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 ϕbcsuperscriptitalic-ϕ𝑏𝑐\phi^bcitalic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT on system HB⊗HCtensor-productsubscript𝐻𝐵subscript𝐻𝐶H_B\otimes H_Citalic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⊗ italic_H start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, the quantum mutual information is I(ϕbc):=SB(ϕb)+SC(ϕc)-SBC(ϕbc)assign𝐼superscriptitalic-ϕ𝑏𝑐subscript𝑆𝐵superscriptitalic-ϕ𝑏subscript𝑆𝐶superscriptitalic-ϕ𝑐subscript𝑆𝐵𝐶superscriptitalic-ϕ𝑏𝑐I(\phi^bc):=S_B(\phi^b)+S_C(\phi^c)-S_BC(\phi^bc)italic_I ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) := italic_S start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) + italic_S start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ), where S(ϕX)=TrϕXlog2(ϕX)𝑆superscriptitalic-ϕ𝑋Trsuperscriptitalic-ϕ𝑋subscript2superscriptitalic-ϕ𝑋S(\phi^X)=\mathrmTr\phi^X\log_2(\phi^X)italic_S ( italic_ϕ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) = roman_Tr italic_ϕ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) is the von Neumann entropy of the quantum state on system X. Set ΠkBsubscriptsuperscriptΠ𝐵𝑘\\Pi^B_k\ roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT to be an one-dimensional von Neumann projection operator on subsystem B𝐵Bitalic_B which satisfies ∑kΠkB=I,(ΠkB)2=ΠkB,ΠkBΠk′B=δkk′formulae-sequencesubscript𝑘superscriptsubscriptΠ𝑘𝐵𝐼formulae-sequencesuperscriptsuperscriptsubscriptΠ𝑘𝐵2superscriptsubscriptΠ𝑘𝐵superscriptsubscriptΠ𝑘𝐵superscriptsubscriptΠsuperscript𝑘′𝐵subscript𝛿𝑘superscript𝑘′\sum_k\Pi_k^B=I,(\Pi_k^B)^2=\Pi_k^B,\Pi_k^B\Pi_k^\prime% ^B=\delta_kk^^\prime∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = italic_I , ( roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT roman_Π start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = italic_δ start_POSTSUBSCRIPT italic_k italic_k start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Then the state ϕbcsuperscriptitalic-ϕ𝑏𝑐\phi^bcitalic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT under the measurement ΠkBsubscriptsuperscriptΠ𝐵𝑘\\Pi^B_k\ roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is changed into
ϕkc=1pkTrB(I⊗ΠkB)ϕbc(I⊗ΠkB)subscriptsuperscriptitalic-ϕ𝑐𝑘1subscript𝑝𝑘subscriptTr𝐵tensor-product𝐼subscriptsuperscriptΠ𝐵𝑘superscriptitalic-ϕ𝑏𝑐tensor-product𝐼subscriptsuperscriptΠ𝐵𝑘\phi^c_k=\frac1p_k\mathrmTr_B(I\otimes\Pi^B_k)\phi^bc(I% \otimes\Pi^B_k)italic_ϕ 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 ⊗ roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ( italic_I ⊗ roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )
with the probability pk=Tr(I⊗ΠkB)ϕbc(I⊗ΠkB)subscript𝑝𝑘Trtensor-product𝐼subscriptsuperscriptΠ𝐵𝑘superscriptitalic-ϕ𝑏𝑐tensor-product𝐼subscriptsuperscriptΠ𝐵𝑘p_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 ⊗ roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ( italic_I ⊗ roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ). For simplicity, we denote by ΠXsuperscriptΠ𝑋\Pi^Xroman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT the measurement ΠkXsuperscriptsubscriptΠ𝑘𝑋\\Pi_k^X\ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT on system X𝑋Xitalic_X. The quantum conditional entropy is simply given by SC|ΠB(ϕbc)=∑kpkS(ϕkc)subscript𝑆conditional𝐶superscriptΠ𝐵superscriptitalic-ϕ𝑏𝑐subscript𝑘subscript𝑝𝑘𝑆subscriptsuperscriptitalic-ϕ𝑐𝑘S_C(\phi^bc)=\sum_kp_kS(\phi^c_k)italic_S start_POSTSUBSCRIPT italic_C | roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_S ( italic_ϕ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ). Then the measurement-induced quantum mutual information is given by
C(ϕbc)=SC(ϕc)-minSC|ΠB(ϕbc).𝐶superscriptitalic-ϕ𝑏𝑐subscript𝑆𝐶superscriptitalic-ϕ𝑐subscript𝑆conditional𝐶superscriptΠ𝐵superscriptitalic-ϕ𝑏𝑐C(\phi^bc)=S_C(\phi^c)-\min S_C(\phi^bc).italic_C ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) - roman_min italic_S start_POSTSUBSCRIPT italic_C | roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) .
By Olliver and Zurek [3], the original definition of the quantum discord Q(ρ)𝑄𝜌Q(\rho)italic_Q ( italic_ρ ) is the difference of the quantum mutual information I(ϕbc)𝐼superscriptitalic-ϕ𝑏𝑐I(\phi^bc)italic_I ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) and the measurement-induced quantum mutual information C(ϕbc)𝐶superscriptitalic-ϕ𝑏𝑐C(\phi^bc)italic_C ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ), i.e.
(2.1) Q(ϕbc)=I(ϕbc)-C(ϕbc)=minΠBSC,𝑄superscriptitalic-ϕ𝑏𝑐𝐼superscriptitalic-ϕ𝑏𝑐𝐶superscriptitalic-ϕ𝑏𝑐subscriptsuperscriptΠ𝐵subscript𝑆conditional𝐶superscriptΠ𝐵superscriptitalic-ϕ𝑏𝑐subscript𝑆conditional𝐶𝐵superscriptitalic-ϕ𝑏𝑐Q(\phi^bc)=I(\phi^bc)-C(\phi^bc)=\min_\Pi^B\S_C(\phi^bc% )-S_B(\phi^bc)\,italic_Q ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = italic_I ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_C ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_C ,
where SC|B(ϕbc)=SBC(ϕbc)-SB(ϕb)subscript𝑆conditional𝐶𝐵superscriptitalic-ϕ𝑏𝑐subscript𝑆𝐵𝐶superscriptitalic-ϕ𝑏𝑐subscript𝑆𝐵superscriptitalic-ϕ𝑏S_B(\phi^bc)=S_BC(\phi^bc)-S_B(\phi^b)italic_S start_POSTSUBSCRIPT italic_C | italic_B end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) is the unmeasured conditional state on subsystem C𝐶Citalic_C.
For the tripartite system HA⊗HB⊗HCtensor-productsubscript𝐻𝐴subscript𝐻𝐵subscript𝐻𝐶H_A\otimes H_B\otimes H_Citalic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⊗ italic_H start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, we consider the BC𝐵𝐶BCitalic_B italic_C composite system as the first subsystem and A𝐴Aitalic_A-system as the second subsystem. The state ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT of system HA⊗HB⊗HCtensor-productsubscript𝐻𝐴subscript𝐻𝐵subscript𝐻𝐶H_A\otimes H_B\otimes H_Citalic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⊗ italic_H start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT gives arise to a state on BC𝐵𝐶BCitalic_B italic_C-subsystem after the von Neumann measurement ΠjAsuperscriptsubscriptΠ𝑗𝐴\\Pi_j^A\ roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT on A𝐴Aitalic_A subsystem. Namely, it takes the following form:
(2.2) ρjbc=1pjbcTrA(ΠjA⊗I)ρabc(ΠjA⊗I)subscriptsuperscript𝜌𝑏𝑐𝑗1subscriptsuperscript𝑝𝑏𝑐𝑗subscriptTr𝐴tensor-productsuperscriptsubscriptΠ𝑗𝐴𝐼superscript𝜌𝑎𝑏𝑐tensor-productsuperscriptsubscriptΠ𝑗𝐴𝐼\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_ρ 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_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I ) end_CELL end_ROW
with probability pjbc=Tr(ΠjA⊗I)ρabc(ΠjA⊗I)subscriptsuperscript𝑝𝑏𝑐𝑗Trtensor-productsuperscriptsubscriptΠ𝑗𝐴𝐼superscript𝜌𝑎𝑏𝑐tensor-productsuperscriptsubscriptΠ𝑗𝐴𝐼p^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_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I ).The measured quantum mutual information of ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT is naturally given by
(2.3) 𝒥(ρabc|ΠA)=SBC(ρbc)-SBC|ΠA(ρabc).𝒥conditionalsuperscript𝜌𝑎𝑏𝑐superscriptΠ𝐴subscript𝑆𝐵𝐶superscript𝜌𝑏𝑐subscript𝑆conditional𝐵𝐶superscriptΠ𝐴superscript𝜌𝑎𝑏𝑐\beginsplit\mathcalJ(\rho^abc|\Pi^A)=S_BC(\rho^bc)-S_\Pi^A(% \rho^abc).\endsplitstart_ROW start_CELL caligraphic_J ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) . end_CELL end_ROW
The quantity of classical correlation of the tripartite state ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT is
(2.4) 𝒞(ρabc)=maxΠA𝒥(ρabc|ΠA)=SBC(ρbc)-minΠASBC|ΠA(ρabc).𝒞superscript𝜌𝑎𝑏𝑐subscriptsuperscriptΠ𝐴𝒥conditionalsuperscript𝜌𝑎𝑏𝑐superscriptΠ𝐴subscript𝑆𝐵𝐶superscript𝜌𝑏𝑐subscriptsuperscriptΠ𝐴subscript𝑆conditional𝐵𝐶superscriptΠ𝐴superscript𝜌𝑎𝑏𝑐\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_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = roman_max start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT caligraphic_J ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) . end_CELL end_ROW
We know that the quantum mutual information I(ρabc)=SA(ρa)+SBC(ρbc)-SABC(ρabc)𝐼superscript𝜌𝑎𝑏𝑐subscript𝑆𝐴superscript𝜌𝑎subscript𝑆𝐵𝐶superscript𝜌𝑏𝑐subscript𝑆𝐴𝐵𝐶superscript𝜌𝑎𝑏𝑐I(\rho^abc)=S_A(\rho^a)+S_BC(\rho^bc)-S_ABC(\rho^abc)italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) + italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ). Similar to Eq.(2.1), the generalized quantum discord of the tripartite state ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT can be defined as
(2.5) 𝒬(ρabc)=I(ρabc)-C(ρabc)=minΠASBC,𝒬superscript𝜌𝑎𝑏𝑐𝐼superscript𝜌𝑎𝑏𝑐𝐶superscript𝜌𝑎𝑏𝑐subscriptsuperscriptΠ𝐴subscript𝑆conditional𝐵𝐶superscriptΠ𝐴superscript𝜌𝑎𝑏𝑐subscript𝑆conditional𝐵𝐶𝐴superscript𝜌𝑎𝑏𝑐\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_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) - italic_C ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) , end_CELL end_ROW
where SBC|A(ρabc)=SABC(ρabc)-SA(ρa)subscript𝑆conditional𝐵𝐶𝐴superscript𝜌𝑎𝑏𝑐subscript𝑆𝐴𝐵𝐶superscript𝜌𝑎𝑏𝑐subscript𝑆𝐴superscript𝜌𝑎S_BC(\rho^abc)=S_ABC(\rho^abc)-S_A(\rho^a)italic_S start_POSTSUBSCRIPT italic_B italic_C | italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) is the unmeasured conditional entropy on BC𝐵𝐶BCitalic_B italic_C-bipartite subsystem.
In order to evaluate the quantity minΠASBC|ΠA(ρabc)subscriptsuperscriptΠ𝐴subscript𝑆conditional𝐵𝐶superscriptΠ𝐴superscript𝜌𝑎𝑏𝑐\min_\Pi^AS_BC(\rho^abc)roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ 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) ΠjkAB=ΠjA⊗Πk|jBsuperscriptsubscriptΠ𝑗𝑘𝐴𝐵tensor-productsuperscriptsubscriptΠ𝑗𝐴superscriptsubscriptΠconditional𝑘𝑗𝐵\beginsplit\Pi_jk^AB=\Pi_j^A\otimes\Pi_j^B\endsplitstart_ROW start_CELL roman_Π start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_k | italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_CELL end_ROW
with the measurement ordering from A𝐴Aitalic_A to B𝐵Bitalic_B. The projector Πk|jBsuperscriptsubscriptΠconditional𝑘𝑗𝐵\Pi_k^Broman_Π start_POSTSUBSCRIPT italic_k | italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT on subsystem B𝐵Bitalic_B is conditional measurement outcome of A𝐴Aitalic_A. These projectors satisfy ∑kΠk|jB=IB,∑jΠjA=IAformulae-sequencesubscript𝑘subscriptsuperscriptΠ𝐵conditional𝑘𝑗superscript𝐼𝐵subscript𝑗superscriptsubscriptΠ𝑗𝐴superscript𝐼𝐴\sum_k\Pi^B_k=I^B,\sum_j\Pi_j^A=I^A∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k | italic_j end_POSTSUBSCRIPT = italic_I start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = italic_I start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT. Then after the measurement ΠjkABsuperscriptsubscriptΠ𝑗𝑘𝐴𝐵\Pi_jk^ABroman_Π start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, the state ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT is collapsed to a state on subsystem C𝐶Citalic_C, i.e.
(2.7) ρjkc=1pjkcTrAB(ΠjkAB⊗I)ρabc(ΠjkAB⊗I)subscriptsuperscript𝜌𝑐𝑗𝑘1subscriptsuperscript𝑝𝑐𝑗𝑘subscriptTr𝐴𝐵tensor-productsuperscriptsubscriptΠ𝑗𝑘𝐴𝐵𝐼superscript𝜌𝑎𝑏𝑐tensor-productsuperscriptsubscriptΠ𝑗𝑘𝐴𝐵𝐼\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_ρ 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_Π start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ⊗ italic_I ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ⊗ italic_I ) end_CELL end_ROW
with the probability pjkc=Tr(ΠjkAB⊗I)ρabc(ΠjkAB⊗I)subscriptsuperscript𝑝𝑐𝑗𝑘Trtensor-productsuperscriptsubscriptΠ𝑗𝑘𝐴𝐵𝐼superscript𝜌𝑎𝑏𝑐tensor-productsuperscriptsubscriptΠ𝑗𝑘𝐴𝐵𝐼p^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_Π start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ⊗ italic_I ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ⊗ italic_I ). The conditional entropy after the AB𝐴𝐵ABitalic_A italic_B-bipartite measurement is
SC|ΠAB(ρabc)=∑jklpjkcλl(jk)log2λl(jk),subscript𝑆conditional𝐶superscriptΠ𝐴𝐵superscript𝜌𝑎𝑏𝑐subscript𝑗𝑘𝑙superscriptsubscript𝑝𝑗𝑘𝑐superscriptsubscript𝜆𝑙𝑗𝑘subscript2superscriptsubscript𝜆𝑙𝑗𝑘S_\Pi^AB(\rho^abc)=\sum_jklp_jk^c\lambda_l^(jk)\log_2% \lambda_l^(jk),italic_S start_POSTSUBSCRIPT italic_C | roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = ∑ 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_λ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j italic_k ) end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j italic_k ) end_POSTSUPERSCRIPT ,
where λl(jk)superscriptsubscript𝜆𝑙𝑗𝑘\lambda_l^(jk)italic_λ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j italic_k ) end_POSTSUPERSCRIPT are eigenvalues of state ρjkcsubscriptsuperscript𝜌𝑐𝑗𝑘\rho^c_jkitalic_ρ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT.
Let ρΠX=∑ΠXΠXρΠXsubscript𝜌superscriptΠ𝑋subscriptsuperscriptΠ𝑋superscriptΠ𝑋𝜌superscriptΠ𝑋\rho_\Pi^X=\sum_\Pi^X\Pi^X\rho\Pi^Xitalic_ρ start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT italic_ρ roman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT be the state after measurement ΠXsuperscriptΠ𝑋\Pi^Xroman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT. Then for a bipartite state ρabsuperscript𝜌𝑎𝑏\rho^abitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT, the conditional entropy on subsystem B𝐵Bitalic_B after the measurement on subsystem A𝐴Aitalic_A is
(2.8) SB|ΠA(ρab)=∑jpjSB(ρjb).subscript𝑆conditional𝐵superscriptΠ𝐴superscript𝜌𝑎𝑏subscript𝑗subscript𝑝𝑗subscript𝑆𝐵subscriptsuperscript𝜌𝑏𝑗\beginsplitS_B(\rho^ab)=\sum_jp_jS_B(\rho^b_j).\endsplitstart_ROW start_CELL italic_S start_POSTSUBSCRIPT italic_B | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ 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(ρΠAab)=SA(ρΠAab)+SB|ΠA(ρab).subscript𝑆𝐴𝐵subscriptsuperscript𝜌𝑎𝑏superscriptΠ𝐴subscript𝑆𝐴subscriptsuperscript𝜌𝑎𝑏superscriptΠ𝐴subscript𝑆conditional𝐵superscriptΠ𝐴superscript𝜌𝑎𝑏S_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_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) + italic_S start_POSTSUBSCRIPT italic_B | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ) .
For the tripartite system, using the measurement ΠABsuperscriptΠ𝐴𝐵\Pi^ABroman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, we have
(2.10) SABC(ρΠABabc)-SAB(ρΠABabc)=SC|ΠAB(ρabc),subscript𝑆𝐴𝐵𝐶subscriptsuperscript𝜌𝑎𝑏𝑐superscriptΠ𝐴𝐵subscript𝑆𝐴𝐵subscriptsuperscript𝜌𝑎𝑏𝑐superscriptΠ𝐴𝐵subscript𝑆conditional𝐶superscriptΠ𝐴𝐵superscript𝜌𝑎𝑏𝑐S_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_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_C | roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) ,
when the measurement on A𝐴Aitalic_A system is ΠAsuperscriptΠ𝐴\Pi^Aroman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT, then we have
(2.11) SABC(ρΠAabc)-SA(ρΠAabc)=SBC|ΠA(ρabc).subscript𝑆𝐴𝐵𝐶subscriptsuperscript𝜌𝑎𝑏𝑐superscriptΠ𝐴subscript𝑆𝐴subscriptsuperscript𝜌𝑎𝑏𝑐superscriptΠ𝐴subscript𝑆conditional𝐵𝐶superscriptΠ𝐴superscript𝜌𝑎𝑏𝑐S_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_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) .
By Eq.(2.9), Eq.(2.10), Eq.(2.11), we have that
SBC|ΠA(ρabc)=SB|ΠA(ρab)+SC|ΠAB(ρabc).subscript𝑆conditional𝐵𝐶superscriptΠ𝐴superscript𝜌𝑎𝑏𝑐subscript𝑆conditional𝐵superscriptΠ𝐴superscript𝜌𝑎𝑏subscript𝑆conditional𝐶superscriptΠ𝐴𝐵superscript𝜌𝑎𝑏𝑐S_\Pi^A(\rho^abc)=S_B(\rho^ab)+S_C(\rho^abc).italic_S start_POSTSUBSCRIPT italic_B italic_C | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT ) + italic_S start_POSTSUBSCRIPT italic_C | roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ) .
Meanwhile, SA(ρΠABabc)=SA(ρΠAabc)subscript𝑆𝐴subscriptsuperscript𝜌𝑎𝑏𝑐superscriptΠ𝐴𝐵subscript𝑆𝐴subscriptsuperscript𝜌𝑎𝑏𝑐superscriptΠ𝐴S_A(\rho^abc_\Pi^AB)=S_A(\rho^abc_\Pi^A)italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ), so the generalization discord of a tripartite state can be written as [10]
(2.12) 𝒬(ρ):=minΠAB[-SBC|A(ρ)+SB|ΠA(ρ)+SC|ΠAB(ρ)].assign𝒬𝜌subscriptsuperscriptΠ𝐴𝐵subscript𝑆conditional𝐵𝐶𝐴𝜌subscript𝑆conditional𝐵superscriptΠ𝐴𝜌subscript𝑆conditional𝐶superscriptΠ𝐴𝐵𝜌\beginsplit\mathcalQ(\rho):=\min_\Pi^AB[-S_BC(\rho)+S_B(% \rho)+S_\Pi^AB(\rho)].\endsplitstart_ROW start_CELL caligraphic_Q ( italic_ρ ) := roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ - italic_S start_POSTSUBSCRIPT italic_B italic_C | italic_A end_POSTSUBSCRIPT ( italic_ρ ) + italic_S start_POSTSUBSCRIPT italic_B | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) + italic_S start_POSTSUBSCRIPT italic_C | roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) ] . 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 𝒬ABC(ρx⊗ρy)=𝒬X(ρx)subscript𝒬𝐴𝐵𝐶tensor-productsuperscript𝜌𝑥superscript𝜌𝑦subscript𝒬𝑋superscript𝜌𝑥\mathcalQ_ABC(\rho^x\otimes\rho^y)=\mathcalQ_X(\rho^x)caligraphic_Q start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ) = caligraphic_Q start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ) for X=AB,BC𝑋𝐴𝐵𝐵𝐶X=AB,BCitalic_X = italic_A italic_B , italic_B italic_C and AC𝐴𝐶ACitalic_A italic_C subsystem. We consider the following tripartite states
(3.1) ρabc=18(I8+a3σ3⊗I4+I2⊗b3σ3⊗I2+I4⊗∑i3ciσi+∑i3riσi⊗σi⊗I2+∑i3siσi⊗I2⊗σi+∑i3Tiσi⊗σi⊗σi),superscript𝜌𝑎𝑏𝑐18subscript𝐼8tensor-productsubscript𝑎3subscript𝜎3subscript𝐼4tensor-producttensor-productsubscript𝐼2subscript𝑏3subscript𝜎3subscript𝐼2tensor-productsubscript𝐼4superscriptsubscript𝑖3subscript𝑐𝑖subscript𝜎𝑖superscriptsubscript𝑖3tensor-productsubscript𝑟𝑖subscript𝜎𝑖subscript𝜎𝑖subscript𝐼2superscriptsubscript𝑖3tensor-productsubscript𝑠𝑖subscript𝜎𝑖subscript𝐼2subscript𝜎𝑖superscriptsubscript𝑖3tensor-productsubscript𝑇𝑖subscript𝜎𝑖subscript𝜎𝑖subscript𝜎𝑖\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_ρ 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_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ⊗ ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , end_CELL end_ROW
where Idsubscript𝐼𝑑I_ditalic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT represents the unit matrix of order d𝑑ditalic_d, and σi(i=1,2,3)subscript𝜎𝑖𝑖123\sigma_i(i=1,2,3)italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i = 1 , 2 , 3 ) are Pauli matrices. The parameters a3,b3,ci,ri,si,Ti∈ℝsubscript𝑎3subscript𝑏3subscript𝑐𝑖subscript𝑟𝑖subscript𝑠𝑖subscript𝑇𝑖ℝa_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 ∈ blackboard_R and they are confined within the internal [-1,1]11[-1,1][ - 1 , 1 ]. Its matrix has the following form:
(3.2) ρ=(**000****00*0**00****0*00*****00*****000****00*0*00****000**).𝜌000missing-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_ρ = ( 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-sequenceket𝑗bra𝑗𝑗01\ italic_j ⟩ ⟨ italic_j be the computational base, then any von Neumann measurement on system X𝑋Xitalic_X can be written as ,j=0,1\V^\dagger , italic_j = 0 , 1 for some unitary matrix V∈SU(2)𝑉SU2V\in\mathrmSU(2)italic_V ∈ roman_SU ( 2 ). Any unitary matrix can be written as V=tI+-1∑kykσk𝑉𝑡𝐼1subscript𝑘subscript𝑦𝑘subscript𝜎𝑘V=tI+\sqrt-1\sum_ky_k\sigma_kitalic_V = italic_t italic_I + square-root start_ARG - 1 end_ARG ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT with t,yk∈ℝ𝑡subscript𝑦𝑘ℝt,y_k\in\mathbbRitalic_t , italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ blackboard_R. When the measurement ΠjXsubscriptsuperscriptΠ𝑋𝑗\\Pi^X_j\ roman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is performed locally on one part of the composite system Y⊗Xtensor-product𝑌𝑋Y\otimes Xitalic_Y ⊗ italic_X, the ensemble ρjY,pjYsuperscriptsubscript𝜌𝑗𝑌superscriptsubscript𝑝𝑗𝑌\\rho_j^Y,p_j^Y\ italic_ρ 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 ρjY=1pjYTrX(I⊗ΠjX)ρYX(I⊗ΠjX)superscriptsubscript𝜌𝑗𝑌1superscriptsubscript𝑝𝑗𝑌subscriptTr𝑋tensor-product𝐼subscriptsuperscriptΠ𝑋𝑗superscript𝜌𝑌𝑋tensor-product𝐼subscriptsuperscriptΠ𝑋𝑗\rho_j^Y=\frac1p_j^Y\mathrmTr_X(I\otimes\Pi^X_j)\rho^YX(% I\otimes\Pi^X_j)italic_ρ 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 ⊗ roman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_Y italic_X end_POSTSUPERSCRIPT ( italic_I ⊗ roman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) with the probability pjY=Tr[ρYX(I⊗ΠjX)]superscriptsubscript𝑝𝑗𝑌Trdelimited-[]superscript𝜌𝑌𝑋tensor-product𝐼superscriptsubscriptΠ𝑗𝑋p_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_ρ start_POSTSUPERSCRIPT italic_Y italic_X end_POSTSUPERSCRIPT ( italic_I ⊗ roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) ].
It follows from symmetry that
(3.3) V†σ1V=(t2+y12-y22-y32)σ1+2(ty3+y1y2)σ2+2(-ty2+y1y3)σ3,V†σ2V=(t2+y22-y32-y12)σ1+2(ty1+y2y3)σ3+2(-ty3+y1y2)σ1,V†σ3V=(t2+y32-y12-y22)σ1+2(ty2+y1y3)σ3+2(-ty1+y2y3)σ2.formulae-sequencesuperscript𝑉†subscript𝜎1𝑉superscript𝑡2superscriptsubscript𝑦12superscriptsubscript𝑦22superscriptsubscript𝑦32subscript𝜎12𝑡subscript𝑦3subscript𝑦1subscript𝑦2subscript𝜎22𝑡subscript𝑦2subscript𝑦1subscript𝑦3subscript𝜎3formulae-sequencesuperscript𝑉†subscript𝜎2𝑉superscript𝑡2superscriptsubscript𝑦22superscriptsubscript𝑦32superscriptsubscript𝑦12subscript𝜎12𝑡subscript𝑦1subscript𝑦2subscript𝑦3subscript𝜎32𝑡subscript𝑦3subscript𝑦1subscript𝑦2subscript𝜎1superscript𝑉†subscript𝜎3𝑉superscript𝑡2superscriptsubscript𝑦32superscriptsubscript𝑦12superscriptsubscript𝑦22subscript𝜎12𝑡subscript𝑦2subscript𝑦1subscript𝑦3subscript𝜎32𝑡subscript𝑦1subscript𝑦2subscript𝑦3subscript𝜎2\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 † end_POSTSUPERSCRIPT italic_σ 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_σ 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_σ 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_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_σ 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_σ 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_σ 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_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_σ 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_σ 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_σ 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_σ 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-sequencesuperscriptsubscript𝑧1𝑋2𝑡subscript𝑦2subscript𝑦1subscript𝑦3formulae-sequencesuperscriptsubscript𝑧2𝑋2𝑡subscript𝑦1subscript𝑦2subscript𝑦3superscriptsubscript𝑧3𝑋superscript𝑡2superscriptsubscript𝑦32superscriptsubscript𝑦12superscriptsubscript𝑦22z_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=1superscriptsuperscriptsubscript𝑧1𝑋2superscriptsuperscriptsubscript𝑧2𝑋2superscriptsuperscriptsubscript𝑧3𝑋21(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 ΠjXσkΠjX=(-1)jzkXΠjXsuperscriptsubscriptΠ𝑗𝑋 Discord servers 𝜎𝑘superscriptsubscriptΠ𝑗𝑋superscript1𝑗superscriptsubscript𝑧𝑘𝑋superscriptsubscriptΠ𝑗𝑋\Pi_j^X\sigma_k\Pi_j^X=(-1)^jz_k^X\Pi_j^Xroman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Π 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_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT for j=0,1𝑗01j=0,1italic_j = 0 , 1 and k=1,2,3𝑘123k=1,2,3italic_k = 1 , 2 , 3.
For the tripartite state ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT, the conditional state on BC𝐵𝐶BCitalic_B italic_C subsystem after measurement ΠjA(j=0,1)superscriptsubscriptΠ𝑗𝐴𝑗01\\Pi_j^A(j=0,1)\ roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_j = 0 , 1 ) on subsystem A𝐴Aitalic_A is
(3.4) ρjbc=1pjbc(Π1A⊗I2⊗I2)ρabc(Π1A⊗I2⊗I2)=1pjbc[(1+(-1)ja3z3A)I2⊗I2+b3σ3⊗I2+(-1)j∑i3riziAσi⊗I2+∑i3(ci+(-1)jsiziA)I2⊗σi+(-1)j∑i3TiziAσi⊗σi],superscriptsubscript𝜌𝑗𝑏𝑐1superscriptsubscript𝑝𝑗𝑏𝑐tensor-productsuperscriptsubscriptΠ1𝐴subscript𝐼2subscript𝐼2superscript𝜌𝑎𝑏𝑐tensor-productsuperscriptsubscriptΠ1𝐴subscript𝐼2subscript𝐼21superscriptsubscript𝑝𝑗𝑏𝑐delimited-[]tensor-product1superscript1𝑗subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝐼2subscript𝐼2tensor-productsubscript𝑏3subscript𝜎3subscript𝐼2superscript1𝑗superscriptsubscript𝑖3tensor-productsubscript𝑟𝑖superscriptsubscript𝑧𝑖𝐴subscript𝜎𝑖subscript𝐼2superscriptsubscript𝑖3tensor-productsubscript𝑐𝑖superscript1𝑗subscript𝑠𝑖superscriptsubscript𝑧𝑖𝐴subscript𝐼2subscript𝜎𝑖superscript1𝑗superscriptsubscript𝑖3tensor-productsubscript𝑇𝑖superscriptsubscript𝑧𝑖𝐴subscript𝜎𝑖subscript𝜎𝑖\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_ρ 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_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ 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 ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∑ 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_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ 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 ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∑ 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_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] , end_CELL end_ROW
where the probabilities are
pjbc=Tr((ΠjA⊗I2⊗I2)ρabc(ΠjA⊗I2⊗I2))=12[1+(-1)ja3z3A],superscriptsubscript𝑝𝑗𝑏𝑐Trtensor-productsuperscriptsubscriptΠ𝑗𝐴subscript𝐼2subscript𝐼2superscript𝜌𝑎𝑏𝑐tensor-productsuperscriptsubscriptΠ𝑗𝐴subscript𝐼2subscript𝐼212delimited-[]1superscript1𝑗subscript𝑎3superscriptsubscript𝑧3𝐴p_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_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ 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 ∑i3(ziA)2=1superscriptsubscript𝑖3superscriptsubscriptsuperscript𝑧𝐴𝑖21\sum_i^3(z^A_i)^2=1∑ 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 ρjbcsuperscriptsubscript𝜌𝑗𝑏𝑐\rho_j^bcitalic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT is
ρjb=TrCρjbc=12(1+(-1)ja3z3A)[(1+(-1)ja3z3A)I2+b3σ3+(-1)j∑i3riziAσi]superscriptsubscript𝜌𝑗𝑏subscriptTr𝐶superscriptsubscript𝜌𝑗𝑏𝑐121superscript1𝑗subscript𝑎3superscriptsubscript𝑧3𝐴delimited-[]1superscript1𝑗subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝐼2subscript𝑏3subscript𝜎3superscript1𝑗superscriptsubscript𝑖3subscript𝑟𝑖superscriptsubscript𝑧𝑖𝐴subscript𝜎𝑖\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_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_ρ 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_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∑ 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_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ]
with the probability pjb=pjbc=12[1+(-1)ja3z3A]superscriptsubscript𝑝𝑗𝑏superscriptsubscript𝑝𝑗𝑏𝑐12delimited-[]1superscript1𝑗subscript𝑎3superscriptsubscript𝑧3𝐴p_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 ρjbsuperscriptsubscript𝜌𝑗𝑏\rho_j^bitalic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT are
λj±=12(1+(-1)ja3z3A)[1+(-1)ja3z3A±(b3+(-1)jr3z3A)2+∑i2(riziA)2].subscriptsuperscript𝜆plus-or-minus𝑗121superscript1𝑗subscript𝑎3superscriptsubscript𝑧3𝐴delimited-[]plus-or-minus1superscript1𝑗subscript𝑎3superscriptsubscript𝑧3𝐴superscriptsubscript𝑏3superscript1𝑗subscript𝑟3superscriptsubscript𝑧3𝐴2superscriptsubscript𝑖2superscriptsubscript𝑟𝑖superscriptsubscript𝑧𝑖𝐴2\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_λ start_POSTSUPERSCRIPT ± 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 ± 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 + ∑ 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) Hε(x)=12[(1+ε+x)log2(1+ε+x)+(1+ε-x)log2(1+ε-x)].subscript𝐻𝜀𝑥12delimited-[]1𝜀𝑥subscript21𝜀𝑥1𝜀𝑥subscript21𝜀𝑥\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_ε end_POSTSUBSCRIPT ( italic_x ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ( 1 + italic_ε + italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_ε + italic_x ) + ( 1 + italic_ε - italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_ε - italic_x ) ] . end_CELL end_ROW
Then measured conditional entropy of B𝐵Bitalic_B subsystem can be obtained as [3, 4, 14, 16, 17, 18]
(3.6) SB|ΠA(ρ)=-∑jpjb(λj+log2λj++λj-log2λj-)=-12[Ha3z3A(A+)+H-a3z3A(A-)-2H(a3z3A)-2],subscript𝑆conditional𝐵superscriptΠ𝐴𝜌subscript𝑗superscriptsubscript𝑝𝑗𝑏subscriptsuperscript𝜆𝑗subscript2subscriptsuperscript𝜆𝑗subscriptsuperscript𝜆𝑗subscript2subscriptsuperscript𝜆𝑗12delimited-[]subscript𝐻subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝐴subscript𝐻subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝐴2𝐻subscript𝑎3superscriptsubscript𝑧3𝐴2\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_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) end_CELL start_CELL = - ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_λ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_λ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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 A±=(b3±r3z3A)2+∑i2(riziA)2subscript𝐴plus-or-minussuperscriptplus-or-minussubscript𝑏3subscript𝑟3superscriptsubscript𝑧3𝐴2superscriptsubscript𝑖2superscriptsubscript𝑟𝑖superscriptsubscript𝑧𝑖𝐴2A_\pm=\sqrt(b_3\pm r_3z_3^A)^2+\sum_i^2(r_iz_i^A)^2italic_A start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = square-root start_ARG ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± 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 + ∑ 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 Πk|jBsuperscriptsubscriptΠconditional𝑘𝑗𝐵\Pi_j^Broman_Π start_POSTSUBSCRIPT italic_k | italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT on BC𝐵𝐶BCitalic_B italic_C system, the state ρjbcsuperscriptsubscript𝜌𝑗𝑏𝑐\rho_j^bcitalic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT is changed to
(3.7) ρjkc=1pjkc[(1+(-1)ja3z3A+(-1)kb3z3B+(-1)j+k∑i3riziAziB)I2+∑i3(ci+(-1)jsiziA+(-1)k+jTiziAziB)σi],(j,k=0,1)fragmentssuperscriptsubscript𝜌𝑗𝑘𝑐1superscriptsubscript𝑝𝑗𝑘𝑐fragments[fragments(1superscriptfragments(1)𝑗subscript𝑎3superscriptsubscript𝑧3𝐴superscriptfragments(1)𝑘subscript𝑏3superscriptsubscript𝑧3𝐵superscriptfragments(1)𝑗𝑘superscriptsubscript𝑖3subscript𝑟𝑖superscriptsubscript𝑧𝑖𝐴superscriptsubscript𝑧𝑖𝐵)subscript𝐼2superscriptsubscript𝑖3fragments(subscript𝑐𝑖superscriptfragments(1)𝑗subscript𝑠𝑖superscriptsubscript𝑧𝑖𝐴superscriptfragments(1)𝑘𝑗subscript𝑇𝑖superscriptsubscript𝑧𝑖𝐴superscriptsubscript𝑧𝑖𝐵)subscript𝜎𝑖],fragments(𝑗,𝑘0,1)\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_ρ 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 ∑ 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 ∑ 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_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] , ( italic_j , italic_k = 0 , 1 ) end_CELL end_ROW
with the probability (k=0,1𝑘01k=0,1italic_k = 0 , 1)
(3.8) p0kc=12(1+a3z3A)(1+αk),p1kc=12(1-a3z3A)(1+βk),formulae-sequencesuperscriptsubscript𝑝0𝑘𝑐121subscript𝑎3superscriptsubscript𝑧3𝐴1subscript𝛼𝑘superscriptsubscript𝑝1𝑘𝑐121subscript𝑎3superscriptsubscript𝑧3𝐴1subscript𝛽𝑘p_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_α 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_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ,
where αk=a3z3A+(-1)k(b3z3B+∑i3riziAziB),βk=-a3z3A+(-1)k(b3z3B-∑i3riziAziB)formulae-sequencesubscript𝛼𝑘subscript𝑎3superscriptsubscript𝑧3𝐴superscript1𝑘subscript𝑏3superscriptsubscript𝑧3𝐵superscriptsubscript𝑖3subscript𝑟𝑖superscriptsubscript𝑧𝑖𝐴superscriptsubscript𝑧𝑖𝐵subscript𝛽𝑘subscript𝑎3superscriptsubscript𝑧3𝐴superscript1𝑘subscript𝑏3superscriptsubscript𝑧3𝐵superscriptsubscript𝑖3subscript𝑟𝑖superscriptsubscript𝑧𝑖𝐴superscriptsubscript𝑧𝑖𝐵\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_α 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 + ∑ 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_β 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 - ∑ 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 ρjkcsubscriptsuperscript𝜌𝑐𝑗𝑘\rho^c_jkitalic_ρ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT are given by
(3.9) λ0k±=12(1+αk)(1+αk±γk),λ1k±=12(1+βk)(1+βk±δk),k=0,1,formulae-sequencesuperscriptsubscript𝜆0𝑘plus-or-minus121subscript𝛼𝑘plus-or-minus1subscript𝛼𝑘subscript𝛾𝑘formulae-sequencesuperscriptsubscript𝜆1𝑘plus-or-minus121subscript𝛽𝑘plus-or-minus1subscript𝛽𝑘subscript𝛿𝑘𝑘01\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_λ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG ( 1 + italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ± italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , italic_λ start_POSTSUBSCRIPT 1 italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG ( 1 + italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ± italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , italic_k = 0 , 1 ,
where
γk=[∑i3(ci+siziA+(-1)kTiziAziB)2]12,δk=[∑i3(-ci+siziA+(-1)kTiziAziB)2]12.formulae-sequencesubscript𝛾𝑘superscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑐𝑖subscript𝑠𝑖superscriptsubscript𝑧𝑖𝐴superscript1𝑘subscript𝑇𝑖superscriptsubscript𝑧𝑖𝐴subscriptsuperscript𝑧𝐵𝑖212subscript𝛿𝑘superscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑐𝑖subscript𝑠𝑖superscriptsubscript𝑧𝑖𝐴superscript1𝑘subscript𝑇𝑖superscriptsubscript𝑧𝑖𝐴subscriptsuperscript𝑧𝐵𝑖212\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_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ∑ 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_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ∑ 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 ∑i3ai2+∑i3bi2+∑i3ri2≤1superscriptsubscript𝑖3superscriptsubscript𝑎𝑖2superscriptsubscript𝑖3superscriptsubscript𝑏𝑖2superscriptsubscript𝑖3superscriptsubscript𝑟𝑖21\sqrt\sum_i^3a_i^2+\sqrt\sum_i^3b_i^2+\sqrt\sum_i^3r_% i^2\leq 1square-root start_ARG ∑ 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 ∑ 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 ∑ 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 ≤ 1.
The entropy of ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT under the measurement ΠABsuperscriptΠ𝐴𝐵\Pi^ABroman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is given by
(3.10) SC|ΠAB(ρ)=-∑j,kpjkc(λjk+log2λjk++λjk-log2λjk-)=-12(1+a3z3A)[Hα0(γ0)+Hα1(γ1)-2Ha3z3A(α0-α12)]-12(1-a3z3A)[Hβ0(δ0)+Hβ1(δ1)-2H-a3z3A(β0-β12)]+2.subscript𝑆conditional𝐶superscriptΠ𝐴𝐵𝜌subscript𝑗𝑘subscriptsuperscript𝑝𝑐𝑗𝑘superscriptsubscript𝜆𝑗𝑘subscript2superscriptsubscript𝜆𝑗𝑘superscriptsubscript𝜆𝑗𝑘subscript2superscriptsubscript𝜆𝑗𝑘121subscript𝑎3superscriptsubscript𝑧3𝐴delimited-[]subscript𝐻subscript𝛼0subscript𝛾0subscript𝐻subscript𝛼1subscript𝛾12subscript𝐻subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝛼0subscript𝛼12121subscript𝑎3superscriptsubscript𝑧3𝐴delimited-[]subscript𝐻subscript𝛽0subscript𝛿0subscript𝐻subscript𝛽1subscript𝛿12subscript𝐻subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝛽0subscript𝛽122\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_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) = end_CELL start_CELL - ∑ 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_λ start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT + italic_λ start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ 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_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_α 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_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ 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_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) ] + 2 . end_CELL end_ROW
In particularly, a3z3A=α0+α12subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝛼0subscript𝛼12a_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_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG.
Let G(z1A,z2A,z3A)=1-SB|ΠA(ρ)𝐺superscriptsubscript𝑧1𝐴superscriptsubscript𝑧2𝐴superscriptsubscript𝑧3𝐴1subscript𝑆conditional𝐵superscriptΠ𝐴𝜌G(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_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) and F(z1A,z2A,z3A,z1B,z2B,z3B)=2-SC|ΠAB(ρ)𝐹superscriptsubscript𝑧1𝐴superscriptsubscript𝑧2𝐴superscriptsubscript𝑧3𝐴subscriptsuperscript𝑧𝐵1subscriptsuperscript𝑧𝐵2subscriptsuperscript𝑧𝐵32subscript𝑆conditional𝐶superscriptΠ𝐴𝐵𝜌F(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_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ), then we have the following result.
For the non-X-states ρ𝜌\rhoitalic_ρ in Eq(3.1) with 14 parameters, the quantum discord is given by
(3.11) 𝒬(ρ)=-SABC(ρ)+SA(ρ)+minΠA(ρ)+SC=3+∑i=18λilog2λi-∑k=12λkalog2λka-maxziX∈[0,1],∑i(ziX)2=1G+F,𝒬𝜌subscript𝑆𝐴𝐵𝐶𝜌subscript𝑆𝐴𝜌subscript𝑆conditional𝐵superscriptΠ𝐴𝜌subscript𝑆conditional𝐶superscriptΠ𝐴𝐵𝜌3superscriptsubscript𝑖18subscript𝜆𝑖subscript2subscript𝜆𝑖superscriptsubscript𝑘12superscriptsubscript𝜆𝑘𝑎subscript2superscriptsubscript𝜆𝑘𝑎subscriptformulae-sequencesubscriptsuperscript𝑧𝑋𝑖01subscript𝑖superscriptsuperscriptsubscript𝑧𝑖𝑋21𝐺𝐹\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_ρ ) = end_CELL start_CELL - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ ) + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ ) + roman_min italic_S start_POSTSUBSCRIPT italic_B end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL 3 + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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 ∈ [ 0 , 1 ] , ∑ 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 λi(i=1,⋯,8)subscript𝜆𝑖𝑖1normal-⋯8\lambda_i(i=1,\cdots,8)italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i = 1 , ⋯ , 8 ) are the eigenvalues of ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT, λka=12[1+(-1)ka3],(k=0,1)superscriptsubscript𝜆𝑘𝑎12delimited-[]1superscript1𝑘subscript𝑎3𝑘01\lambda_k^a=\frac12[1+(-1)^ka_3],(k=0,1)italic_λ 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 ρabcsuperscript𝜌𝑎𝑏𝑐\rho^abcitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b italic_c end_POSTSUPERSCRIPT on subsystem A𝐴Aitalic_A and X𝑋Xitalic_X represents subsystem A,B𝐴𝐵A,Bitalic_A , italic_B.
Theorem 3.2.
Let r=maxr2𝑟subscript𝑟1subscript𝑟2r=\max\r_1italic_r = roman_max italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , then maxziX∈[0,1],∑i(ziX)2=1G+Fsubscriptformulae-sequencesubscriptsuperscript𝑧𝑋𝑖01subscript𝑖superscriptsuperscriptsubscript𝑧𝑖𝑋21𝐺𝐹\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 ∈ [ 0 , 1 ] , ∑ 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 a3b3r3≤0,r32-r2≥a3b3r3formulae-sequencesubscript𝑎3subscript𝑏3subscript𝑟30superscriptsubscript𝑟32superscript𝑟2subscript𝑎3subscript𝑏3subscript𝑟3a_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 ≤ 0 , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 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)≤0subscript𝑏3subscript𝑟3subscript𝑐3subscript𝑠30(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 ) ≤ 0, we have
(3.12) maxziX∈[0,1],∑i(ziX)2=1G+F=G(0,0,1)+F(0,0,1,0,0,1),subscriptformulae-sequencesubscriptsuperscript𝑧𝑋𝑖01subscript𝑖superscriptsuperscriptsubscript𝑧𝑖𝑋21𝐺𝐹𝐺001𝐹001001\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 ∈ [ 0 , 1 ] , ∑ 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)]𝐺00112delimited-[]subscript𝐻subscript𝑎3subscript𝑏3subscript𝑟3subscript𝐻subscript𝑎3subscript𝑏3subscript𝑟32𝐻subscript𝑎3\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)[Hα0(γ0)+Hα1(γ1)-2Ha3(b3+r3)]+12(1-a3)[Hβ0(δ0)+Hβ1(δ1)-2H-a3(b3-r3)].𝐹001001121subscript𝑎3delimited-[]subscript𝐻subscript𝛼0subscript𝛾0subscript𝐻subscript𝛼1subscript𝛾12subscript𝐻subscript𝑎3subscript𝑏3subscript𝑟3121subscript𝑎3delimited-[]subscript𝐻subscript𝛽0subscript𝛿0subscript𝐻subscript𝛽1subscript𝛿12subscript𝐻subscript𝑎3subscript𝑏3subscript𝑟3\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_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ 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_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ 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)𝑘01(k=0,1)( italic_k = 0 , 1 )
αk=a3+(-1)k(b3+r3),γk=[∑i3ci2+s32+T32+2(c3s3+(-1)k(c3T3+s3T3))]12,formulae-sequencesubscript𝛼𝑘subscript𝑎3superscript1𝑘subscript𝑏3subscript𝑟3subscript𝛾𝑘superscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑐𝑖2superscriptsubscript𝑠32superscriptsubscript𝑇322subscript𝑐3subscript𝑠3superscript1𝑘subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇312\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_α 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_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ ∑ 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 ,
βk=-a3+(-1)k(b3-r3),δk=[∑i3ci2+s32+T32+2(-c3s3+(-1)k(s3T3-c3T3))]12.formulae-sequencesubscript𝛽𝑘subscript𝑎3superscript1𝑘subscript𝑏3subscript𝑟3subscript𝛿𝑘superscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑐𝑖2superscriptsubscript𝑠32superscriptsubscript𝑇322subscript𝑐3subscript𝑠3superscript1𝑘subscript𝑠3subscript𝑇3subscript𝑐3subscript𝑇312\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_β 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_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ ∑ 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,c1s1≤0,s1≤|c1|formulae-sequencesubscript𝑏30formulae-sequencesubscript𝑐1subscript𝑠10subscript𝑠1subscript𝑐1b_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 ≤ 0 , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ | italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | and max=|r1|subscript𝑟1subscript𝑟2subscript𝑟3subscript𝑟1\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) maxziX∈[0,1],∑i(ziX)2=1G+F=G(1,0,0)+F(1,0,0,1,0,0),subscriptformulae-sequencesubscriptsuperscript𝑧𝑋𝑖01subscript𝑖superscriptsuperscriptsubscript𝑧𝑖𝑋21𝐺𝐹𝐺100𝐹100100\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 ∈ [ 0 , 1 ] , ∑ 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)]𝐺10012delimited-[]subscript𝐻subscript𝑎3subscript𝑟1subscript𝐻subscript𝑎3subscript𝑟12𝐻subscript𝑎3G(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(γ0)+H-r1(γ1)+Hr1(δ0)+H-r1(δ1)-4H(r1)].𝐹10010012delimited-[]subscript𝐻subscript𝑟1subscript𝛾0subscript𝐻subscript𝑟1subscript𝛾1subscript𝐻subscript𝑟1subscript𝛿0subscript𝐻subscript𝑟1subscript𝛿14𝐻subscript𝑟1\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_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ 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)𝑘01(k=0,1)( italic_k = 0 , 1 )
γk=[∑i3ci2+s12+T12+2(c1s1+(-1)k(c1T1+s1T1))]12,δk=[∑i3ci2+s12+T12+2(-c1s1+(-1)k(c1T1-s1T1))]12.formulae-sequencesubscript𝛾𝑘superscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑐𝑖2superscriptsubscript𝑠12superscriptsubscript𝑇122subscript𝑐1subscript𝑠1superscript1𝑘subscript𝑐1subscript𝑇1subscript𝑠1subscript𝑇112subscript𝛿𝑘superscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑐𝑖2superscriptsubscript𝑠12superscriptsubscript𝑇122subscript𝑐1subscript𝑠1superscript1𝑘subscript𝑐1subscript𝑇1subscript𝑠1subscript𝑇112\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_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ∑ 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_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ∑ 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,c1s1≤0,s1≤|c1|formulae-sequencesubscript𝑏30formulae-sequencesubscript𝑐1subscript𝑠10subscript𝑠1subscript𝑐1b_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 ≤ 0 , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ | italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | and maxr2=|r2|subscript𝑟1subscript𝑟2subscript𝑟3subscript𝑟2\max\=|r_2|roman_max italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = | italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT |, we have
(3.18) maxziX∈[0,1],∑i(ziX)2=1G+F=G(0,1,0)+F(0,1,0,0,1,0),subscriptformulae-sequencesubscriptsuperscript𝑧𝑋𝑖01subscript𝑖superscriptsuperscriptsubscript𝑧𝑖𝑋21𝐺𝐹𝐺010𝐹010010\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 ∈ [ 0 , 1 ] , ∑ 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)]𝐺01012delimited-[]subscript𝐻subscript𝑎3subscript𝑟2subscript𝐻subscript𝑎3subscript𝑟22𝐻subscript𝑎3G(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(γ0)+H-r2(γ1)+Hr2(δ0)+H-r2(δ1)-4H(r2)].𝐹01001012delimited-[]subscript𝐻subscript𝑟2subscript𝛾0subscript𝐻subscript𝑟2subscript𝛾1subscript𝐻subscript𝑟2subscript𝛿0subscript𝐻subscript𝑟2subscript𝛿14𝐻subscript𝑟2\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_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ 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)𝑘01(k=0,1)( italic_k = 0 , 1 )
γk=[∑i3ci2+s22+T22+2(c2s2+(-1)kc2T2+s2T2)]12;δk=[∑i3ci2+s22+T22+2(-c2s2+(-1)kc2T2-s2T2)]12.formulae-sequencesubscript𝛾𝑘superscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑐𝑖2superscriptsubscript𝑠22superscriptsubscript𝑇222subscript𝑐2subscript𝑠2superscript1𝑘subscript𝑐2subscript𝑇2subscript𝑠2subscript𝑇212subscript𝛿𝑘superscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑐𝑖2superscriptsubscript𝑠22superscriptsubscript𝑇222subscript𝑐2subscript𝑠2superscript1𝑘subscript𝑐2subscript𝑇2subscript𝑠2subscript𝑇212\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_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ∑ 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_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = [ ∑ 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)[Hα0(γ0)+Hα1(γ1)-2Ha3z3A(α0-α12)]+12(1-a3z3A)[Hβ0(δ0)+Hβ1(δ1)-2H-a3z3A(β0-β12)].𝐺𝐹subscript𝐻subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝐵subscript𝐻subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝐵2𝐻subscript𝑎3superscriptsubscript𝑧3𝐴121subscript𝑎3superscriptsubscript𝑧3𝐴delimited-[]subscript𝐻subscript𝛼0subscript𝛾0subscript𝐻subscript𝛼1subscript𝛾12subscript𝐻subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝛼0subscript𝛼12121subscript𝑎3superscriptsubscript𝑧3𝐴delimited-[]subscript𝐻subscript𝛽0subscript𝛿0subscript𝐻subscript𝛽1subscript𝛿12subscript𝐻subscript𝑎3superscriptsubscript𝑧3𝐴subscript𝛽0subscript𝛽12\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_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ 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_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_α 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_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ 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_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) ] . end_CELL end_ROW
Note that F𝐹Fitalic_F is a function of six variables and the first three are exactly the variables of G𝐺Gitalic_G. Our strategy of locating the extremal points of G+F𝐺𝐹G+Fitalic_G + italic_F is first finding the critical points z1A,z2A,z3Asuperscriptsubscript𝑧1𝐴superscriptsubscript𝑧2𝐴superscriptsubscript𝑧3𝐴z_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 G𝐺Gitalic_G and verify that at those points the critical points of G𝐺Gitalic_G are attainable, then we can find the maximal points of F+G𝐹𝐺F+Gitalic_F + italic_G.
For case 1: a3b3r3≤0subscript𝑎3subscript𝑏3subscript𝑟30a_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 ≤ 0 and r32-r2≥a3b3r3superscriptsubscript𝑟32superscript𝑟2subscript𝑎3subscript𝑏3subscript𝑟3r_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 ≥ 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)𝐺superscriptsubscript𝑧1𝐴superscriptsubscript𝑧2𝐴superscriptsubscript𝑧3𝐴𝐺001\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 F𝐹Fitalic_F are degenerated into (k=0,1𝑘01k=0,1italic_k = 0 , 1)
αk=a3+(-1)k(b3z3B+r3z3B),βk=-a3+(-1)k(b3z3B-r3z3B),γk=∑i3ci2+s32+T32(z3B)2+2[c3s3+(-1)k(s3T3z3B+c3T3z3B)]12,δk=∑i3ci2+s32+T32(z3B)2+2[-c3s3+(-1)k(s3T3z3B-c3T3z3B)]12.formulae-sequencesubscript𝛼𝑘subscript𝑎3superscript1𝑘subscript𝑏3superscriptsubscript𝑧3𝐵subscript𝑟3superscriptsubscript𝑧3𝐵formulae-sequencesubscript𝛽𝑘subscript𝑎3superscript1𝑘subscript𝑏3superscriptsubscript𝑧3𝐵subscript𝑟3superscriptsubscript𝑧3𝐵formulae-sequencesubscript𝛾𝑘superscriptsuperscriptsubscript𝑖3superscriptsubscript𝑐𝑖2superscriptsubscript𝑠32superscriptsubscript𝑇32superscriptsubscriptsuperscript𝑧𝐵322delimited-[]subscript𝑐3subscript𝑠3superscript1𝑘subscript𝑠3subscript𝑇3subscriptsuperscript𝑧𝐵3subscript𝑐3subscript𝑇3subscriptsuperscript𝑧𝐵312subscript𝛿𝑘superscriptsuperscriptsubscript𝑖3superscriptsubscript𝑐𝑖2superscriptsubscript𝑠32superscriptsubscript𝑇32superscriptsubscriptsuperscript𝑧𝐵322delimited-[]subscript𝑐3subscript𝑠3superscript1𝑘subscript𝑠3subscript𝑇3subscriptsuperscript𝑧𝐵3subscript𝑐3subscript𝑇3subscriptsuperscript𝑧𝐵312\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_α 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_β 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_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = ∑ 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_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = ∑ 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)[Hα0(γ0)+Hα1(γ1)]+12(1-a3)[Hβ0(γ0)+Hβ1(γ1)].𝐹subscriptsuperscript𝑧𝐴1subscriptsuperscript𝑧𝐴2subscriptsuperscript𝑧𝐴3subscriptsuperscript𝑧𝐵1subscriptsuperscript𝑧𝐵2subscriptsuperscript𝑧𝐵3𝐹001subscriptsuperscript𝑧𝐵3121subscript𝑎3delimited-[]subscript𝐻subscript𝛼0subscript𝛾0subscript𝐻subscript𝛼1subscript𝛾1121subscript𝑎3delimited-[]subscript𝐻subscript𝛽0subscript𝛾0subscript𝐻subscript𝛽1subscript𝛾1\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_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ 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_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] . end_CELL end_ROW
When (b3+r3)(c3+s3)≤0subscript𝑏3subscript𝑟3subscript𝑐3subscript𝑠30(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 ) ≤ 0, it can be observed that F𝐹Fitalic_F is an even function for z3B∈[-1,1]superscriptsubscript𝑧3𝐵11z_3^B\in[-1,1]italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ∈ [ - 1 , 1 ], so we just need to consider z3B∈[0,1]superscriptsubscript𝑧3𝐵01z_3^B\in[0,1]italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ∈ [ 0 , 1 ]. The derivative of F𝐹Fitalic_F on z3Bsuperscriptsubscript𝑧3𝐵z_3^Bitalic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT is given by
(3.23) ∂F∂z3B=14(1+a3)(b3+r3)log2(1+α1)2[(1+α0)2-γ02](1+α0)2[(1+α1)2-γ12]+-c3T3-s3T3+T32z3Bγ1log21+α1+γ11+α1-γ1+c3T3+s3T3+T32z3Bγ0log21+α0+γ01+α0-γ0+14(1-a3)(b3-r3)log2(1+β1)2[(1+β0)2-δ02](1+β0)2[(1+β1)2-δ12]+c3T3-s3T3+T32z3Bδ1log21+β1+δ11+β1-δ1+-c3T3+s3T3+T32z3Bδ0log21+β0+δ01+β0-δ0;𝐹superscriptsubscript𝑧3𝐵141subscript𝑎3subscript𝑏3subscript𝑟3subscript2superscript1subscript𝛼12delimited-[]superscript1subscript𝛼02superscriptsubscript𝛾02superscript1subscript𝛼02delimited-[]superscript1subscript𝛼12superscriptsubscript𝛾12subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛾1subscript21subscript𝛼1subscript𝛾11subscript𝛼1subscript𝛾1subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛾0subscript21subscript𝛼0subscript𝛾01subscript𝛼0subscript𝛾0141subscript𝑎3subscript𝑏3subscript𝑟3subscript2superscript1subscript𝛽12delimited-[]superscript1subscript𝛽02superscriptsubscript𝛿02superscript1subscript𝛽02delimited-[]superscript1subscript𝛽12superscriptsubscript𝛿12subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛿1subscript21subscript𝛽1subscript𝛿11subscript𝛽1subscript𝛿1subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛿0subscript21subscript𝛽0subscript𝛿01subscript𝛽0subscript𝛿0\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 ∂ italic_F end_ARG start_ARG ∂ 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_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_ARG ( 1 + italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ 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_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_γ 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_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_γ 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_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_ARG ( 1 + italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_δ 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_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_δ 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_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ; end_CELL end_ROW
If b3+r3≤0subscript𝑏3subscript𝑟30b_3+r_3\leq 0italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≤ 0 and c3+s3≥0subscript𝑐3subscript𝑠30c_3+s_3\geq 0italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≥ 0, we have γ0≥γ1subscript𝛾0subscript𝛾1\gamma_0\geq\gamma_1italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, α1≥α0subscript𝛼1subscript𝛼0\alpha_1\geq\alpha_0italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, δ0≥δ1subscript𝛿0subscript𝛿1\delta_0\geq\delta_1italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and β1≥β0subscript𝛽1subscript𝛽0\beta_1\geq\beta_0italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, then
(3.24) (b3+r3)log2(1+α1)2[(1+α0)2-γ02](1+α0)2[(1+α1)2-γ12]≥0;(b3-r3)log2(1+β1)2[(1+β0)2-δ02](1+β0)2[(1+β1)2-δ12]≥0;formulae-sequencesubscript𝑏3subscript𝑟3subscript2superscript1subscript𝛼12delimited-[]superscript1subscript𝛼02superscriptsubscript𝛾02superscript1subscript𝛼02delimited-[]superscript1subscript𝛼12superscriptsubscript𝛾120subscript𝑏3subscript𝑟3subscript2superscript1subscript𝛽12delimited-[]superscript1subscript𝛽02superscriptsubscript𝛿02superscript1subscript𝛽02delimited-[]superscript1subscript𝛽12superscriptsubscript𝛿120(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_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_ARG ( 1 + italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG ≥ 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_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_ARG ( 1 + italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG ≥ 0 ;
(3.25) -c3T3-s3T3+T32z3Bγ1log21+α1+γ11+α1-γ1+c3T3+s3T3+T32z3Bγ0log21+α0+γ01+α0-γ0≥-c3T3-s3T3+T32z3Bγ0log21+α1+γ11+α1-γ1+c3T3+s3T3+T32z3Bγ0log21+α1+γ11+α1-γ1=2T32z3Bγ0log21+α1+γ11+α1-γ1≥0;subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛾1subscript21subscript𝛼1subscript𝛾11subscript𝛼1subscript𝛾1subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛾0subscript21subscript𝛼0subscript𝛾01subscript𝛼0subscript𝛾0subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛾0subscript21subscript𝛼1subscript𝛾11subscript𝛼1subscript𝛾1subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛾0subscript21subscript𝛼1subscript𝛾11subscript𝛼1subscript𝛾12superscriptsubscript𝑇32subscriptsuperscript𝑧𝐵3subscript𝛾0subscript21subscript𝛼1subscript𝛾11subscript𝛼1subscript𝛾10\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_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_γ 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_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT 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_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_γ 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_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_γ 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_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ≥ 0 ; end_CELL end_ROW
(3.26) c3T3-s3T3+T32z3Bδ1log21+β1+δ11+β1-δ1+-c3T3+s3T3+T32z3Bδ0log21+β0+δ01+β0-δ0≥c3T3-s3T3+T32z3Bδ0log21+β1+δ11+β1-δ1+-c3T3+s3T3+T32z3Bδ0log21+β1+δ11+β1-δ1=2T32z3Bδ0log21+β1+δ11+β1-δ1≥0.subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛿1subscript21subscript𝛽1subscript𝛿11subscript𝛽1subscript𝛿1subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛿0subscript21subscript𝛽0subscript𝛿01subscript𝛽0subscript𝛿0subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛿0subscript21subscript𝛽1subscript𝛿11subscript𝛽1subscript𝛿1subscript𝑐3subscript𝑇3subscript𝑠3subscript𝑇3superscriptsubscript𝑇32superscriptsubscript𝑧3𝐵subscript𝛿0subscript21subscript𝛽1subscript𝛿11subscript𝛽1subscript𝛿12superscriptsubscript𝑇32subscriptsuperscript𝑧𝐵3subscript𝛿0subscript21subscript𝛽1subscript𝛿11subscript𝛽1subscript𝛿10\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_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_δ 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_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT 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_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_δ 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_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_δ 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_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ≥ 0 . end_CELL end_ROW
Hence in this case we get ∂F∂z3B≥0𝐹subscriptsuperscript𝑧𝐵30\frac\partialF\partialz^B_3\geq 0divide start_ARG ∂ italic_F end_ARG start_ARG ∂ italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ≥ 0 when z3B∈[0,1]superscriptsubscript𝑧3𝐵01z_3^B\in[0,1]italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ∈ [ 0 , 1 ].
If b3+r3≥0subscript𝑏3subscript𝑟30b_3+r_3\geq 0italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≥ 0 and c3+s3≤0subscript𝑐3subscript𝑠30c_3+s_3\leq 0italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≤ 0, we also can show that ∂F∂z3B≥0𝐹superscriptsubscript𝑧3𝐵0\frac\partial F\partial z_3^B\geq 0divide start_ARG ∂ italic_F end_ARG start_ARG ∂ italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG ≥ 0 similarly. So F𝐹Fitalic_F is a strictly monotonically increasing function with z3B∈[0,1]subscriptsuperscript𝑧𝐵301z^B_3\in[0,1]italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ [ 0 , 1 ]. Similarly we can check that F𝐹Fitalic_F is a strictly monotonically increasing function with respect to z1B∈[0,1]subscriptsuperscript𝑧𝐵101z^B_1\in[0,1]italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ [ 0 , 1 ] or z2B∈[0,1]subscriptsuperscript𝑧𝐵201z^B_2\in[0,1]italic_z start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ [ 0 , 1 ] in case 2. ∎
Theorem 3.3.
For the Werner-GHZ state ρw=c|ψ⟩⟨ψ|+(1-c)I8subscript𝜌𝑤𝑐ket𝜓bra𝜓1𝑐𝐼8\rho_w=c|\psi\rangle\langle\psi|+(1-c)\fracI8italic_ρ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = italic_c | italic_ψ ⟩ ⟨ italic_ψ | + ( 1 - italic_c ) divide start_ARG italic_I end_ARG start_ARG 8 end_ARG, where |ψ⟩=|000⟩+|111⟩2ket𝜓ket000ket1112|\psi\rangle=\frac2| italic_ψ ⟩ = divide start_ARG | 000 ⟩ + | 111 ⟩ end_ARG start_ARG 2 end_ARG, the quantum discord is
(3.27) 𝒬=18(1-c)log2(1-c)+1+7c8log2(1+7c)-14(1+3c)log2(1+3c).𝒬181𝑐𝑙𝑜subscript𝑔21𝑐17𝑐8𝑙𝑜subscript𝑔217𝑐1413𝑐𝑙𝑜subscript𝑔213𝑐\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).𝐺superscriptsubscript𝑧1𝐴superscriptsubscript𝑧2𝐴superscriptsubscript𝑧3𝐴𝐻𝑐\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 θ=cz3B𝜃𝑐superscriptsubscript𝑧3𝐵\theta=cz_3^Bitalic_θ = 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(θ)=12[Hθ(|c+θ|)+Hθ(|c-θ|)+H-θ(|c+θ|)+H-θ(|c-θ|)]-2H(θ).𝐹superscriptsubscript𝑧1𝐴superscriptsubscript𝑧2𝐴superscriptsubscript𝑧3𝐴superscriptsubscript𝑧1𝐵superscriptsubscript𝑧2𝐵superscriptsubscript𝑧3𝐵𝐹𝜃12delimited-[]subscript𝐻𝜃𝑐𝜃subscript𝐻𝜃𝑐𝜃subscript𝐻𝜃𝑐𝜃subscript𝐻𝜃𝑐𝜃2𝐻𝜃\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_θ ) 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_θ end_POSTSUBSCRIPT ( | italic_c + italic_θ | ) + italic_H start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( | italic_c - italic_θ | ) + italic_H start_POSTSUBSCRIPT - italic_θ end_POSTSUBSCRIPT ( | italic_c + italic_θ | ) + italic_H start_POSTSUBSCRIPT - italic_θ end_POSTSUBSCRIPT ( | italic_c - italic_θ | ) ] - 2 italic_H ( italic_θ ) . end_CELL end_ROW
It is easy to see that F(θ)𝐹𝜃F(\theta)italic_F ( italic_θ ) is monotonically increasing with respect to θ∈[0,1]𝜃01\theta\in[0,1]italic_θ ∈ [ 0 , 1 ]. So maxF(θ)=F(maxθ)=F(c)𝐹𝜃𝐹𝜃𝐹𝑐\max\F(\theta)\=F(\max\\theta\)=F(c)roman_max italic_F ( italic_θ ) = italic_F ( roman_max italic_θ ) = italic_F ( italic_c ). Fig.1 shows the behavior of the function 𝒬𝒬\mathcalQcaligraphic_Q.
Next, we consider the following general tripartite state
(3.29) ρ=18(I8+∑i3aiσi⊗I4+I2⊗∑i3biσi⊗I2+I4⊗∑i3ciσi+∑i3riσi⊗σi⊗I2+∑i3siσi⊗I2⊗σi+∑i3viI2⊗σi⊗σi+∑i3Tiσi⊗σi⊗σi).𝜌18subscript𝐼8superscriptsubscript𝑖3tensor-productsubscript𝑎𝑖subscript𝜎𝑖subscript𝐼4tensor-productsubscript𝐼2superscriptsubscript𝑖3tensor-productsubscript𝑏𝑖subscript𝜎𝑖subscript𝐼2tensor-productsubscript𝐼4superscriptsubscript𝑖3subscript𝑐𝑖subscript𝜎𝑖superscriptsubscript𝑖3tensor-productsubscript𝑟𝑖subscript𝜎𝑖subscript𝜎𝑖subscript𝐼2superscriptsubscript𝑖3tensor-productsubscript𝑠𝑖subscript𝜎𝑖subscript𝐼2subscript𝜎𝑖superscriptsubscript𝑖3tensor-productsubscript𝑣𝑖subscript𝐼2subscript𝜎𝑖subscript𝜎𝑖superscriptsubscript𝑖3tensor-productsubscript𝑇𝑖subscript𝜎𝑖subscript𝜎𝑖subscript𝜎𝑖\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_ρ end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 8 end_ARG ( italic_I start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ⊗ ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ 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 ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) . end_CELL end_ROW
Let a=∑i3ai2𝑎superscriptsubscript𝑖3superscriptsubscript𝑎𝑖2a=\sqrt\sum_i^3a_i^2italic_a = square-root start_ARG ∑ 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=∑i3bi2𝑏superscriptsubscript𝑖3superscriptsubscript𝑏𝑖2b=\sqrt\sum_i^3b_i^2italic_b = square-root start_ARG ∑ 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 ρ𝜌\rhoitalic_ρ in Eq.(3.29), we have the following results:
Case 1: when ai=vi=Ti=0,r1=r2=r3=rformulae-sequencesubscript𝑎𝑖subscript𝑣𝑖subscript𝑇𝑖0subscript𝑟1subscript𝑟2subscript𝑟3𝑟a_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) 𝒬(ρ)=∑i8λilog2λi+4+Hb(r)+H-b(r)-H(|b+r|)-12[Hb+B(r)+Hb-B(r)+H-b+B(r)+H-b-B(r)],𝒬𝜌superscriptsubscript𝑖8subscript𝜆𝑖subscript2subscript𝜆𝑖4subscript𝐻𝑏𝑟subscript𝐻𝑏𝑟𝐻𝑏𝑟12delimited-[]subscript𝐻𝑏B𝑟subscript𝐻𝑏B𝑟subscript𝐻𝑏B𝑟subscript𝐻𝑏B𝑟\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_ρ ) = end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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=[∑i3(sibib+ci)2]12normal-Bsuperscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑠𝑖subscript𝑏𝑖𝑏subscript𝑐𝑖212\mathrmB=[\sum_i^3(s_i\fracb_ib+c_i)^2]^\frac12roman_B = [ ∑ 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-sequencesubscript𝑏𝑖subscript𝑣𝑖subscript𝑇𝑖0subscript𝑟1subscript𝑟2subscript𝑟3𝑟b_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) 𝒬(ρ)=∑i8λilog2λi+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)],𝒬𝜌superscriptsubscript𝑖8subscript𝜆𝑖subscript2subscript𝜆𝑖3𝐻superscript𝑎212delimited-[]subscript𝐻𝑎𝑟subscript𝐻𝑎𝑟2𝐻𝑎121𝑎delimited-[]subscript𝐻𝑎A𝑟subscript𝐻𝑎A𝑟2subscript𝐻𝑎𝑟121𝑎delimited-[]subscript𝐻𝑎A𝑟subscript𝐻𝑎A𝑟2subscript𝐻𝑎𝑟\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_ρ ) end_CELL start_CELL = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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=[∑i3(siaia+ci)2]12normal-Asuperscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑠𝑖subscript𝑎𝑖𝑎subscript𝑐𝑖212\mathrmA=[\sum_i^3(s_i\fraca_ia+c_i)^2]^\frac12roman_A = [ ∑ 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=0subscript𝑟𝑖subscript𝑇𝑖subscript𝑣𝑖0r_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) 𝒬(ρ)=∑i8λilog2λi+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)],𝒬𝜌superscriptsubscript𝑖8subscript𝜆𝑖subscript2subscript𝜆𝑖3𝐻superscript𝑎212delimited-[]subscript𝐻𝑏𝑎subscript𝐻𝑏𝑎2𝐻𝑎121𝑎delimited-[]subscript𝐻𝑎A𝑏subscript𝐻𝑎A𝑏2subscript𝐻𝑎𝑏121𝑎delimited-[]subscript𝐻𝑏A𝑏subscript𝐻𝑎A𝑏2subscript𝐻𝑎𝑏\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_ρ ) end_CELL start_CELL = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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=[∑i3(siaia+ci)2]12normal-Asuperscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑠𝑖subscript𝑎𝑖𝑎subscript𝑐𝑖212\mathrmA=[\sum_i^3(s_i\fraca_ia+c_i)^2]^\frac12roman_A = [ ∑ 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-sequencesubscript𝑎𝑖subscript𝑐𝑖subscript𝑠𝑖subscript𝑇𝑖0subscript𝑟1subscript𝑟2subscript𝑟3𝑟subscript𝑣1subscript𝑣2subscript𝑣3𝑣a_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) 𝒬(ρ)=∑i8λilog2λi+Hb(r)+H-b(r)+4-H(|b+r|)-12[Hb+v(r)+Hb-v(r)+H-b+v(r)H-b-v(r)].𝒬𝜌superscriptsubscript𝑖8subscript𝜆𝑖subscript2subscript𝜆𝑖subscript𝐻𝑏𝑟subscript𝐻𝑏𝑟4𝐻𝑏𝑟12delimited-[]subscript𝐻𝑏𝑣𝑟subscript𝐻𝑏𝑣𝑟subscript𝐻𝑏𝑣𝑟subscript𝐻𝑏𝑣𝑟\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_ρ ) = end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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-sequencesubscript𝑟𝑖subscript𝑇𝑖subscript𝑠𝑖subscript𝑐𝑖0subscript𝑣1subscript𝑣2subscript𝑣3𝑣r_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) 𝒬(ρ)=∑i8λilog2λi+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)].𝒬𝜌superscriptsubscript𝑖8subscript𝜆𝑖subscript2subscript𝜆𝑖3𝐻superscript𝑎212delimited-[]subscript𝐻𝑏𝑎subscript𝐻𝑏𝑎2𝐻𝑎121𝑎delimited-[]subscript𝐻𝑎𝑣𝑏subscript𝐻𝑎𝑣𝑏2subscript𝐻𝑎𝑏121𝑎delimited-[]subscript𝐻𝑎𝑣𝑏subscript𝐻𝑎𝑣𝑏2subscript𝐻𝑎𝑏\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_ρ ) end_CELL start_CELL = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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-sequencesubscript𝑏𝑖subscript𝑠𝑖subscript𝑐𝑖subscript𝑇𝑖0subscript𝑟1subscript𝑟2subscript𝑟3𝑟subscript𝑣1subscript𝑣2subscript𝑣3𝑣b_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) 𝒬(ρ)=∑i8λilog2λi+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)].𝒬𝜌superscriptsubscript𝑖8subscript𝜆𝑖subscript2subscript𝜆𝑖3𝐻superscript𝑎212delimited-[]subscript𝐻𝑎𝑟subscript𝐻𝑎𝑟2𝐻𝑎121𝑎delimited-[]subscript𝐻𝑎𝑣𝑟subscript𝐻𝑎𝑣𝑟2subscript𝐻𝑎𝑟121𝑎delimited-[]subscript𝐻𝑎𝑣𝑟subscript𝐻𝑎𝑣𝑟2subscript𝐻𝑎𝑟\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_ρ ) end_CELL start_CELL = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ 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. Let’s consider case 1: ai=vi=Ti=0,r1=r2=r3=rformulae-sequencesubscript𝑎𝑖subscript𝑣𝑖subscript𝑇𝑖0subscript𝑟1subscript𝑟2subscript𝑟3𝑟a_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)𝐺superscriptsubscript𝑧1𝐴superscriptsubscript𝑧2𝐴superscriptsubscript𝑧3𝐴𝐺subscript𝑏1𝑏subscript𝑏2𝑏subscript𝑏3𝑏\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 θ=∑i3rziBbib𝜃superscriptsubscript𝑖3𝑟superscriptsubscript𝑧𝑖𝐵subscript𝑏𝑖𝑏\theta=\sum_i^3rz_i^B\fracb_ibitalic_θ = ∑ 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,θ)=12[Hb+B(θ)+Hb-B(θ)+H-b+B(θ)+H-b-B(θ)]-Hb(θ)-H-b(θ)-2,𝐹superscriptsubscript𝑧1𝐴superscriptsubscript𝑧2𝐴superscriptsubscript𝑧3𝐴superscriptsubscript𝑧1𝐵superscriptsubscript𝑧2𝐵superscriptsubscript𝑧3𝐵𝐹subscript𝑏1𝑏subscript𝑏2𝑏subscript𝑏3𝑏𝜃12delimited-[]subscript𝐻𝑏B𝜃subscript𝐻𝑏B𝜃subscript𝐻𝑏B𝜃subscript𝐻𝑏B𝜃subscript𝐻𝑏𝜃subscript𝐻𝑏𝜃2\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_θ ) 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_θ ) + italic_H start_POSTSUBSCRIPT italic_b - roman_B end_POSTSUBSCRIPT ( italic_θ ) + italic_H start_POSTSUBSCRIPT - italic_b + roman_B end_POSTSUBSCRIPT ( italic_θ ) + italic_H start_POSTSUBSCRIPT - italic_b - roman_B end_POSTSUBSCRIPT ( italic_θ ) ] - italic_H start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_θ ) - italic_H start_POSTSUBSCRIPT - italic_b end_POSTSUBSCRIPT ( italic_θ ) - 2 , end_CELL end_ROW
where B=[∑i3(sibib+ci)2]12Bsuperscriptdelimited-[]superscriptsubscript𝑖3superscriptsubscript𝑠𝑖subscript𝑏𝑖𝑏subscript𝑐𝑖212\mathrmB=[\sum_i^3(s_i\fracb_ib+c_i)^2]^\frac12roman_B = [ ∑ 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 F𝐹Fitalic_F over θ𝜃\thetaitalic_θ is equal to
(3.37) ∂F∂θ=14[log2(1+b+B+θ)(1+b-B+θ)(1+b-θ)2(1+b+B-θ)(1+b-B-θ)(1+b+θ)2+log2(1-b-B+θ)(1-b+B+θ)(1-b-θ)2(1-b-B-θ)(1-b+B-θ)(1-b+θ)2].𝐹𝜃14delimited-[]subscript21𝑏B𝜃1𝑏B𝜃superscript1𝑏𝜃21𝑏B𝜃1𝑏B𝜃superscript1𝑏𝜃2subscript21𝑏B𝜃1𝑏B𝜃superscript1𝑏𝜃21𝑏B𝜃1𝑏B𝜃superscript1𝑏𝜃2\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 ∂ italic_F end_ARG start_ARG ∂ italic_θ 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_θ ) ( 1 + italic_b - roman_B + italic_θ ) ( 1 + italic_b - italic_θ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 + italic_b + roman_B - italic_θ ) ( 1 + italic_b - roman_B - italic_θ ) ( 1 + italic_b + italic_θ ) 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_θ ) ( 1 - italic_b + roman_B + italic_θ ) ( 1 - italic_b - italic_θ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_b - roman_B - italic_θ ) ( 1 - italic_b + roman_B - italic_θ ) ( 1 - italic_b + italic_θ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] . end_CELL end_ROW
Obviously, ∂F∂θ≥0𝐹𝜃0\frac\partial F\partial\theta\geq 0divide start_ARG ∂ italic_F end_ARG start_ARG ∂ italic_θ end_ARG ≥ 0 when θ∈[0,1]𝜃01\theta\in[0,1]italic_θ ∈ [ 0 , 1 ]. Then F(θ)𝐹𝜃F(\theta)italic_F ( italic_θ ) is a strictly increasing function and maxF(θ)=F(maxθ)𝐹𝜃𝐹𝜃\maxF(\theta)=F(\max\\theta\)roman_max italic_F ( italic_θ ) = italic_F ( roman_max italic_θ ).
Let Y=θ+μ[1-(z1B)2-(z2B)2-(z3B)2]𝑌𝜃𝜇delimited-[]1superscriptsuperscriptsubscript𝑧1𝐵2superscriptsuperscriptsubscript𝑧2𝐵2superscriptsuperscriptsubscript𝑧3𝐵2Y=\theta+\mu[1-(z_1^B)^2-(z_2^B)^2-(z_3^B)^2]italic_Y = italic_θ + italic_μ [ 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 ], ∂Y∂z1B=rb1b-2μz1B𝑌superscriptsubscript𝑧1𝐵𝑟subscript𝑏1𝑏2𝜇superscriptsubscript𝑧1𝐵\frac\partial Y\partial z_1^B=r\fracb_1b-2\mu z_1^Bdivide start_ARG ∂ italic_Y end_ARG start_ARG ∂ 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_μ italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, ∂Y∂z2B=rb2b-2μz2B𝑌superscriptsubscript𝑧2𝐵𝑟subscript𝑏2𝑏2𝜇superscriptsubscript𝑧2𝐵\frac\partial Y\partial z_2^B=r\fracb_2b-2\mu z_2^Bdivide start_ARG ∂ italic_Y end_ARG start_ARG ∂ 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_μ italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, ∂Y∂z3B=rb3b-2μz3B𝑌superscriptsubscript𝑧3𝐵𝑟subscript𝑏3𝑏2𝜇superscriptsubscript𝑧3𝐵\frac\partial Y\partial z_3^B=r\fracb_3b-2\mu z_3^Bdivide start_ARG ∂ italic_Y end_ARG start_ARG ∂ 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_μ italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, ∂Y∂μ=1-(z1B)2-(z2B)2-(z3B)2𝑌𝜇1superscriptsuperscriptsubscript𝑧1𝐵2superscriptsuperscriptsubscript𝑧2𝐵2superscriptsuperscriptsubscript𝑧3𝐵2\frac\partial Y\partial\mu=1-(z_1^B)^2-(z_2^B)^2-(z_3^B)^2divide start_ARG ∂ italic_Y end_ARG start_ARG ∂ italic_μ 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 ∂Y∂z1B=0,∂Y∂z2B=0,∂Y∂z3B=0,∂Y∂μ=0formulae-sequence𝑌superscriptsubscript𝑧1𝐵0formulae-sequence𝑌superscriptsubscript𝑧2𝐵0formulae-sequence𝑌superscriptsubscript𝑧3𝐵0𝑌𝜇0\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 ∂ italic_Y end_ARG start_ARG ∂ italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = 0 , divide start_ARG ∂ italic_Y end_ARG start_ARG ∂ italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = 0 , divide start_ARG ∂ italic_Y end_ARG start_ARG ∂ italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG = 0 , divide start_ARG ∂ italic_Y end_ARG start_ARG ∂ italic_μ end_ARG = 0, we have ziB=bibsuperscriptsubscript𝑧𝑖𝐵subscript𝑏𝑖𝑏z_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 maxθ=r𝜃𝑟\max\\theta\=rroman_max italic_θ = italic_r and maxF(θ)=F(r)𝐹𝜃𝐹𝑟\maxF(\theta)=F(r)roman_max italic_F ( italic_θ ) = italic_F ( italic_r ), then case 1 is shown. ∎
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-sequencesubscript𝑎10formulae-sequencesubscript𝑎20formulae-sequencesubscript𝑎30.03formulae-sequencesubscript𝑏10formulae-sequencesubscript𝑏20formulae-sequencesubscript𝑏30.25formulae-sequencesubscript𝑐10.12formulae-sequencesubscript𝑐20.12formulae-sequencesubscript𝑐30.01formulae-sequencesubscript𝑟10.1formulae-sequencesubscript𝑟20.1formulae-sequencesubscript𝑟30.3formulae-sequencesubscript𝑠10.13formulae-sequencesubscript𝑠20.13formulae-sequencesubscript𝑠30.26formulae-sequencesubscript𝑣10formulae-sequencesubscript𝑣20formulae-sequencesubscript𝑣30formulae-sequencesubscript𝑇10.02formulae-sequencesubscript𝑇20.02subscript𝑇30.36a_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 𝒬=0.8889𝒬0.8889\mathcalQ=0.8889caligraphic_Q = 0.8889. Fig. 2 shows the behavior of the quantum discord 𝒬𝒬\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-sequencesubscript𝑎1subscript𝑎2subscript𝑎30formulae-sequencesubscript𝑏10.2formulae-sequencesubscript𝑏20.05formulae-sequencesubscript𝑏30.1formulae-sequencesubscript𝑐10.04formulae-sequencesubscript𝑐20.06formulae-sequencesubscript𝑐30.11subscript𝑟1subscript𝑟2subscript𝑟30.17formulae-sequencesubscript𝑠10.08formulae-sequencesubscript𝑠20.15formulae-sequencesubscript𝑠30.25subscript𝑣1subscript𝑣2subscript𝑣3subscript𝑇1subscript𝑇2subscript𝑇30a_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 𝒬=0.9970𝒬0.9970\mathcalQ=0.9970caligraphic_Q = 0.9970. Fig. 3 and Fig. 4 show the behavior of the function G𝐺Gitalic_G and F𝐹Fitalic_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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