Discord measures quantum correlation by comparing the quantum mutual information with the maximal amount of mutual information accessible to a quantum measurement. This paper analyzes the properties of diagonal discord, which compares quantum mutual information with the mutual information revealed by a measurement that correspond to the eigenstates of the local density matrices. In contrast to optimized discord, diagonal discord is easily computable; it is relevant to a variety of fields such as thermodynamics and resource theory. Here we show that for the generic case of non-degenerate local density matrices, diagonal discord exhibits good properties. We use the theory of resource destroying maps to prove that diagonal discord is monotonically nonincreasing under the operation of local discord non-generating qudit channels, d>22d>2italic_d >2, and provide numerical evidence that such monotonicity holds for qubit channels as well. We also show that it is continuous, and derive a Fannes-like continuity bound. Our results hold for a variety of simple discord measures generalized from diagonal discord.
preprint: MIT-CTP/4902
Quantum discord measures a very general form of non-classical correlation, which can be present in quantum systems even in the absence of entanglement. Since the first expositions of this concept more than a decade ago Ollivier and Zurek (2001); Henderson and Vedral (2001), a substantial amount of research effort has been devoted to understanding the mathematical properties and physical meanings of discord and similar quantities. Comprehensive surveys of the properties of discord can be found in Modi et al. (2012a); Adesso et al. (2016), and references Modi (2014); Brodutch and Terno (2016); Adesso et al. (2016); Vedral (2017) provide recent perspectives on the field.
The study of discord presents many challenges and open questions. One major difficulty with discord-like quantities is that they are typically hard to compute. The canonical version of quantum discord is defined to be the difference between quantum mutual information (total correlation) and the maximum possible amount of correlation that is locally accessible (classical correlation), which involves optimization over all possible local measurements. Such optimization renders the problem of estimating discord (along with other optimized quantities such as deficit, geometric discord) computationally intractable (NP-complete) Huang (2014). Moreover, the analytic formulas for these optimized quantities are only known for very limited cases, such as two-qubit X-states Chen et al. (2011) and highly symmetric states Chitambar (2012). So it is generally very difficult to evaluate discord or to analyze the properties of discord. Another problem is that there appear to be both conceptual and technical obstacles in studying discord under operational frameworks, in particular, resource theory. The connection of discord-type quantities to resource theory remains unclear.
Diagonal discord is a natural simplification of discord, in which, rather than maximizing mutual information over all measurements, one looks at the mutual information revealed by a measurement with respect to the eigenbasis of the reduced density matrix of the subsystem under study Lloyd et al. (2015). That is, we allow the local density matrix to choose to define mutual information by the locally minimally disturbing measurement. Because such measurement does not disturb the local states, diagonal discord truly represents the property of correlation. By definition, diagonal discord is an upper bound for discord as originally defined, and is zero for states with with zero discord. Different entropic measures of discord (the original discord and deficit Modi et al. (2012b); Zurek (2003)) coincide with diagonal discord when the optimization procedure leads to measurements with respect to the local eigenbasis. We note that quantities defined by a similar local measurement strategy have been considered earlier: for instance, the so-called measurement induced disturbance Luo (2008a) and nonlocality Luo and Fu (2011a) are close variants of diagonal discord defined by a local eigenbasis as well. Diagonal discord has been shown to play key roles in thermodynamic scenarios, such as energy transport Lloyd et al. (2015) and work extraction Brodutch and Terno (2010).
In contrast to optimized discord-type quantities, diagonal discord is generically efficiently computable. We were inspired by the example of the introduction of negativity as a measure of entanglement Vidal and Werner (2002): because it is efficiently computable, negativity greatly simplifies the study of entanglement in a wide variety of scenarios. Furthermore, diagonal discord naturally emerges from the theory of resource destroying maps Liu et al. (2016), a recent framework for analyzing resource theories that can also be applied to nonconvex theories. Indeed, local measurement in an eigenbasis is a canonical discord destroying map, and the diagonal discord of a state is just the relative entropy of the state to its discord-destroyed counterpart. We believe that the study of diagonal discord may forge new links between discord and resource theory.
Because diagonal is defined without optimization over local measurements, however, several of its important mathematical properties must be verified. The purpose of this paper is to provide such verification. First, monotonicity under operations that are considered free is a defining feature of resource measures; identifying such monotones is a central theme of resource theory. A curious property of discord is that it can even be created by some local operations Streltsov et al. (2011); Hu et al. (2012). One can easily show that the minimum (contractive) distances to classical states (sometimes known as geometric measures of discord) are monotonically nonincreasing under all local discord nongenerating operations, but it is not clear whether such a property holds for diagonal discord. Note that the monotonicity under all nongenerating operations is arguably an overly strong requirement Adesso et al. (2016), which automatically implies monotonicity under all stricter theories (with less free operations). Second, continuity is also a desirable feature Brodutch and Modi (2012); Adesso et al. (2016), which indicates that the measure does not see a sudden jump under arbitrarily small perturbations. Again, continuity holds for optimized discord Brodutch and Modi (2012); Xi et al. (2011), but from examples given in Brodutch and Modi (2012); Wu et al. (2009), where the local states are both maximally mixed qubits, we know that diagonal discord can generally be discontinuous at degeneracies. However, the continuity properties otherwise remains unexplored. These two unclear features represent the most important concerns of restricting to local eigenbases.
In this paper, we address the above concerns by providing more complete analysis on the monotonicity and continuity of diagonal discord.DISCORD SERVERfind that, rather surprisingly, diagonal discord exhibits good monotonicity properties. We do so by showing that local isotropic channels commute with the canonical discord destroying map, which implies that diagonal discord is monotone under them by Liu et al. (2016). By the classification of commutativity-preserving operations Streltsov et al. (2011); Hu et al. (2012); Guo and Hou (2013), we conclude that monotonicity holds for all local commutativity-preserving operations except for unital qubit channels that are not isotropic. However, numerical studies imply that monotonicity holds for these channels as well. Then, we prove that, when the local density operator is nondegenerate, diagonal discord is continuous. We derive a Fannes-type continuity bound, which diverges as the minimum gap between eigenvalues tends to zero as expected. Proofs and some detailed discussions are left to the appendix.
Diagonal discord.-Here, we define the notion of diagonal discord more formally. Without loss of generality, we mainly study the one-sided discord of a bipartite state ABsubscriptho_ABitalic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, where the local measurements are made on subsystem AAitalic_A. It is straightforward to generalize the results to two-sided measurements or multipartite cases.
Let subscriptsuperscriptsubscriptketbra\iangle_A\langle i italic_i start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ibe a local eigenbasis of AAitalic_A, i.e., suppose A=trBABsubscriptsubscripttrsubscriptho_A=m tr_Bho_ABitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = roman_tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT admits spectral decomposition A=ipiiAsubscriptsubscriptsubscriptsuperscriptsubscriptho_A=\sum_ip_i\Pi_i^Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT. Note that the eigenbasis is not uniquely defined in the presence of degeneracy. Define A(AB)=i(iAIB)AB(iAIB)=iiAi|AB|isubscriptsubscriptsubscripttensor-productsubscriptsuperscriptsubscriptsubscripttensor-productsubscriptsuperscriptsubscriptsubscripttensor-productsuperscriptsubscriptquantum-operator-productsubscript\pi_A(ho_AB)=\sum_i(\Pi^A_i\otimes I_B)ho_AB(\Pi^A_i% \otimes I_B)=\sum_i\Pi_i^A\otimes\langle i|ho_AB|iangleitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( roman_ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( roman_ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) = start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_i | italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT | italic_i , which describes the local measurement in eigenbasis iAsuperscriptsubscript\\Pi_i^A\ roman_ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPTand the canonical discord destroying map Liu et al. (2016).
Diagonal discord of ABsubscriptho_ABitalic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT as measured by AAitalic_A, denoted as DA(AB)subscriptsubscript\barD_A(ho_AB)over start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ), quantifies the reduction in mutual information induced by Asubscript\pi_Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. Since Asubscript\pi_Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT does not perturb Asubscriptho_Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, DAsubscript\barD_Aover start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT equals the increase in the global entropy. So diagonal discord represents a unified simplification of discord and deficit. Formally,
DA(AB)subscriptsubscript\displaystyle\barD_A(ho_AB)over start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) :=assign\displaystyle:=:= I(AB)-supAI(A(AB))subscriptsubscriptsupremumsubscriptsubscriptsubscript\displaystyle I(ho_AB)-\sup_\pi_AI(\pi_A(ho_AB))italic_I ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) - roman_sup start_POSTSUBSCRIPT italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I ( italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ) (1)
=\displaystyle== infAS(A(AB))-S(AB).subscriptinfimumsubscriptsubscriptsubscriptsubscript\displaystyle\inf_\pi_AS(\pi_A(ho_AB))-S(ho_AB).roman_inf start_POSTSUBSCRIPT italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_S ( italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ) - italic_S ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) . (2)
The optimized versions of the first and second line are respectively discord and deficit, which are inequivalent in general. It is crucial that diagonal discord can also take the form of relative entropy (a straightforward derivation in appendix):
DA(AB)=infAS(ABA(AB)).subscriptsubscriptsubscriptinfimumsubscriptconditionalsubscriptsubscriptsubscript\barD_A(ho_AB)=\inf_\pi_AS(ho_AB\parallel\pi_A(ho_AB)).over start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = roman_inf start_POSTSUBSCRIPT italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_S ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ) . (3)
From Eq. (3), it can be seen that diagonal discord indeed only vanishes for classical-quantum states, the fixed points of Asubscript\pi_Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, which is a necessary condition for a discord measure. Note that, in general, optimization is still needed within degenerate subspaces. It can be seen that, as long as the degenerate subspace is small, diagonal discord can be efficiently computed. In this paper, we are mostly concerned with the nondegenerate case, where Asubscript\pi_Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is unique.
Structure of \piitalic_ theory and monotonicity.-Local discord non-generating channels (XA(A)subscriptsubscript\barX_A(\pi_A)over start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT )), i.e. commutativity-preserving channels Hu et al. (2012), consist of unital (equivalent to mixed-unitary Watrous (2016)) channels for qubits (dA=2subscript2d_A=2italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = 2) and isotropic channels for qudits (dA>2subscript2d_A>2italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT >2) Streltsov et al. (2011); Hu et al. (2012); Guo and Hou (2013), in addition to all semiclassical channels, which always destroy discord. In general, identifying operations under which some measure behaves as a monotone is a highly nontrivial task. Due to Eq. (3), the general monotonicity condition emerged from the theory of resource destroying maps Liu et al. (2016) can be applied to diagonal discord: D\barDover start_ARG italic_D end_ARG is monotonically nonincreasing under XA(A)subscriptsubscriptX_A(\pi_A)italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ), the set of local operations that commute with Asubscript\pi_Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. We use MU, ISO, and SC to denote the sets of mixed-unitary, isotropic and semiclassical channels (without the completely depolarizing channel, so that =\mathbfSC\cap\mathbfISO=\emptysetbold_SC bold_ISO = ) respectively. For simplicity, we assume that Asubscriptho_Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is nondegenerate.
Now we identify operations that belong to XA(A)subscriptsubscriptX_A(\pi_A)italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ). First, it is known that XA(A)not-subset-ofsubscriptsubscript\mathbfSCot\subsetX_A(\pi_A)bold_SC italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) Liu et al. (2016). Also recall that isotropic channels take the form ISO()=(1-)W()+I/dsuperscriptISO1\mathcalE^\mathrmISO(ho)=(1-\gamma)W(ho)+\gamma I/dcaligraphic_E start_POSTSUPERSCRIPT roman_ISO end_POSTSUPERSCRIPT ( italic_ ) = ( 1 - italic_ ) italic_W ( italic_ ) + italic_ italic_I / italic_d, where WWitalic_W is either unitary or antiunitary. Ref. Liu et al. (2016) showed that unitary-isotropic channels are in XA(A)subscriptsubscriptX_A(\pi_A)italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ). We analyze the remaining cases for qubits and qudits separately, since they exhibit different structures in the theory of \piitalic_.
For the qubit case (dA=2subscript2d_A=2italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = 2), we derive an explicit local condition that determines if a local mixed-unitary channel commute with Asubscript\pi_Aitalic_ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT:
Consider the qubit mixed-unitary channel MU()=pUUsuperscriptnormal-MUsubscriptsubscriptsubscript
