Quantum discord in a bipartite system can be dynamically revealed and quantified through purely local operations on one of the two subsystems. To achieve this, the local detection method harnesses the influence of initial correlations on the reduced dynamics of an interacting bipartite system. This article’s aim is to provide an accessible introduction to this method and to review recent theoretical and experimental progress.
Experimental achievements in the last decades have established the precise quantum control of individual quantum systems Wineland (2013); Haroche (2013). Furthermore, recent efforts are focussed on the assembly and monitoring of interacting quantum systems, with various applications in the context of quantum information Bloch et al. (2008); Häffner et al. (2008); Schneider et al. (2012); Blatt and Roos (2012). The efficient characterization of the underlying quantum states in high-dimensional state spaces, however, remains a challenge due to the large number of parameters.
One possible strategy for the analysis of systems whose size, complexity or structure is beyond the reach of a detailed microscopic examination is therefore to restrict access to a small, easily controllable subsystem Haikka and Maniscalco (2014); Gessner (2015). By interaction with the remaining system, the locally observable quantities of the subsystem may be able to convey information about the global properties of the interacting system. While in general it is not always clear whether sufficient information about a possibly complex surounding system can be obtained from the few variables of the accessible subsystem, such an approach has proven to be suitable for probing the presence of correlations between the probe and its environment in a variety of situations Laine et al. (2010); Gessner (2015). In the present article, we review recent progress in the local detection method Gessner and Breuer (2011, 2013a)-an interaction-assisted method, able to reveal quantum discord of the global system through the dynamics of a local subsystem. The method can be implemented when access is restricted to a controllable subsystem, and it has been tested in various different experimental settings. With the help of the examples reviewed in this article we discuss under which physical circumstances a successful local detection based on this method can generally be expected.
The concept of quantum discord can be intuitively understood in terms of local measurements of a bipartite quantum system. Measurements usually induce disturbances of the quantum system under observation von Neumann (1955); Lüders (1951). An exception to this textbook rule is found if the system is initially prepared in an eigenstate of the measured observable. More generally, if observable M𝑀Mitalic_M and quantum state ρ𝜌\rhoitalic_ρ commute, i.e., [M,ρ]=0𝑀𝜌0[M,\rho]=0[ italic_M , italic_ρ ] = 0, a non-selective measurement of M𝑀Mitalic_M will leave the quantum state ρ𝜌\rhoitalic_ρ unchanged von Neumann (1955); Lüders (1951). Such a measurement projects the system into the eigenstate |φm⟩ketsubscript𝜑𝑚|\varphi_m\rangle| italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ with probability pm=⟨φm|ρ|φm⟩subscript𝑝𝑚quantum-operator-productsubscript𝜑𝑚𝜌subscript𝜑𝑚p_m=\langle\varphi_m|\rho|\varphi_m\rangleitalic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_ρ | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩, where we assume a non-degenerate observable with spectral decomposition M=∑mλm|φm⟩⟨φm|𝑀subscript𝑚subscript𝜆𝑚ketsubscript𝜑𝑚brasubscript𝜑𝑚M=\sum_m\lambda_m|\varphi_m\rangle\langle\varphi_m|italic_M = ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |. The state at the outcome of the projective measurement is consequently given as
Φ(ρ)Φ𝜌\displaystyle\Phi(\rho)roman_Φ ( italic_ρ ) =∑mpm|φm⟩⟨φm|absentsubscript𝑚subscript𝑝𝑚ketsubscript𝜑𝑚brasubscript𝜑𝑚\displaystyle=\sum_mp_m|\varphi_m\rangle\langle\varphi_m|= ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT |
=∑m|φm⟩⟨φm|ρ|φm⟩⟨φm|.absentsubscript𝑚ketsubscript𝜑𝑚quantum-operator-productsubscript𝜑𝑚𝜌subscript𝜑𝑚brasubscript𝜑𝑚\displaystyle=\sum_m|\varphi_m\rangle\langle\varphi_m|\rho|\varphi_m% \rangle\langle\varphi_m|.= ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_ρ | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | . (1)
In fact, we find that Φ(ρ)=ρΦ𝜌𝜌\Phi(\rho)=\rhoroman_Φ ( italic_ρ ) = italic_ρ if and only if [M,ρ]=0𝑀𝜌0[M,\rho]=0[ italic_M , italic_ρ ] = 0. Hence, for any given quantum state ρ𝜌\rhoitalic_ρ, we can construct a family of observables M𝑀Mitalic_M which can be measured non-selectively without disturbance. This family is comprised of all observables with the same eigenvectors as ρ𝜌\rhoitalic_ρ, assuming no degeneracies.
Let us now consider the case of a bipartite quantum system, described by a tensor product of Hilbert spaces ℋ=ℋA⊗ℋBℋtensor-productsubscriptℋ𝐴subscriptℋ𝐵\mathcalH=\mathcalH_A\otimes\mathcalH_Bcaligraphic_H = caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ caligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. Under which circumstances is it possible to construct local observables whose non-selective measurement does not disturb the total quantum state ρ𝜌\rhoitalic_ρ? The post-measurement state of a non-selective measurement of a local observable MA⊗𝕀B=∑mλm|φm⟩⟨φm|⊗𝕀Btensor-productsubscript𝑀𝐴subscript𝕀𝐵subscript𝑚tensor-productsubscript𝜆𝑚ketsubscript𝜑𝑚brasubscript𝜑𝑚subscript𝕀𝐵M_A\otimes\mathbbI_B=\sum_m\lambda_m|\varphi_m\rangle\langle% \varphi_m|\otimes\mathbbI_Bitalic_M start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is given by
(Φ⊗𝕀)ρtensor-productΦ𝕀𝜌\displaystyle(\Phi\otimes\mathbbI)\rho( roman_Φ ⊗ blackboard_I ) italic_ρ =∑m(|φm⟩⟨φm|⊗𝕀B)ρ(|φm⟩⟨φm|⊗𝕀B),absentsubscript𝑚tensor-productketsubscript𝜑𝑚brasubscript𝜑𝑚subscript𝕀𝐵𝜌tensor-productketsubscript𝜑𝑚brasubscript𝜑𝑚subscript𝕀𝐵\displaystyle=\sum_m(|\varphi_m\rangle\langle\varphi_m|\otimes\mathbbI% _B)\rho(|\varphi_m\rangle\langle\varphi_m|\otimes\mathbbI_B),= ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ ( | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) , (2)
with 𝕀Bsubscript𝕀𝐵\mathbbI_Bblackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT the identity on ℋBsubscriptℋ𝐵\mathcalH_Bcaligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. Again considering only non-degenerate observables, one finds that (Φ⊗𝕀)ρ=ρtensor-productΦ𝕀𝜌𝜌(\Phi\otimes\mathbbI)\rho=\rho( roman_Φ ⊗ blackboard_I ) italic_ρ = italic_ρ is indeed equivalent to [MA⊗𝕀B,ρ]=0tensor-productsubscript𝑀𝐴subscript𝕀𝐵𝜌0[M_A\otimes\mathbbI_B,\rho]=0[ italic_M start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , italic_ρ ] = 0. The above question can thus be reformulated as: Which quantum states ρ𝜌\rhoitalic_ρ commute with at least one local, non-degenerate observable? Obviously, if systems A𝐴Aitalic_A and B𝐵Bitalic_B are completely uncorrelated, i.e., if the total quantum state factorizes as ρ=ρA⊗ρB𝜌tensor-productsubscript𝜌𝐴subscript𝜌𝐵\rho=\rho_A\otimes\rho_Bitalic_ρ = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, then we can conclude that a family of local observables, e.g. in system A𝐴Aitalic_A, can always be constructed from the eigenvectors of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. The presence of correlations between the two systems, however, changes the situation.
Only a certain set of quantum states admit the existence of a non-degenerate observable MAsubscript𝑀𝐴M_Aitalic_M start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, such that [MA⊗𝕀B,ρ]=0tensor-productsubscript𝑀𝐴subscript𝕀𝐵𝜌0[M_A\otimes\mathbbI_B,\rho]=0[ italic_M start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , italic_ρ ] = 0. This family is known as the states of zero discord. They can always be written as Ollivier and Zurek (2001); Henderson and Vedral (2001); Modi et al. (2012)
ρzd=∑mpm|φm⟩⟨φm|⊗ρBm,subscript𝜌𝑧𝑑subscript𝑚tensor-productsubscript𝑝𝑚ketsubscript𝜑𝑚brasubscript𝜑𝑚superscriptsubscript𝜌𝐵𝑚\displaystyle\rho_zd=\sum_mp_m|\varphi_m\rangle\langle\varphi_m|% \otimes\rho_B^m,italic_ρ start_POSTSUBSCRIPT italic_z italic_d end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT , (3)
where pmsubscript𝑝𝑚p_mitalic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT is a probability distribution and ρBmsuperscriptsubscript𝜌𝐵𝑚\rho_B^mitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT are density operators on system B𝐵Bitalic_B. It is important to note that the |φm⟩ketsubscript𝜑𝑚|\varphi_m\rangle| italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ form an orthonormal basis of ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT since they are the eigenvectors of the Hermitian operator MAsubscript𝑀𝐴M_Aitalic_M start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. This distinguishes states of zero discord from separable states with the general form Werner (1989)
ρsep=∑ipiρAi⊗ρBi,subscript𝜌𝑠𝑒𝑝subscript𝑖tensor-productsubscript𝑝𝑖superscriptsubscript𝜌𝐴𝑖superscriptsubscript𝜌𝐵𝑖\displaystyle\rho_sep=\sum_ip_i\rho_A^i\otimes\rho_B^i,italic_ρ start_POSTSUBSCRIPT italic_s italic_e italic_p end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , (4)
where the states ρAisuperscriptsubscript𝜌𝐴𝑖\rho_A^iitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT are arbitrary and need neither be pure nor orthogonal. Furthermore, unlike entanglement, discord is an asymmetric property, requiring specification of the system which is measured. Throughout this article this will always be system A𝐴Aitalic_A.
Quantum discord therefore characterizes the presence or absence of a local observable which commutes with the full quantum state. Nonzero discord can only be observed in correlated (i.e. not factorizing) quantum states, however, even some separable states exhibit discord. For pure states, the concepts of discord and entanglement coincide. Hence, in general, discord is a concept closely connected to correlations but does not itself measure correlations. In particular, a local operation on system A𝐴Aitalic_A may change the orthogonality properties of the |φm⟩ketsubscript𝜑𝑚|\varphi_m\rangle| italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ⟩ in Eq. (3) Dakić et al. (2010); Ciccarello and Giovannetti (2012); Streltsov et al. (2011a) and thereby create discord without creating correlations Gessner et al. (2012); Lanyon et al. (2013).
The inability to commute with any local observable renders quantum states of nonzero discord furthermore suitable for certain technological tasks Modi et al. (2012). For instance, a phase shift φ𝜑\varphiitalic_φ, imprinted by a local unitary transformation e-iMAφ⊗𝕀Btensor-productsuperscript𝑒𝑖subscript𝑀𝐴𝜑subscript𝕀𝐵e^-iM_A\varphi\otimes\mathbbI_Bitalic_e start_POSTSUPERSCRIPT - italic_i italic_M start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_φ end_POSTSUPERSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT can be estimated with high precision Helstrom (1976) only if the initial quantum state ρ𝜌\rhoitalic_ρ is strongly affected by this transformation Girolami et al. (2014). Conversely, if MA⊗𝕀Btensor-productsubscript𝑀𝐴subscript𝕀𝐵M_A\otimes\mathbbI_Bitalic_M start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT happens to commute with ρ𝜌\rhoitalic_ρ, the state is completely invariant under this transformation and, consequently, an estimation of the phase shift φ𝜑\varphiitalic_φ is impossible. While states of zero discord are insensitive to the action of certain local operators, this can be excluded for all states of nonzero discord, since no local operator commutes with these states Girolami et al. (2014).
Among other applications Modi et al. (2012); Girolami et al. (2013), discord was further shown to be useful for the distribution Cubitt et al. (2003); Streltsov et al. (2012); Chuan et al. (2012); Fedrizzi et al. (2013); Vollmer et al. (2013); Peuntinger et al. (2013) and activation Streltsov et al. (2011b); Piani et al. (2011); Adesso et al. (2014) of entanglement. To exploit these phenomena experimentally, one needs to first find convenient ways to generate sufficiently robust discordant quantum states Ciccarello and Giovannetti (2012); Streltsov et al. (2011a); Lanyon et al. (2013); Carnio et al. (2015); Orieux et al. (2015); Carnio et al. (2016). Second, methods to recognize the presence of discord, and perhaps to even quantify discord in experimentally relevant situations are required Zhang et al. (2011); Girolami et al. (2011); Rahimi-Keshari et al. (2013); Modi et al. (2012). The local detection method Gessner and Breuer (2011, 2013a), to be introduced in the next section, is a dynamical method which allows to detect and quantify discord in a bipartite system with limited experimental requirements.
II The local detection method
The efficient detection of properties such as entanglement or discord is a challenging task Mintert et al. (2005); Gühne and Tóth (2009); Horodecki et al. (2009), which usually requires measurements of correlated observables. A popular approach for low-dimensional systems is to first obtain full information about the quantum state, and then to calculate a suitable quantifier based on the measured entries of the density matrix Häffner et al. (2005); Lanyon et al. (2013). Not only require the results of tomographic reconstructions of quantum states careful statistical analysis Schwemmer et al. (2015), their experimental realization soon becomes prohibitively expensive when high-dimensional or multipartite systems are of interest. In these cases, a complete characterization of the full quantum state can no longer be of interest. Alternatively, the measurement of carefully designed observables (such as entanglement witnesses) may reveal the presence of entanglement Duan et al. (2000); Sørensen et al. (2001); Giovannetti et al. (2003); Pezzé and Smerzi (2009); Gühne and Tóth (2009) or discord Zhang et al. (2011); Girolami et al. (2011); Silva et al. (2013); Rahimi-Keshari et al. (2013); Wecker et al. (2014); Hosseini et al. (2014) without knowledge of the full quantum state. Yet, such procedures are often restricted to Hilbert spaces of a certain dimension and structure (e.g. qubit systems or two-mode systems of continuous variables), cf. Gessner et al. . Their implementation furthermore often requires a high degree of control over the full quantum system (or even multiple copies thereof Walborn et al. (2006); Schmid et al. (2008); Islam et al. (2015)) which is difficult to achieve with increasing system size.
We may also encounter situations in which the experimenter may not even have full access to the complete quantum state of systems A𝐴Aitalic_A and B𝐵Bitalic_B but instead may be limited to measurements and operations on the subsystem A𝐴Aitalic_A. This limitation may be due to a fundamental inability to access the second system, when, for instance party B𝐵Bitalic_B is spatially separated from the experimenter at A𝐴Aitalic_A or describes degrees of freedom that cannot be measured experimentally. It may also be a deliberate choice such as to restrict the dimension of the quantum system which is to be controlled. In either case one may consider the subsystem B𝐵Bitalic_B an ancilla system or environment to system A𝐴Aitalic_A. Since the reduced dynamics of system A𝐴Aitalic_A may be strongly influenced by correlations with system B𝐵Bitalic_B, a dynamical witness for discord may be observable, even by restricting to local measurements of system A𝐴Aitalic_A.
II.1 Witnessing discord with local operations
To introduce the basic idea of the local detection method Gessner and Breuer (2011, 2013a) let us recall that states of zero discord are characterized by their invariance under non-selective measurements, i.e., a state ρ𝜌\rhoitalic_ρ has zero discord if and only if there exists a complete set of one-dimensional orthogonal projectors 𝚷=Π1,Π2,…,fragmentsΠfragmentssubscriptΠ1,subscriptΠ2,…,\boldsymbol\Pi=\\Pi_1,\Pi_2,\dots,\bold_Π = roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_Π start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , , such that ρ=(Φ𝚷⊗𝕀)ρ𝜌tensor-productsubscriptΦ𝚷𝕀𝜌\rho=(\Phi_\boldsymbol\Pi\otimes\mathbbI)\rhoitalic_ρ = ( roman_Φ start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ⊗ blackboard_I ) italic_ρ, where
(Φ𝚷⊗𝕀)ρ=∑i(Πi⊗𝕀B)ρ(Πi⊗𝕀B).tensor-productsubscriptΦ𝚷𝕀𝜌subscript𝑖tensor-productsubscriptΠ𝑖subscript𝕀𝐵𝜌tensor-productsubscriptΠ𝑖subscript𝕀𝐵\displaystyle(\Phi_\boldsymbol\Pi\otimes\mathbbI)\rho=\sum_i(\Pi_i% \otimes\mathbbI_B)\rho(\Pi_i\otimes\mathbbI_B).( roman_Φ start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ⊗ blackboard_I ) italic_ρ = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ ( roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) . (5)
This operation (5) is a purely local operation on system A𝐴Aitalic_A and its implementation does not require any knowledge of system B𝐵Bitalic_B. It furthermore describes complete dephasing in the basis 𝚷𝚷\boldsymbol\Pibold_Π and is therefore called a local dephasing operation Gessner and Breuer (2011).
Let us assume the state ρ𝜌\rhoitalic_ρ has zero discord. A local dephasing operation that leaves the state invariant thus exists, but in which particular local basis 𝚷𝚷\boldsymbol\Pibold_Π? Let us first express the full quantum state ρ𝜌\rhoitalic_ρ in terms of families of completely arbitrary local operator bases as ρ=∑αAα⊗Bα𝜌subscript𝛼tensor-productsubscript𝐴𝛼subscript𝐵𝛼\rho=\sum_\alphaA_\alpha\otimes B_\alphaitalic_ρ = ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⊗ italic_B start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT. To answer the above question, we study the reduced density matrix of system A𝐴Aitalic_A, which is obtained by performing the partial trace over B𝐵Bitalic_B, after the dephasing operation. We obtain
TrB(Φ𝚷⊗𝕀B)ρsubscriptTr𝐵tensor-productsubscriptΦ𝚷subscript𝕀𝐵𝜌\displaystyle\mathrmTr_B\(\Phi_\boldsymbol\Pi\otimes\mathbbI_B)\rho\roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ =∑i∑αΠiAαΠiTrBαabsentsubscript𝑖subscript𝛼subscriptΠ𝑖subscript𝐴𝛼subscriptΠ𝑖Trsubscript𝐵𝛼\displaystyle=\sum_i\sum_\alpha\Pi_iA_\alpha\Pi_i\mathrmTr\B_% \alpha\= ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_Tr italic_B start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT
=∑ipiΠi,absentsubscript𝑖subscript𝑝𝑖subscriptΠ𝑖\displaystyle=\sum_ip_i\Pi_i,= ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , (6)
where pi=∑αTrΠiAαTrBα=Tr(Πi⊗𝕀B)ρsubscript𝑝𝑖subscript𝛼TrsubscriptΠ𝑖subscript𝐴𝛼Trsubscript𝐵𝛼Trtensor-productsubscriptΠ𝑖subscript𝕀𝐵𝜌p_i=\sum_\alpha\mathrmTr\\Pi_iA_\alpha\\mathrmTr\B_\alpha\=% \mathrmTr\(\Pi_i\otimes\mathbbI_B)\rho\italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT roman_Tr roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT roman_Tr italic_B start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = roman_Tr ( roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ . The invariance property ρA=TrBρ=TrB(Φ𝚷⊗𝕀)ρ=∑ipiΠisubscript𝜌𝐴subscriptTr𝐵𝜌subscriptTr𝐵tensor-productsubscriptΦ𝚷𝕀𝜌subscript𝑖subscript𝑝𝑖subscriptΠ𝑖\rho_A=\mathrmTr_B\\rho\=\mathrmTr_B\(\Phi_\boldsymbol\Pi% \otimes\mathbbI)\rho\=\sum_ip_i\Pi_iitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( roman_Φ start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ⊗ blackboard_I ) italic_ρ = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT shows that the basis 𝚷𝚷\boldsymbol\Pibold_Π under which a local dephasing operation has no effect on the total quantum state must coincide with the eigenbasis of the reduced density matrix ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT Gessner and Breuer (2011). For now, we do not consider the case of degeneracies, which may complicate the situation Gessner and Breuer (2013a).
Hence, one may test for the presence of discord by realizing a local dephasing operation in the eigenbasis of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. The full quantum state is invariant under this operation if and only if it contains no discord. However, a possible change of the full quantum state under the local dephasing operation cannot be directly observed in system A𝐴Aitalic_A: Even if the state contains discord, i.e., it changes under the local dephasing, the resulting reduced density matrix ρA′=TrBρ′subscriptsuperscript𝜌′𝐴subscriptTr𝐵superscript𝜌′\rho^\prime_A=\mathrmTr_B\rho^\primeitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT with ρ′=(Φ𝚷⊗𝕀B)ρsuperscript𝜌′tensor-productsubscriptΦ𝚷subscript𝕀𝐵𝜌\rho^\prime=(\Phi_\boldsymbol\Pi\otimes\mathbbI_B)\rhoitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( roman_Φ start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ will always coincide with ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT Gessner and Breuer (2011). This can be seen by realizing that the result of Eq. (II.1) was derived without making any assumption about the full quantum state ρ𝜌\rhoitalic_ρ. We will therefore always observe that ρA=ρA′subscript𝜌𝐴subscriptsuperscript𝜌′𝐴\rho_A=\rho^\prime_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, regardless of whether ρ=ρ′𝜌superscript𝜌′\rho=\rho^\primeitalic_ρ = italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT holds or not.
Moreover, since the local dephasing operation does not act on system B𝐵Bitalic_B, one further finds that also ρB′=ρBsubscriptsuperscript𝜌′𝐵subscript𝜌𝐵\rho^\prime_B=\rho_Bitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT must always hold Gessner and Breuer (2013a). In fact, we can conclude that if any difference between the original state ρ𝜌\rhoitalic_ρ and the locally dephased reference state ρ′superscript𝜌′\rho^\primeitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT exists, it must be contained in the correlations between the two subsystems-their respective local descriptions are always unchanged by the local dephasing. Does this mean that it is impossible to observe ρ≠ρ′𝜌superscript𝜌′\rho eq\rho^\primeitalic_ρ ≠ italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT (which would constitute a witness for discord) by purely local measurements of any of the two systems? A solution can be found by considering the dynamics of this bipartite system. In fact, a change of the correlation properties of the initial state can have a strong observable impact on the reduced dynamics of one of the subsystems. This is especially well known in the theory of open quantum systems Breuer and Petruccione (2002), where the influence of initial system-environment correlations poses a considerable theoretical challenge Pechukas (1994); Alicki (1995); Pechukas (1995). Here, however, it can be exploited to reveal a change of the correlations between the subsystems to the local dynamics. Thus, even if ρ𝜌\rhoitalic_ρ and ρ′superscript𝜌′\rho^\primeitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are indistinguishable to measurements of system A𝐴Aitalic_A at some initial time t=0𝑡0t=0italic_t = 0, they may become locally distinguishable after the two systems have been interacting for a time t>0𝑡0t>0italic_t >0.
Let us assume that systems A𝐴Aitalic_A and B𝐵Bitalic_B are subject to some interaction. For simplicity, we consider a unitary evolution of the composite system 111This assumption is not essential for the local detection method Gessner and Breuer (2013a)., such that the evolution in subsystem A𝐴Aitalic_A, given the initial state ρ𝜌\rhoitalic_ρ, is governed by
ρA(t)=TrBU(t)ρU†(t).subscript𝜌𝐴𝑡subscriptTr𝐵𝑈𝑡𝜌superscript𝑈†𝑡\displaystyle\rho_A(t)=\mathrmTr_B\U(t)\rho U^\dagger(t)\.italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_U ( italic_t ) italic_ρ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) . (7)
If the state ρ𝜌\rhoitalic_ρ was subject to local dephasing before the time evolution, the state of system A𝐴Aitalic_A at time t𝑡titalic_t instead reads
ρA′(t)=TrBU(t)ρ′U†(t).subscriptsuperscript𝜌′𝐴𝑡subscriptTr𝐵𝑈𝑡superscript𝜌′superscript𝑈†𝑡\displaystyle\rho^\prime_A(t)=\mathrmTr_B\U(t)\rho^\primeU^\dagger% (t)\.italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_U ( italic_t ) italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) . (8)
We had already noted that ρA(0)=ρA′(0)subscript𝜌𝐴0subscriptsuperscript𝜌′𝐴0\rho_A(0)=\rho^\prime_A(0)italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( 0 ) = italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( 0 ), regardless of the properties of ρ𝜌\rhoitalic_ρ. However, if we observe
ρA(t)≠ρA′(t)subscript𝜌𝐴𝑡subscriptsuperscript𝜌′𝐴𝑡\displaystyle\rho_A(t) eq\rho^\prime_A(t)italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) ≠ italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) (9)
at a later time t>0𝑡0t>0italic_t >0 we can safely conclude that ρ≠ρ′𝜌superscript𝜌′\rho eq\rho^\primeitalic_ρ ≠ italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT which implies the presence of discord in the state ρ𝜌\rhoitalic_ρ Gessner and Breuer (2011, 2013a). All of the necessary steps, i.e.,
•
Finding the eigenbasis of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT
•
Observing the time evolution of ρA(t)subscript𝜌𝐴𝑡\rho_A(t)italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t )
•
Local dephasing of the initial state in the eigenbasis of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT
•
Observing the time evolution of ρA′(t)subscriptsuperscript𝜌′𝐴𝑡\rho^\prime_A(t)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t )
can be carried out with strictly local access only to system A𝐴Aitalic_A, whereas no control or even knowledge of system B𝐵Bitalic_B is required. The local detection protocol is illustrated in the diagram in Fig. 1.
II.2 Quantifying discord with local operations
Detecting the mere presence of discord by following the protocol outlined above does not provide any quantitative information about the discord of the state ρ𝜌\rhoitalic_ρ. Considering that discord is a rather ubiquitous phenomenon Ferraro et al. (2010), it is also relevant to estimate how strongly discordant a given initial state is. Certain quantifiers furthermore allow for an operational interpretation and therefore directly quantify how well a certain quantum information task can be carried out Streltsov et al. (2012); Chuan et al. (2012); Streltsov et al. (2011b); Piani et al. (2011); Girolami et al. (2014, 2013).
A straight-forward way to quantify discord emerges from the local dephasing operation. From the discussion above, we know that ρ′superscript𝜌′\rho^\primeitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT differs from ρ𝜌\rhoitalic_ρ if and only if ρ𝜌\rhoitalic_ρ contains discord. A simple quantifier of discord is thus given by the dephasing disturbance Luo (2008); Gessner (2015), expressed by the trace distance
D(ρ)=∥ρ-ρ′∥,𝐷𝜌norm𝜌superscript𝜌′\displaystyle D(\rho)=\|\rho-\rho^\prime\|,italic_D ( italic_ρ ) = ∥ italic_ρ - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ , (10)
where ∥X∥=1/2TrX†Xnorm𝑋12Trsuperscript𝑋†𝑋\|X\|=1/2\mathrmTr\sqrtX^\daggerX∥ italic_X ∥ = 1 / 2 roman_T roman_r square-root start_ARG italic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_X end_ARG Nielsen and Chuang (2000). The trace distance has several appealing properties, most notably for our purposes is its contractivity under trace-preserving and positive operations Ruskai (1994). The unitary time evolution and the partial trace operation are both positive operations (they map positive operators, such as the density operator, to positive operators) Nielsen and Chuang (2000). Thus, using the contractivity property we find that the locally observable distance between the reduced density matrices ρA(t)subscript𝜌𝐴𝑡\rho_A(t)italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) and ρA′(t)subscriptsuperscript𝜌′𝐴𝑡\rho^\prime_A(t)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) provides a lower bound for the dephasing disturbance Gessner and Breuer (2013a):
d(t)=∥ρA(t)-ρA′(t)∥𝑑𝑡normsubscript𝜌𝐴𝑡subscriptsuperscript𝜌′𝐴𝑡\displaystyle d(t)=\|\rho_A(t)-\rho^\prime_A(t)\|italic_d ( italic_t ) = ∥ italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) ∥ =∥TrBU(t)(ρ-ρ′)U†(t)∥absentnormsubscriptTr𝐵𝑈𝑡𝜌superscript𝜌′superscript𝑈†𝑡\displaystyle=\|\mathrmTr_B\U(t)(\rho-\rho^\prime)U^\dagger(t)\\|= ∥ roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_U ( italic_t ) ( italic_ρ - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ∥
≤∥U(t)(ρ-ρ′)U†(t)∥absentnorm𝑈𝑡𝜌superscript𝜌′superscript𝑈†𝑡\displaystyle\leq\|U(t)(\rho-\rho^\prime)U^\dagger(t)\|≤ ∥ italic_U ( italic_t ) ( italic_ρ - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ∥
=∥ρ-ρ′∥.absentnorm𝜌superscript𝜌′\displaystyle=\|\rho-\rho^\prime\|.= ∥ italic_ρ - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ . (11)
The above inequality holds for all t≥0𝑡0t\geq 0italic_t ≥ 0, hence one may optimize the locally accessible lower bound by observing the time evolution of the local system for as long as possible and then taking the maximum distance Gessner et al. (2014a)
dmax=maxt∥ρS(t)-ρS′(t)∥≤∥ρ-ρ′∥.subscript𝑑subscript𝑡normsubscript𝜌𝑆𝑡subscriptsuperscript𝜌′𝑆𝑡norm𝜌superscript𝜌′\displaystyle d_\max=\max_t\|\rho_S(t)-\rho^\prime_S(t)\|\leq\|\rho-% \rho^\prime\|.italic_d start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT = roman_max start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ italic_ρ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_t ) - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_t ) ∥ ≤ ∥ italic_ρ - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ . (12)
The above definition assumes that the local dephasing operation is unique, which requires that the eigenbasis of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is unambiguous. This, however, is not the case if degeneracies are present in the spectrum of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. In these cases the dephasing disturbance does not produce a suitable quantifier of discord Girolami et al. (2011); Campbell et al. (2011). This problem can be circumvented by including a minimization over all possible dephasing bases. Recalling the 𝚷𝚷\boldsymbol\Pibold_Π-dependent definition (5) of a general dephasing operation, we introduce the minimal dephasing disturbance Gessner et al. (2014b); Gessner (2015)
Dmin(ρ)=min𝚷∥ρ-(Φ𝚷⊗𝕀)ρ∥.subscript𝐷𝜌subscript𝚷norm𝜌tensor-productsubscriptΦ𝚷𝕀𝜌\displaystyle D_\min(\rho)=\min_\boldsymbol\Pi\|\rho-(\Phi_\boldsymbol% \Pi\otimes\mathbbI)\rho\|.italic_D start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_ρ ) = roman_min start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ∥ italic_ρ - ( roman_Φ start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ⊗ blackboard_I ) italic_ρ ∥ . (13)
This measure in fact quantifies the maximal amount of entanglement which can be activated from discord in a measurement (minimal entanglement potential) Streltsov et al. (2011b); Piani et al. (2011); Nakano et al. (2013); Adesso et al. (2014) when the accessible subsystem is a qubit, i.e.,ℋA=ℂ2subscriptℋ𝐴superscriptℂ2\mathcalH_A=\mathbbC^2caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = blackboard_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.
A locally accessible bound for the minimal dephasing disturbance can be obtained by dephasing over different local bases instead of just the eigenbasis of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT Gessner et al. (2014b). We introduce
ρA𝚷(t)=TrBU(t)(Φ𝚷⊗𝕀)ρU†(t)subscriptsuperscript𝜌𝚷𝐴𝑡subscriptTr𝐵𝑈𝑡tensor-productsubscriptΦ𝚷𝕀𝜌superscript𝑈†𝑡\displaystyle\rho^\boldsymbol\Pi_A(t)=\mathrmTr_B\U(t)(\Phi_% \boldsymbol\Pi\otimes\mathbbI)\rho U^\dagger(t)\italic_ρ start_POSTSUPERSCRIPT bold_Π end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_U ( italic_t ) ( roman_Φ start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ⊗ blackboard_I ) italic_ρ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) (14)
and the corresponding local trace distance
d𝚷(t)=∥ρA(t)-ρA𝚷(t)∥.subscript𝑑𝚷𝑡normsubscript𝜌𝐴𝑡subscriptsuperscript𝜌𝚷𝐴𝑡\displaystyle d_\boldsymbol\Pi(t)=\|\rho_A(t)-\rho^\boldsymbol\Pi_A% (t)\|.italic_d start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ( italic_t ) = ∥ italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) - italic_ρ start_POSTSUPERSCRIPT bold_Π end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) ∥ . (15)
A more rigorous bound for discord than Eq. (12) is then obtained as Gessner et al. (2014b)
dmin(ρ)=maxtmin𝚷d𝚷(t)≤Dmin(ρ).subscript𝑑𝜌subscript𝑡subscript𝚷subscript𝑑𝚷𝑡subscript𝐷𝜌\displaystyle d_\min(\rho)=\max_t\min_\boldsymbol\Pid_\boldsymbol\Pi% (t)\leq D_\min(\rho).italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_ρ ) = roman_max start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT roman_min start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ( italic_t ) ≤ italic_D start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_ρ ) . (16)
To measure the above quantity, one first records the time evolution of ρS𝚷(t)subscriptsuperscript𝜌𝚷𝑆𝑡\rho^\boldsymbol\Pi_S(t)italic_ρ start_POSTSUPERSCRIPT bold_Π end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_t ) for different dephasing bases 𝚷𝚷\boldsymbol\Pibold_Π. At each time t𝑡titalic_t, the minimum of all d𝚷(t)subscript𝑑𝚷𝑡d_\boldsymbol\Pi(t)italic_d start_POSTSUBSCRIPT bold_Π end_POSTSUBSCRIPT ( italic_t ) is obtained, the minimum being taken over all 𝚷𝚷\boldsymbol\Pibold_Π. Then, within the set of minima one finds the maximum value over all times t𝑡titalic_t to obtain the strongest available local witness. Ideally, the optimization over 𝚷𝚷\boldsymbol\Pibold_Π should be carried out over all possible bases, which is experimentally impossible. In a realistic situation a systematic sampling over a sufficiently closely spaced grid of basis vectors can yield a good estimate with reasonable overhead, see, e.g., Adesso et al. (2014).
II.3 Pure states: Locally accessible lower bound for negativity
Let us consider the simple case of a pure state with a controllable qubit subspace, ℋA=ℂ2subscriptℋ𝐴superscriptℂ2\mathcalH_A=\mathbbC^2caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = blackboard_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. As mentioned before, the concept of discord reduces for pure states to entanglement-in the absence of classical mixing, this is the only form of correlation that can be present. In this case, the dephasing disturbance (10) can be evaluated analytically and yields Gessner et al. (2014b); Gessner (2015)
D(ρ)=𝒩(ρ),𝐷𝜌𝒩𝜌\displaystyle D(\rho)=\mathcalN(\rho),italic_D ( italic_ρ ) = caligraphic_N ( italic_ρ ) , (17)
where 𝒩𝒩\mathcalNcaligraphic_N denotes the negativity Vidal and Werner (2002),
𝒩(ρ)=∥ρΓ∥-12,𝒩𝜌normsuperscript𝜌Γ12\displaystyle\mathcalN(\rho)=\frac\rho^\Gamma\2,caligraphic_N ( italic_ρ ) = divide start_ARG ∥ italic_ρ start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT ∥ - 1 end_ARG start_ARG 2 end_ARG , (18)
and ρΓsuperscript𝜌Γ\rho^\Gammaitalic_ρ start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT is the partial transpose of ρ𝜌\rhoitalic_ρ .
On the other hand, the minimal dephasing disturbance (13) coincides with the minimal entanglement potential, which for pure states also reduces to the negativity Piani and Adesso (2012). Hence, in the above scenario, the dephasing disturbance coincides (10) with the minimal dephasing disturbance (13), and the minimum is achieved by dephasing in the eigenbasis of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT Gessner et al. (2014b). Moreover, the local distance (12) yields a locally accessible lower bound for the negativity.
II.4 Efficacy of the local detection method
Discord can ultimately be traced back to those two-body coherences that are present in ρ𝜌\rhoitalic_ρ but are no longer found in ρ′superscript𝜌′\rho^\primeitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, i.e., after local dephasing. Those coherences are not detectable in either of the two subsystems. Therefore, the performance of the local detection method depends crucially on the interacting dynamics between the two subsystems. Its role is to map these initially hidden two-body coherences to measurable elements of the reduced density matrix of system A𝐴Aitalic_A at a later time.
Certainly some dynamical processes will work better than others in detecting these correlations. For instance, if no interaction between the two systems were present, i.e., U(t)=UA(t)⊗UB(t)𝑈𝑡tensor-productsubscript𝑈𝐴𝑡subscript𝑈𝐵𝑡U(t)=U_A(t)\otimes U_B(t)italic_U ( italic_t ) = italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t ) ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_t ), this task could never be achieved Gessner and Breuer (2013a). In the course of this article we will observe a number of different time evolutions and thereby explore the limitations of the local detection method based on these examples. Let us already mention that early estimates for the efficacy of the method have been obtained based on a formulation in terms of the Hilbert-Schmidt distance, which allows for analytical evaluation of measure-theoretic averages over the unitary group Gessner and Breuer (2013b). It was shown that Gessner and Breuer (2011)
∫𝑑μ(U)∥TrBU(ρ-ρ′)U†∥22=dA2dB-dBdA2dB2-1∥ρ-ρ′∥22,differential-d𝜇𝑈subscriptsuperscriptnormsubscriptTr𝐵𝑈𝜌superscript𝜌′superscript𝑈†22superscriptsubscript𝑑𝐴2subscript𝑑𝐵subscript𝑑𝐵superscriptsubscript𝑑𝐴2superscriptsubscript𝑑𝐵21subscriptsuperscriptnorm𝜌superscript𝜌′22\displaystyle\int d\mu(U)\|\mathrmTr_B\U(\rho-\rho^\prime)U^\dagger\% \|^2_2=\fracd_A^2d_B-d_Bd_A^2d_B^2-1\|\rho-\rho^% \prime\|^2_2,∫ italic_d italic_μ ( italic_U ) ∥ roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_U ( italic_ρ - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = divide start_ARG italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 end_ARG ∥ italic_ρ - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (19)
with the Hilbert-Schmidt norm ∥X∥22=TrX†Xsuperscriptsubscriptnorm𝑋22Trsuperscript𝑋†𝑋\|X\|_2^2=\mathrmTrX^\daggerX∥ italic_X ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = roman_Tr italic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_X, dAsubscript𝑑𝐴d_Aitalic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and dBsubscript𝑑𝐵d_Bitalic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT being the dimensions of the Hilbert spaces ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and ℋBsubscriptℋ𝐵\mathcalH_Bcaligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, respectively, and dμ𝑑𝜇d\muitalic_d italic_μ representing the Haar measure on the unitary group. These group-theoretic methods further allow for analytical evaluation of the variance corresponding to the above average Gessner and Breuer (2013a), as well as time-dependent averages with respect to more realistic random matrix ensembles Gessner and Breuer (2013b). These results show that the locally observable signal, obtained from a generic dynamical system, is directly proportional to the discord of the initial state. Hence a generic unitary evolution is expected to reveal the quantum discord based on the local detection method. For further details on the Haar-measure integration techniques and additional numerical and analytical studies of the local detection method in this context, we refer to Refs. Gessner (2011); Gessner and Breuer (2011, 2013a, 2013b). Note, however, that the Hilbert-Schmidt distance is not contractive under positive maps. This can lead to unphysical behavior of Hilbert-Schmidt based quantifiers for discord and related quantities Ozawa (2000); Piani (2012). For this reason, it is generally recommended to use the trace distance instead Paula et al. (2013).
We also observe that the proportionality factor on the right-hand side of Eq. (19) shrinks to zero as dBsubscript𝑑𝐵d_Bitalic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT increases. This might suggest the conclusion that the signal becomes undetectably small when the observable system is coupled to a truly infinite environment. The result (19) however makes statements about generic evolutions which are well represented by the average over all unitaries. Generally speaking, systems that lead to such a dynamics are typically strongly chaotic, and deviations from the average result (19) are certainly expected. In common situations one might as well encounter highly non-generic evolutions, in particular in quantum optical systems. Later in this article we will discuss a successful experimental detection of system-environment discord by means of the local detection method where the accessible system couples to an infinite-dimensional, Markovian (memoryless) environment Tang et al. (2015).
III Experiments
The local detection method has been used to reveal discord in experiments with photons Tang et al. (2015); Cialdi et al. (2014) and trapped ions Gessner et al. (2014a). In all experimental applications reported so far, the controllable subspace was two-dimensional, whereas correlations were detected with ancilla systems ranging from two-dimensional systems to continuous variables.
III.1 Trapped-ion experiment
An experimental realization of the local detection method was first reported in Gessner et al. (2014a). A single trapped ion is used to simulate both a two-dimensional, locally accessible quantum system by means of its electronic degree of freedom, and a bosonic ancilla system, comprised of the same ion’s motional degree of freedom. Since the ion is confined in a harmonic trapping potential, the ancilla system is described by a quantum harmonic oscillator Wineland et al. (1998); Leibfried et al. (2003). When the ion is driven by a laser whose detuning from the ion’s resonance transition coincides with the frequency of the harmonic motion, an interaction between the two degrees of freedom, described to a good approximation by the anti-Jaynes-Cummings Hamiltonian
H=ℏΩη2(σ+a†+σ-a).𝐻Planck-constant-over-2-piΩ𝜂2subscript𝜎superscript𝑎†subscript𝜎𝑎\displaystyle H=\frac\hbar\Omega\eta2(\sigma_+a^\dagger+\sigma_-a).italic_H = divide start_ARG roman_ℏ roman_Ω italic_η end_ARG start_ARG 2 end_ARG ( italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT + italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT italic_a ) . (20)
is evoked Wineland et al. (1998, 1992); Leibfried et al. (2003); Blockley et al. (1992); Häffner et al. (2008); Gessner (2015). Here, a𝑎aitalic_a and a†superscript𝑎†a^\daggeritalic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT are the bosonic creation and annihilation operators of the harmonic oscillator mode, σ+subscript𝜎\sigma_+italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT, σ-subscript𝜎\sigma_-italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT denote the ladder operators for the electronic qubit system, ΩΩ\Omegaroman_Ω denotes the Rabi frequency and η𝜂\etaitalic_η is the Lamb-Dicke parameter Leibfried et al. (2003). In the above expression we assumed, for ease of notation, the Lamb-Dicke regime, i.e., η⟨(a+a†)2⟩≪1much-less-than𝜂delimited-⟨⟩superscript𝑎superscript𝑎†21\eta\sqrt\langle(a+a^\dagger)^2\rangle\ll 1italic_η square-root start_ARG ⟨ ( italic_a + italic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ end_ARG ≪ 1; the analysis of the trapped-ion experiment, however, can be extended beyond this regime and beyond ideal experimental conditions Gessner et al. (2014a); for a detailed account see Gessner (2015). The Hamiltonian (20) induces a coherent coupling between the states |g,n⟩ket𝑔𝑛|g,n\rangle| italic_g , italic_n ⟩ and |e,n+1⟩ket𝑒𝑛1|e,n+1\rangle| italic_e , italic_n + 1 ⟩, where |g⟩ket𝑔|g\rangle| italic_g ⟩ and |e⟩ket𝑒|e\rangle| italic_e ⟩ describe the electronic ground- and excited states, respectively, and |n⟩ket𝑛|n\rangle| italic_n ⟩ is a Fock state of the harmonic motion. An ion initially prepared in the state |g,n⟩ket𝑔𝑛|g,n\rangle| italic_g , italic_n ⟩ hence undergoes a Rabi oscillation of the form
|Ψ(t)⟩=U(t)|g,n⟩=cos(Ωn2t)|g,n⟩+sin(Ωn2t)|e,n+1⟩,ketΨ𝑡𝑈𝑡ket𝑔𝑛subscriptΩ𝑛2𝑡ket𝑔𝑛subscriptΩ𝑛2𝑡ket𝑒𝑛1\displaystyle|\Psi(t)\rangle=U(t)|g,n\rangle=\cos\left(\frac\Omega_n2t% \right)|g,n\rangle+\sin\left(\frac\Omega_n2t\right)|e,n+1\rangle,| roman_Ψ ( italic_t ) ⟩ = italic_U ( italic_t ) | italic_g , italic_n ⟩ = roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t ) | italic_g , italic_n ⟩ + roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t ) | italic_e , italic_n + 1 ⟩ , (21)
where U(t)=exp(-iHt/ℏ)𝑈𝑡𝑖𝐻𝑡Planck-constant-over-2-piU(t)=\exp(-iHt/\hbar)italic_U ( italic_t ) = roman_exp ( - italic_i italic_H italic_t / roman_ℏ ) and Ωn=n+1ηΩsubscriptΩ𝑛𝑛1𝜂Ω\Omega_n=\sqrtn+1\eta\Omegaroman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = square-root start_ARG italic_n + 1 end_ARG italic_η roman_Ω.
Initially, the system is prepared by optical pumping of the electronic level to the ground state and laser cooling of the motional degree of freedom, leading to the product state
ρ0=|g⟩⟨g|⊗∑n=0∞pn|n⟩⟨n|.subscript𝜌0tensor-productket𝑔bra𝑔superscriptsubscript𝑛0subscript𝑝𝑛ket𝑛bra𝑛\displaystyle\rho_0=|g\rangle\langle g|\otimes\sum_n=0^\inftyp_n|n% \rangle\langle n|.italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = | italic_g ⟩ ⟨ italic_g | ⊗ ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_n ⟩ ⟨ italic_n | . (22)
The thermal populations pn=n¯n/(n¯+1)n+1subscript𝑝𝑛superscript¯𝑛𝑛superscript¯𝑛1𝑛1p_n=\barn^n/(\barn+1)^n+1italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = over¯ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT / ( over¯ start_ARG italic_n end_ARG + 1 ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT can be given in terms of the mean phonon number n¯¯𝑛\barnover¯ start_ARG italic_n end_ARG; see, e.g., Leibfried et al. (2003). When this initial state is exposed to the laser interaction for a duration t0subscript𝑡0t_0italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (state preparation), it evolves as
ρ(t0)𝜌subscript𝑡0\displaystyle\rho(t_0)italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) =U(t0)ρ0U†(t0)absent𝑈subscript𝑡0subscript𝜌0superscript𝑈†subscript𝑡0\displaystyle=U(t_0)\rho_0U^\dagger(t_0)= italic_U ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )
=∑n=0∞pn[cos(Ωn2t0)2|g,n⟩⟨g,n|+sin(Ωn2t0)cos(Ωn2t0)|g,n⟩⟨e,n+1|fragmentssuperscriptsubscript𝑛0subscript𝑝𝑛fragments[superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2|g,n⟩fragments⟨g,n|fragments(subscriptΩ𝑛2subscript𝑡0)fragments(subscriptΩ𝑛2subscript𝑡0)|g,n⟩fragments⟨e,n1|\displaystyle=\sum_n=0^\inftyp_n\left[\cos\left(\frac\Omega_n2t_0% \right)^2|g,n\rangle\langle g,n|+\sin\left(\frac\Omega_n2t_0\right)% \cos\left(\frac\Omega_n2t_0\right)|g,n\rangle\langle e,n+1|\right.= ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_g , italic_n ⟩ ⟨ italic_g , italic_n | + roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) | italic_g , italic_n ⟩ ⟨ italic_e , italic_n + 1 |
+sin(Ωn2t0)cos(Ωn2t0)|e,n+1⟩⟨g,n|+sin(Ωn2t0)2|e,n+1⟩⟨e,n+1|].fragmentsfragments(subscriptΩ𝑛2subscript𝑡0)fragments(subscriptΩ𝑛2subscript𝑡0)|e,n1⟩⟨g,n|superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2|e,n1⟩⟨e,n1|].\displaystyle\hskip 42.67912pt\left.+\sin\left(\frac\Omega_n2t_0\right% )\cos\left(\frac\Omega_n2t_0\right)|e,n+1\rangle\langle g,n|+\sin\left% (\frac\Omega_n2t_0\right)^2|e,n+1\rangle\langle e,n+1|\right].+ roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) | italic_e , italic_n + 1 ⟩ ⟨ italic_g , italic_n | + roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_e , italic_n + 1 ⟩ ⟨ italic_e , italic_n + 1 | ] . (23)
The goal is now to detect the presence of discord between the electronic and motional degrees of freedom in the state ρ(t0)𝜌subscript𝑡0\rho(t_0)italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). To this end, the local detection method is employed, which allows us to limit experimental access to the electronic degree of freedom. The first task is to obtain the eigenbasis of the reduced density matrix, which determines the basis for the local dephasing operation. By tracing over the motional degrees of freedom (system B𝐵Bitalic_B), we obtain the quantum state of the qubit (system A𝐴Aitalic_A),
ρA(t0)=TrBρ(t0)subscript𝜌𝐴subscript𝑡0subscriptTr𝐵𝜌subscript𝑡0\displaystyle\rho_A(t_0)=\mathrmTr_B\rho(t_0)italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) =∑n=0∞pn[cos(Ωn2t0)2|g⟩⟨g|+sin(Ωn2t0)2|e⟩⟨e|],fragmentssuperscriptsubscript𝑛0subscript𝑝𝑛fragments[superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2|g⟩fragments⟨g|superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2|e⟩fragments⟨e|],\displaystyle=\sum_n=0^\inftyp_n\left[\cos\left(\frac\Omega_n2t_0% \right)^2|g\rangle\langle g|+\sin\left(\frac\Omega_n2t_0\right)^2% |e\rangle\langle e|\right],= ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_g ⟩ ⟨ italic_g | + roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_e ⟩ ⟨ italic_e | ] , (24)
which is diagonal in the basis g⟩,ket𝑔ket𝑒\ italic_e ⟩ at all times.
To detect the discord of the state at time t0subscript𝑡0t_0italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the dynamical evolution is interrupted and a local dephasing is performed by projecting onto the local subspaces, spanned by |g⟩ket𝑔|g\rangle| italic_g ⟩ and |e⟩ket𝑒|e\rangle| italic_e ⟩, respectively. Experimentally this is achieved by inducing a weak ac-Stark shift on the ground state for a well-controlled period of time. To Discord servers , another laser which off-resonantly addresses a transition between the ground state and another short-lived excited state is used. This adds a relative phase shift to any superposition that involves the states |e⟩ket𝑒|e\rangle| italic_e ⟩ and |g⟩ket𝑔|g\rangle| italic_g ⟩. By performing an average over a suitably chosen family of phase shifts, the relative phase relation between the states |e⟩ket𝑒|e\rangle| italic_e ⟩ and |g⟩ket𝑔|g\rangle| italic_g ⟩ can be completely removed, thereby effectively realizing the local dephasing operation Gessner et al. (2014a); Gessner (2015). This technique can be combined with local, coherent laser manipulations to achieve dephasing in an arbitrary basis Gessner et al. (2014a); Gessner (2015).
The total state after local dephasing is then given as
ρ′(t0)superscript𝜌′subscript𝑡0\displaystyle\rho^\prime(t_0)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) =(Φ⊗𝕀)ρ(t0)absenttensor-productΦ𝕀𝜌subscript𝑡0\displaystyle=(\Phi\otimes\mathbbI)\rho(t_0)= ( roman_Φ ⊗ blackboard_I ) italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )
=∑i∈e,g(|i⟩⟨i|⊗𝕀B)ρ(t0)(|i⟩⟨i|⊗𝕀B)absentsubscript𝑖𝑒𝑔tensor-productket𝑖bra𝑖subscript𝕀𝐵𝜌subscript𝑡0tensor-productket𝑖bra𝑖subscript𝕀𝐵\displaystyle=\sum_i\in\e,g\(|i\rangle\langle i|\otimes\mathbbI_B)\rho% (t_0)(|i\rangle\langle i|\otimes\mathbbI_B)= ∑ start_POSTSUBSCRIPT italic_i ∈ italic_e , italic_g end_POSTSUBSCRIPT ( | italic_i ⟩ ⟨ italic_i | ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( | italic_i ⟩ ⟨ italic_i | ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT )
=∑n=0∞pn[cos(Ωn2t0)2|g,n⟩⟨g,n|+sin(Ωn2t0)2|e,n+1⟩⟨e,n+1|].fragmentssuperscriptsubscript𝑛0subscript𝑝𝑛fragments[superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2|g,n⟩fragments⟨g,n|superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2|e,n1⟩fragments⟨e,n1|].\displaystyle=\sum_n=0^\inftyp_n\left[\cos\left(\frac\Omega_n2t_0% \right)^2|g,n\rangle\langle g,n|+\sin\left(\frac\Omega_n2t_0\right)% ^2|e,n+1\rangle\langle e,n+1|\right].= ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_g , italic_n ⟩ ⟨ italic_g , italic_n | + roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_e , italic_n + 1 ⟩ ⟨ italic_e , italic_n + 1 | ] . (25)
By construction, ρ(t0)𝜌subscript𝑡0\rho(t_0)italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and ρ′(t0)superscript𝜌′subscript𝑡0\rho^\prime(t_0)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) only differ inasmuch as ρ(t0)𝜌subscript𝑡0\rho(t_0)italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) contains discord while ρ′(t0)superscript𝜌′subscript𝑡0\rho^\prime(t_0)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) does not, thus, comparison with Eq. (III.1) now allows us to precisely identify those terms that produce the discord in ρ(t0)𝜌subscript𝑡0\rho(t_0)italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). As anticipated, these are the two-body coherences |e,n⟩⟨g,n+1|ket𝑒𝑛bra𝑔𝑛1|e,n\rangle\langle g,n+1|| italic_e , italic_n ⟩ ⟨ italic_g , italic_n + 1 | (and its adjoint counterpart). Since these matrix elements are indeed off-diagonal in both of the sub-systems, any local measurement of the qubit or the ions’ motion will be unable to detect their presence. One readily confirms that ρ′(t0)superscript𝜌′subscript𝑡0\rho^\prime(t_0)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) has the same reduced density matrices as ρ(t0)𝜌subscript𝑡0\rho(t_0)italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Before we proceed to study the signature of discord in the subsequent qubit dynamics, we evaluate the dephasing disturbance, which reads Gessner et al. (2014a),
D(ρ(t0))=∑n=0∞pn|sin(Ωn2t0)cos(Ωn2t0)|.𝐷𝜌subscript𝑡0superscriptsubscript𝑛0subscript𝑝𝑛subscriptΩ𝑛2subscript𝑡0subscriptΩ𝑛2subscript𝑡0\displaystyle D(\rho(t_0))=\sum_n=0^\inftyp_n\left|\sin\left(\frac% \Omega_n2t_0\right)\cos\left(\frac\Omega_n2t_0\right)\right|.italic_D ( italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) | . (26)
As expected, this quantifies precisely the magnitude of the above-mentioned two-body coherences.
Despite being hidden from local measurements, the discord contained in the state ρ(t0)𝜌subscript𝑡0\rho(t_0)italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) can be detected at a later time by observing deviating evolutions of the reduced density matrices ρA(t0+t1)subscript𝜌𝐴subscript𝑡0subscript𝑡1\rho_A(t_0+t_1)italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and ρA′(t0+t1)subscriptsuperscript𝜌′𝐴subscript𝑡0subscript𝑡1\rho^\prime_A(t_0+t_1)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). The state is again subjected to the laser interaction, for a duration t1subscript𝑡1t_1italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (detection). The evolution of the unperturbed state was given in Eq. (24), whereas the dephased state evolves as
ρA′(t1+t0)subscriptsuperscript𝜌′𝐴subscript𝑡1subscript𝑡0\displaystyle\rho^\prime_A(t_1+t_0)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) =TrBU(t1)ρ′(t0)U†(t1)absentsubscriptTr𝐵𝑈subscript𝑡1superscript𝜌′subscript𝑡0superscript𝑈†subscript𝑡1\displaystyle=\mathrmTr_B\U(t_1)\rho^\prime(t_0)U^\dagger(t_1)\= roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_U ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )
=∑n=0∞pn[(cos(Ωn2t0)2cos(Ωn2t1)2+sin(Ωn2t0)2sin(Ωn2t1)2)|g⟩⟨g|fragmentssuperscriptsubscript𝑛0subscript𝑝𝑛fragments[fragments(superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2superscriptfragments(subscriptΩ𝑛2subscript𝑡1)2superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2superscriptfragments(subscriptΩ𝑛2subscript𝑡1)2)|g⟩fragments⟨g|\displaystyle=\sum_n=0^\inftyp_n\left[\left(\cos\left(\frac\Omega_n% 2t_0\right)^2\cos\left(\frac\Omega_n2t_1\right)^2+\sin\left(% \frac\Omega_n2t_0\right)^2\sin\left(\frac\Omega_n2t_1\right)% ^2\right)|g\rangle\langle g|\right.= ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT [ ( roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) | italic_g ⟩ ⟨ italic_g |
+(cos(Ωn2t0)2sin(Ωn2t1)2+sin(Ωn2t0)2cos(Ωn2t1)2)|e⟩⟨e|].fragmentsfragments(superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2superscriptfragments(subscriptΩ𝑛2subscript𝑡1)2superscriptfragments(subscriptΩ𝑛2subscript𝑡0)2superscriptfragments(subscriptΩ𝑛2subscript𝑡1)2)|e⟩⟨e|].\displaystyle\hskip 36.98866pt+\left.\left(\cos\left(\frac\Omega_n2t_0% \right)^2\sin\left(\frac\Omega_n2t_1\right)^2+\sin\left(\frac% \Omega_n2t_0\right)^2\cos\left(\frac\Omega_n2t_1\right)^2% \right)|e\rangle\langle e|\right].+ ( roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + roman_sin ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_cos ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) | italic_e ⟩ ⟨ italic_e | ] . (27)
The difference between the two evolutions can be observed by measuring the excited-state population. In the trapped-ion experiment, this is realized by a highly efficient fluorescence readout method Wineland et al. (1998); Leibfried et al. (2003). We observe the difference
de(t0,t1)subscript𝑑𝑒subscript𝑡0subscript𝑡1\displaystyle d_e(t_0,t_1)italic_d start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) =⟨e|ρA(t1+t0)-ρA′(t1+t0)|e⟩absentquantum-operator-product𝑒subscript𝜌𝐴subscript𝑡1subscript𝑡0subscriptsuperscript𝜌′𝐴subscript𝑡1subscript𝑡0𝑒\displaystyle=\langle e|\rho_A(t_1+t_0)-\rho^\prime_A(t_1+t_0)|e\rangle= ⟨ italic_e | italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) | italic_e ⟩
=12∑n=0∞pnsin(Ωnt0)sin(Ωnt1),absent12superscriptsubscript𝑛0subscript𝑝𝑛subscriptΩ𝑛subscript𝑡0subscriptΩ𝑛subscript𝑡1\displaystyle=\frac12\sum_n=0^\inftyp_n\sin\left(\Omega_nt_0% \right)\sin\left(\Omega_nt_1\right),= divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT roman_sin ( roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) roman_sin ( roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , (28)
where we have used the identity 2sinαcosα=sin2α2𝛼𝛼2𝛼2\sin\alpha\cos\alpha=\sin 2\alpha2 roman_sin italic_α roman_cos italic_α = roman_sin 2 italic_α. The fact that both states ρA(t0)subscript𝜌𝐴subscript𝑡0\rho_A(t_0)italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and ρA′(t0)subscriptsuperscript𝜌′𝐴subscript𝑡0\rho^\prime_A(t_0)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) are diagonal in the basis g⟩,ket𝑔ket𝑒\g\rangle, italic_e ⟩ allows us to determine the trace distance directly from the excited-state deviations as
d(t0,t1)=∥ρA(t1+t0)-ρA′(t1+t0)∥=|de(t0,t1)|.𝑑subscript𝑡0subscript𝑡1normsubscript𝜌𝐴subscript𝑡1subscript𝑡0subscriptsuperscript𝜌′𝐴subscript𝑡1subscript𝑡0subscript𝑑𝑒subscript𝑡0subscript𝑡1\displaystyle d(t_0,t_1)=\|\rho_A(t_1+t_0)-\rho^\prime_A(t_1+t% _0)\|=|d_e(t_0,t_1)|.italic_d ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = ∥ italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) - italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∥ = | italic_d start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) | . (29)
This quantity is locally measurable in subsystem A𝐴Aitalic_A and provides a lower bound to the dephasing disturbance (26), a global property of the full quantum state. Whenever a nonzero deviation (29) is observed, we can conclude that ρ(t0)𝜌subscript𝑡0\rho(t_0)italic_ρ ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) contained nonzero discord, and the magnitude of the local trace distance further allows for a quantification of the initial discord.
The experimental protocol and the measured local witness for discord is shown in Fig. 2. The experiment was performed for two different initial temperatures of the ion’s motion. The theoretical description used for the plot includes the effects of experimental imperfections, such as small detunings and fluctuating parameters Gessner et al. (2014a); Gessner (2015). For both environmental temperatures, a strong signature of the initial discord is observed.
Finally, the tightest bound to the dephasing disturbance (26) is obtained by the largest deviation dmaxsubscript𝑑d_\maxitalic_d start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT, as introduced in Eq. (12). This quantity is plotted for different t0subscript𝑡0t_0italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT in Fig. 3. Comparison with the predictions show that the locally recovered signatures of the initial discord are as large as theoretically possible, and almost saturate the inequality (12) in the case of the low-temperature initial state.
The signal of the higher-temperature state is not as pronounced as the one obtained from the low-temperature distribution. The question arises whether the local signal would vanish completely if the temperature was increased even further, as one might expect if the usability of the local detection method was limited to effectively finite-dimensional environments. This, is however not the case Gessner (2015). A simple estimate of the signal for higher temperatures can be obtained by fixing the preparation and detection pulse durations to the value Ω0t0=Ω0t1=π/2subscriptΩ0subscript𝑡0subscriptΩ0subscript𝑡1𝜋2\Omega_0t_0=\Omega_0t_1=\pi/2roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_π / 2. This leads to the maximum local signal of 1/2121/21 / 2 if the initial state of motion has temperature zero.
From Eq. (29), we obtain
d(t0,t1)|t0=t1=π/(2Ω0)=12∑n=0∞n¯n(n¯+1)n+1sin2(πn+12).evaluated-at𝑑subscript𝑡0subscript𝑡1subscript𝑡0subscript𝑡1𝜋2subscriptΩ012superscriptsubscript𝑛0superscript¯𝑛𝑛superscript¯𝑛1𝑛1superscript2𝜋𝑛12\displaystyle\left.d(t_0,t_1)\right|_t_0=t_1=\pi/(2\Omega_0)=\frac% 12\sum_n=0^\infty\frac\barn^n(\barn+1)^n+1\sin^2\left(% \frac\pi\sqrtn+12\right).italic_d ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_π / ( 2 roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG over¯ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG ( over¯ start_ARG italic_n end_ARG + 1 ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT end_ARG roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_π square-root start_ARG italic_n + 1 end_ARG end_ARG start_ARG 2 end_ARG ) . (30)
The signal is shown in Fig. 4 as a function of n¯¯𝑛\barnover¯ start_ARG italic_n end_ARG. After an initial drop, the signal remains close to the finite value of 1/4141/41 / 4 for much higher average phonon numbers than those tested in the experiment. This shows that even for a high-temperature thermal distribution, the local detection method is able to reveal the qubit-motion discord dynamically under an evolution governed by the Hamiltonian (20). We remark that the derivation presented in this section was based on the Lamb-Dicke limit. For sufficiently small values of η𝜂\etaitalic_η, the expression (30) still represents a valid approximation for the exact expression even for large values of n¯¯𝑛\barnover¯ start_ARG italic_n end_ARG. In particular, values up to n¯≈50¯𝑛50\barn\approx 50over¯ start_ARG italic_n end_ARG ≈ 50 can be adequately described as long as η≲0.05less-than-or-similar-to𝜂0.05\eta\lesssim 0.05italic_η ≲ 0.05, which applies to the parameter reported in the experiment Gessner et al. (2014a). An exact expression for the effective Rabi frequency ΩnsubscriptΩ𝑛\Omega_nroman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, beyond the Lamb-Dicke limit, can be given in terms of the generalized Laguerre polynomials Ln(α)(x)subscriptsuperscript𝐿𝛼𝑛𝑥L^(\alpha)_n(x)italic_L start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) as Ωn=ηe-η2(n+1)-1/2Ln(1)(η2)subscriptΩ𝑛𝜂superscript𝑒superscript𝜂2superscript𝑛112subscriptsuperscript𝐿1𝑛superscript𝜂2\Omega_n=\eta e^-\eta^2(n+1)^-1/2L^(1)_n(\eta^2)roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_η italic_e start_POSTSUPERSCRIPT - italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_n + 1 ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_L start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) Wineland et al. (1998); Leibfried et al. (2003); Gessner (2015). Numerical simulations with the exact expression produce nonzero values of the local signal even when n¯¯𝑛\barnover¯ start_ARG italic_n end_ARG and η𝜂\etaitalic_η attain values outside of the Lamb-Dicke limit. For very high values of n¯¯𝑛\barnover¯ start_ARG italic_n end_ARG, however, the validity of the effective description of the laser-ion interaction through Eq. (20) reaches its limits, since the fast-moving ion can no longer be laser-addressed with sufficient precision; hence the truly infinite limit n¯→∞→¯𝑛\barn\rightarrow\inftyover¯ start_ARG italic_n end_ARG → ∞ cannot be tested with this ansatz.
III.2 Photonic experiment with continuous-variable ancilla
The first optical realization of the local detection method was reported in Tang et al. (2015). The accessible system here is represented by a photon’s polarization degrees of freedom, which interact with the same photon’s frequency degree of freedom when passing through a birefringent material. In contrast to the trapped-ion experiment, the ancilla system is no longer described by a single harmonic oscillator mode, but instead by a continuum of modes.
The experimental setup is summarized in Fig. 5. Initially single photons are created in the quantum state
ρpisubscript𝜌𝑝𝑖\displaystyle\rho_piitalic_ρ start_POSTSUBSCRIPT italic_p italic_i end_POSTSUBSCRIPT =∑ωΔωG(ω)(12|H,ω⟩⟨H,ω|+βeiφ|H,ω⟩⟨V,ω|fragmentssubscript𝜔ΔωGfragments(ω)fragments(12|H,ω⟩fragments⟨H,ω|βsuperscript𝑒𝑖𝜑|H,ω⟩fragments⟨V,ω|\displaystyle=\sum_\omega\Delta\omega G(\omega)\left(\frac12|H,\omega% \rangle\langle H,\omega|+\beta e^i\varphi|H,\omega\rangle\langle V,\omega|\right.= ∑ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT roman_Δ italic_ω italic_G ( italic_ω ) ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG | italic_H , italic_ω ⟩ ⟨ italic_H , italic_ω | + italic_β italic_e start_POSTSUPERSCRIPT italic_i italic_φ end_POSTSUPERSCRIPT | italic_H , italic_ω ⟩ ⟨ italic_V , italic_ω |
+βe-iφ|V,ω⟩⟨H,ω|+12|V,ω⟩⟨V,ω|),fragmentsβsuperscript𝑒𝑖𝜑|V,ω⟩⟨H,ω|12|V,ω⟩⟨V,ω|),\displaystyle\left.\hskip 76.82234pt+\>\beta e^-i\varphi|V,\omega\rangle% \langle H,\omega|+\frac12|V,\omega\rangle\langle V,\omega|\right),+ italic_β italic_e start_POSTSUPERSCRIPT - italic_i italic_φ end_POSTSUPERSCRIPT | italic_V , italic_ω ⟩ ⟨ italic_H , italic_ω | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | italic_V , italic_ω ⟩ ⟨ italic_V , italic_ω | ) , (31)
where the mixed frequency distribution is described by a Lorentzian line shape,
G(ω)=1πδωδω2+(ω-ω0)2.𝐺𝜔1𝜋𝛿𝜔𝛿superscript𝜔2superscript𝜔subscript𝜔02\displaystyle G(\omega)=\frac1\pi\frac\delta\omega\delta\omega^2+(% \omega-\omega_0)^2.italic_G ( italic_ω ) = divide start_ARG 1 end_ARG start_ARG italic_π end_ARG divide start_ARG italic_δ italic_ω end_ARG start_ARG italic_δ italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_ω - italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG . (32)
Here, we have discretized the frequency space by introducing a small frequency interval ΔωΔ𝜔\Delta\omegaroman_Δ italic_ω; later on we will consider the continuum limit Δω→0→Δ𝜔0\Delta\omega\rightarrow 0roman_Δ italic_ω → 0. A basis for the polarization state is defined by the states H⟩,ket𝐻ket𝑉\ italic_H ⟩ , , describing horizontal and vertical polarization, respectively. When passing through a birefringent material, states with a specific polarization direction travel with a modified velocity, and, due to a different dwell time inside the material, experience a modified phase shift. Formally, we find
Uc(t):
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