hb(r)+H-b(r)+4-H(|b+r|)-12[Hb+v(r)+Hb-v(r)+H-b+v(r)H-b-v(r)].š¬šsuperscriptsubscriptš8subscriptššsubscript2subscriptššsubscriptš»ššsubscriptš»šš4š»šš12delimited-[]subscriptš»šš£šsubscriptš»šš£šsubscriptš»šš£šsubscriptš»šš£š\beginsplit\mathcalQ(\rho)=&\sum_i^8\lambda_i\log_2\lambda_i+H% _b(r)+H_-b(r)+4-H(|b+r|)\\ -&\frac12[H_b+v(r)+H_b-v(r)+H_-b+v(r)H_-b-v(r)].\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ ) = end_CELL start_CELL ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_H start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_b end_POSTSUBSCRIPT ( italic_r ) + 4 - italic_H ( | italic_b + italic_r | ) end_CELL end_ROW start_ROW start_CELL - end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_b + italic_v end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT italic_b - italic_v end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_b + italic_v end_POSTSUBSCRIPT ( italic_r ) italic_H start_POSTSUBSCRIPT - italic_b - italic_v end_POSTSUBSCRIPT ( italic_r ) ] . end_CELL end_ROW
Case 5: when ri=Ti=si=ci=0,v1=v2=v3=vformulae-sequencesubscriptššsubscriptššsubscriptš šsubscriptšš0subscriptš£1subscriptš£2subscriptš£3š£r_i=T_i=s_i=c_i=0,v_1=v_2=v_3=vitalic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_v, we have that
(3.34) š¬(Ļ)=āi8Ī»ilog2Ī»i+3-H(a2)-12[Hb(a)+H-b(a)-2H(a)]-12(1+a)[Ha+v(b)+Ha-v(b)-2Ha(b)]-12(1-a)[H-a+v(b)+H-a-v(b)-2H-a(b)].š¬šsuperscriptsubscriptš8subscriptššsubscript2subscriptšš3š»superscriptš212delimited-[]subscriptš»ššsubscriptš»šš2š»š121šdelimited-[]subscriptš»šš£šsubscriptš»šš£š2subscriptš»šš121šdelimited-[]subscriptš»šš£šsubscriptš»šš£š2subscriptš»šš\beginsplit\mathcalQ(\rho)&=\sum_i^8\lambda_i\log_2\lambda_i+3% -H(a^2)-\frac12[H_b(a)+H_-b(a)-2H(a)]\\ &-\frac12(1+a)[H_a+v(b)+H_a-v(b)-2H_a(b)]\\ &-\frac12(1-a)[H_-a+v(b)+H_-a-v(b)-2H_-a(b)].\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ ) end_CELL start_CELL = ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 3 - italic_H ( italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_a ) + italic_H start_POSTSUBSCRIPT - italic_b end_POSTSUBSCRIPT ( italic_a ) - 2 italic_H ( italic_a ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_a + italic_v end_POSTSUBSCRIPT ( italic_b ) + italic_H start_POSTSUBSCRIPT italic_a - italic_v end_POSTSUBSCRIPT ( italic_b ) - 2 italic_H start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_b ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a ) end_ARG [ italic_H start_POSTSUBSCRIPT - italic_a + italic_v end_POSTSUBSCRIPT ( italic_b ) + italic_H start_POSTSUBSCRIPT - italic_a - italic_v end_POSTSUBSCRIPT ( italic_b ) - 2 italic_H start_POSTSUBSCRIPT - italic_a end_POSTSUBSCRIPT ( italic_b ) ] . end_CELL end_ROW
Case 6: when bi=si=ci=Ti=0,r1=r2=r3=r,v1=v2=v3=vformulae-sequencesubscriptššsubscriptš šsubscriptššsubscriptšš0subscriptš1subscriptš2subscriptš3šsubscriptš£1subscriptš£2subscriptš£3š£b_i=s_i=c_i=T_i=0,r_1=r_2=r_3=r,v_1=v_2=v_3=vitalic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_r , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_v, we have that
(3.35) š¬(Ļ)=āi8Ī»ilog2Ī»i+3-H(a2)-12[Ha(r)+H-a(r)-2H(a)]-12(1+a)[Ha+v(r)+Ha-v(r)-2Ha(r)]-12(1-a)[H-a+v(r)+H-a-v(r)-2H-a(r)].š¬šsuperscriptsubscriptš8subscriptššsubscript2subscriptšš3š»superscriptš212delimited-[]subscriptš»ššsubscriptš»šš2š»š121šdelimited-[]subscriptš»šš£šsubscriptš»šš£š2subscriptš»šš121šdelimited-[]subscriptš»šš£šsubscriptš»šš£š2subscriptš»šš\beginsplit\mathcalQ(\rho)&=\sum_i^8\lambda_i\log_2\lambda_i+3% -H(a^2)-\frac12[H_a(r)+H_-a(r)-2H(a)]\\ &-\frac12(1+a)[H_a+v(r)+H_a-v(r)-2H_a(r)]\\ &-\frac12(1-a)[H_-a+v(r)+H_-a-v(r)-2H_-a(r)].\endsplitstart_ROW start_CELL caligraphic_Q ( italic_Ļ ) end_CELL start_CELL = ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Ī» start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 3 - italic_H ( italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_a end_POSTSUBSCRIPT ( italic_r ) - 2 italic_H ( italic_a ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_a ) end_ARG [ italic_H start_POSTSUBSCRIPT italic_a + italic_v end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT italic_a - italic_v end_POSTSUBSCRIPT ( italic_r ) - 2 italic_H start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_r ) ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_a ) end_ARG [ italic_H start_POSTSUBSCRIPT - italic_a + italic_v end_POSTSUBSCRIPT ( italic_r ) + italic_H start_POSTSUBSCRIPT - italic_a - italic_v end_POSTSUBSCRIPT ( italic_r ) - 2 italic_H start_POSTSUBSCRIPT - italic_a end_POSTSUBSCRIPT ( italic_r ) ] . end_CELL end_ROW
All cases can be shown similarly. Letās consider case 1: ai=vi=Ti=0,r1=r2=r3=rformulae-sequencesubscriptššsubscriptš£šsubscriptšš0subscriptš1subscriptš2subscriptš3ša_i=v_i=T_i=0,r_1=r_2=r_3=ritalic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_r, maxG(z1A,z2A,z3A)=G(b1b,b2b,b3b)šŗsuperscriptsubscriptš§1š“superscriptsubscriptš§2š“superscriptsubscriptš§3š“šŗsubscriptš1šsubscriptš2šsubscriptš3š\max\G(z_1^A,z_2^A,z_3^A)\=G(\fracb_1b,\fracb_2b,% \fracb_3b)roman_max italic_G ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_G ( divide start_ARG italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG , divide start_ARG italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG , divide start_ARG italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG ). Let Īø=āi3rziBbibšsuperscriptsubscriptš3šsuperscriptsubscriptš§ššµsubscriptššš\theta=\sum_i^3rz_i^B\fracb_ibitalic_Īø = ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT divide start_ARG italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG, then
(3.36) F(z1A,z2A,z3A,z1B,z2B,z3B)=F(b1b,b2b,b3b,Īø)=12[Hb+B(Īø)+Hb-B(Īø)+H-b+B(Īø)+H-b-B(Īø)]-Hb(Īø)-H-b(Īø)-2,š¹superscriptsubscriptš§1š“superscriptsubscriptš§2š“superscriptsubscriptš§3š“superscriptsubscriptš§1šµsuperscriptsubscriptš§2šµsuperscriptsubscriptš§3šµš¹subscriptš1šsubscriptš2šsubscriptš3šš12delimited-[]subscriptš»šBšsubscriptš»šBšsubscriptš»šBšsubscriptš»šBšsubscriptš»ššsubscriptš»šš2\beginsplit&F(z_1^A,z_2^A,z_3^A,z_1^B,z_2^B,z_3^B)=F% (\fracb_1b,\fracb_2b,\fracb_3b,\theta)\\ =&\frac12[H_b+\mathrmB(\theta)+H_b-\mathrmB(\theta)+H_-b+\mathrm% B(\theta)+H_-b-\mathrmB(\theta)]-H_b(\theta)-H_-b(\theta)-2,\endsplitstart_ROW start_CELL end_CELL start_CELL italic_F ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) = italic_F ( divide start_ARG italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG , divide start_ARG italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG , divide start_ARG italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG , italic_Īø ) end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_b + roman_B end_POSTSUBSCRIPT ( italic_Īø ) + italic_H start_POSTSUBSCRIPT italic_b - roman_B end_POSTSUBSCRIPT ( italic_Īø ) + italic_H start_POSTSUBSCRIPT - italic_b + roman_B end_POSTSUBSCRIPT ( italic_Īø ) + italic_H start_POSTSUBSCRIPT - italic_b - roman_B end_POSTSUBSCRIPT ( italic_Īø ) ] - italic_H start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_Īø ) - italic_H start_POSTSUBSCRIPT - italic_b end_POSTSUBSCRIPT ( italic_Īø ) - 2 , end_CELL end_ROW
where B=[āi3(sibib+ci)2]12Bsuperscriptdelimited-[]superscriptsubscriptš3superscriptsubscriptš šsubscriptšššsubscriptšš212\mathrmB=[\sum_i^3(s_i\fracb_ib+c_i)^2]^\frac12roman_B = [ ā start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG + italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT.
The derivative of Fš¹Fitalic_F over Īøš\thetaitalic_Īø is equal to
(3.37) āFāĪø=14[log2(1+b+B+Īø)(1+b-B+Īø)(1+b-Īø)2(1+b+B-Īø)(1+b-B-Īø)(1+b+Īø)2+log2(1-b-B+Īø)(1-b+B+Īø)(1-b-Īø)2(1-b-B-Īø)(1-b+B-Īø)(1-b+Īø)2].š¹š14delimited-[]subscript21šBš1šBšsuperscript1šš21šBš1šBšsuperscript1šš2subscript21šBš1šBšsuperscript1šš21šBš1šBšsuperscript1šš2\beginsplit\frac\partial F\partial\theta=&\frac14[\log_2\frac(1+b% +\mathrmB+\theta)(1+b-\mathrmB+\theta)(1+b-\theta)^2(1+b+\mathrmB-% \theta)(1+b-\mathrmB-\theta)(1+b+\theta)^2\\ +&\log_2\frac(1-b-\mathrmB+\theta)(1-b+\mathrmB+\theta)(1-b-\theta)^2% (1-b-\mathrmB-\theta)(1-b+\mathrmB-\theta)(1-b+\theta)^2].\endsplitstart_ROW start_CELL divide start_ARG ā italic_F end_ARG start_ARG ā italic_Īø end_ARG = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + italic_b + roman_B + italic_Īø ) ( 1 + italic_b - roman_B + italic_Īø ) ( 1 + italic_b - italic_Īø ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 + italic_b + roman_B - italic_Īø ) ( 1 + italic_b - roman_B - italic_Īø ) ( 1 + italic_b + italic_Īø ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL + end_CELL start_CELL roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 - italic_b - roman_B + italic_Īø ) ( 1 - italic_b + roman_B + italic_Īø ) ( 1 - italic_b - italic_Īø ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_b - roman_B - italic_Īø ) ( 1 - italic_b + roman_B - italic_Īø ) ( 1 - italic_b + italic_Īø ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] . end_CELL end_ROW
Obviously, āFāĪøā„0š¹š0\frac\partial F\partial\theta\geq 0divide start_ARG ā italic_F end_ARG start_ARG ā italic_Īø end_ARG ā„ 0 when Īøā[0,1]š01\theta\in[0,1]italic_Īø ā [ 0 , 1 ]. Then F(Īø)š¹šF(\theta)italic_F ( italic_Īø ) is a strictly increasing function and maxF(Īø)=F(maxĪø)š¹šš¹š\maxF(\theta)=F(\max\\theta\)roman_max italic_F ( italic_Īø ) = italic_F ( roman_max italic_Īø ).
Let Y=Īø+Ī¼[1-(z1B)2-(z2B)2-(z3B)2]šššdelimited-[]1superscriptsuperscriptsubscriptš§1šµ2superscriptsuperscriptsubscriptš§2šµ2superscriptsuperscriptsubscriptš§3šµ2Y=\theta+\mu[1-(z_1^B)^2-(z_2^B)^2-(z_3^B)^2]italic_Y = italic_Īø + italic_Ī¼ [ 1 - ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ], āYāz1B=rb1b-2Ī¼z1Bšsuperscriptsubscriptš§1šµšsubscriptš1š2šsuperscriptsubscriptš§1šµ\frac\partial Y\partial z_1^B=r\fracb_1b-2\mu z_1^Bdivide start_ARG ā italic_Y end_ARG start_ARG ā italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = italic_r divide start_ARG italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG - 2 italic_Ī¼ italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, āYāz2B=rb2b-2Ī¼z2Bšsuperscriptsubscriptš§2šµšsubscriptš2š2šsuperscriptsubscriptš§2šµ\frac\partial Y\partial z_2^B=r\fracb_2b-2\mu z_2^Bdivide start_ARG ā italic_Y end_ARG start_ARG ā italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = italic_r divide start_ARG italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG - 2 italic_Ī¼ italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, āYāz3B=rb3b-2Ī¼z3Bšsuperscriptsubscriptš§3šµšsubscriptš3š2šsuperscriptsubscriptš§3šµ\frac\partial Y\partial z_3^B=r\fracb_3b-2\mu z_3^Bdivide start_ARG ā italic_Y end_ARG start_ARG ā italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = italic_r divide start_ARG italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG - 2 italic_Ī¼ italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, āYāĪ¼=1-(z1B)2-(z2B)2-(z3B)2šš1superscriptsuperscriptsubscriptš§1šµ2superscriptsuperscriptsubscriptš§2šµ2superscriptsuperscriptsubscriptš§3šµ2\frac\partial Y\partial\mu=1-(z_1^B)^2-(z_2^B)^2-(z_3^B)^2divide start_ARG ā italic_Y end_ARG start_ARG ā italic_Ī¼ end_ARG = 1 - ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Imposing āYāz1B=0,āYāz2B=0,āYāz3B=0,āYāĪ¼=0formulae-sequencešsuperscriptsubscriptš§1šµ0formulae-sequencešsuperscriptsubscriptš§2šµ0formulae-sequencešsuperscriptsubscriptš§3šµ0šš0\frac\partial Y\partial z_1^B=0,\frac\partial Y\partial z_2^B=% 0,\frac\partial Y\partial z_3^B=0,\frac\partial Y\partial\mu=0divide start_ARG ā italic_Y end_ARG start_ARG ā italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = 0 , divide start_ARG ā italic_Y end_ARG start_ARG ā italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = 0 , divide start_ARG ā italic_Y end_ARG start_ARG ā italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = 0 , divide start_ARG ā italic_Y end_ARG start_ARG ā italic_Ī¼ end_ARG = 0, we have ziB=bibsuperscriptsubscriptš§ššµsubscriptšššz_i^B=\fracb_ibitalic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = divide start_ARG italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_b end_ARG. So maxĪø=ršš\max\\theta\=rroman_max italic_Īø = italic_r and maxF(Īø)=F(r)š¹šš¹š\maxF(\theta)=F(r)roman_max italic_F ( italic_Īø ) = italic_F ( italic_r ), then case 1 is shown. ā
Example 1. For a state in Eq.(3.1), when a1=0,a2=0,a3=0.03,b1=0,b2=0,b3=0.25,c1=0.12,c2=0.12,c3=0.01,r1=0.1,r2=0.1,r3=-0.3,s1=0.13,s2=0.13,s3=-0.26,v1=0,v2=0,v3=0,T1=-0.02,T2=-0.02,T3=-0.36formulae-sequencesubscriptš10formulae-sequencesubscriptš20formulae-sequencesubscriptš30.03formulae-sequencesubscriptš10formulae-sequencesubscriptš20formulae-sequencesubscriptš30.25formulae-sequencesubscriptš10.12formulae-sequencesubscriptš20.12formulae-sequencesubscriptš30.01formulae-sequencesubscriptš10.1formulae-sequencesubscriptš20.1formulae-sequencesubscriptš30.3formulae-sequencesubscriptš 10.13formulae-sequencesubscriptš 20.13formulae-sequencesubscriptš 30.26formulae-sequencesubscriptš£10formulae-sequencesubscriptš£20formulae-sequencesubscriptš£30formulae-sequencesubscriptš10.02formulae-sequencesubscriptš20.02subscriptš30.36a_1=0,a_2=0,a_3=0.03,b_1=0,b_2=0,b_3=0.25,c_1=0.12,c_2=0.12,c_% 3=0.01,r_1=0.1,r_2=0.1,r_3=-0.3,s_1=0.13,s_2=0.13,s_3=-0.26,v_1% =0,v_2=0,v_3=0,T_1=-0.02,T_2=-0.02,T_3=-0.36italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.03 , italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 , italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 , italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.25 , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.12 , italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.12 , italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.01 , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.1 , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.1 , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = - 0.3 , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.13 , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.13 , italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = - 0.26 , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 , italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - 0.02 , italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - 0.02 , italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = - 0.36. According to the case 1 of Theorem 3.2, we have š¬=0.8889š¬0.8889\mathcalQ=0.8889caligraphic_Q = 0.8889. Fig. 2 shows the behavior of the quantum discord š¬š¬\mathcalQcaligraphic_Q.
Example 2. For a state of the case 1 in Theorem 3.4, when a1=a2=a3=0,b1=0.2,b2=0.05,b3=0.1,c1=0.04,c2=0.06,c3=0.11,r1=r2=r3=0.17,s1=0.08,s2=0.15,s3=0.25,v1=v2=v3=T1=T2=T3=0formulae-sequencesubscriptš1subscriptš2subscriptš30formulae-sequencesubscriptš10.2formulae-sequencesubscriptš20.05formulae-sequencesubscriptš30.1formulae-sequencesubscriptš10.04formulae-sequencesubscriptš20.06formulae-sequencesubscriptš30.11subscriptš1subscriptš2subscriptš30.17formulae-sequencesubscriptš 10.08formulae-sequencesubscriptš 20.15formulae-sequencesubscriptš 30.25subscriptš£1subscriptš£2subscriptš£3subscriptš1subscriptš2subscriptš30a_1=a_2=a_3=0,b_1=0.2,b_2=0.05,b_3=0.1,c_1=0.04,c_2=0.06,c_3% =0.11,r_1=r_2=r_3=0.17,s_1=0.08,s_2=0.15,s_3=0.25,v_1=v_2=v_3% =T_1=T_2=T_3=0italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.2 , italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.05 , italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.1 , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.04 , italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.06 , italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.11 , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.17 , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.08 , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.15 , italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.25 , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0. Then the quantum discord is š¬=0.9970š¬0.9970\mathcalQ=0.9970caligraphic_Q = 0.9970. Fig. 3 and Fig. 4 show the behavior of the function GšŗGitalic_G and Fš¹Fitalic_F respectively.
Quantum discord is one of the important correlations in studying quantum systems. It is well-known that the quantum discord is hard to compute explicitly, and only sporadic formulas are known, for instance, the Bell state and the X-state etc. Recently important progresses are made to generalize the notion to multipartite quantum systems [10], and their explicit formulas are expectedly not easy to find. In this work, we have found explicit formulas of the quantum discord for tripartite non X-states with 14 parameters, including some famous states such as the Werner-GHZ state.
The research is supported in part by the NSFC grants 11871325 and 12126351, and Natural Science Foundation of Hubei Province grant no. 2020CFB538 as well as Simons Foundation grant no. 523868.
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