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arXiv:2304.05388 · Journal of Mathematical Physics

Correlation measures of a quantum state and information characteristics of a quantum channel

Максим Евгеньевич Широков

Published Nov 1, 2023• 5 citations

Abstract

We discuss the interconnections between basic correlation measures of a bipartite quantum state and basic information characteristics of a quantum channel, focusing on the benefits of these interconnections for solving specific problems concerning the characteristics of both types. We describe properties of the (unoptimized and optimized) quantum discord in infinite-dimensional bipartite systems. In particular, using the generalized Koashi–Winter relation, a simple condition is obtained that guarantees that a state with zero quantum discord is quantum-classical. Two possible definitions of the quantum discord for states with infinite one-way classical correlation are proposed and analysed. The generalized versions of Koashi–Winter and Xi–Lu–Wang–Li relations are used to obtain advanced continuity bounds for the Holevo information at the outputs of a channel and its complementary channel (as functions of a channel for a given ensemble of input states), for the Holevo capacity and the unregularized private capacity of a quantum channel depending either on the input dimension or on the input energy bound. We also discuss the properties of quantum channels which are “doppelgangers” of the monotonicity of the quantum discord and the entropy reduction of a local measurement under quantum channels acting on an unmeasured subsystem.

DOI: 10.1063/5.0160409

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We discuss the interconnections between basic correlation measures of a bipartite quantum state and basic information characteristics of a quantum channel, focusing on the benefits of these interconnections for solving specific problems concerning the characteristics of both types. We describe properties of the (unoptimized and optimized) quantum discord in infinite-dimensional bipartite systems. In particular, using the generalized Koashi–Winter relation, a simple condition is obtained that guarantees that a state with zero quantum discord is quantum-classical. Two possible definitions of the quantum discord for states with infinite one-way classical correlation are proposed and analysed. The generalized versions of Koashi–Winter and Xi–Lu–Wang–Li relations are used to obtain advanced continuity bounds for the Holevo information at the outputs of a channel and its complementary channel (as functions of a channel for a given ensemble of input states), for the Holevo capacity and the unregularized private capacity of a quantum channel depending either on the input dimension or on the input energy bound. We also discuss the properties of quantum channels which are “doppelgangers” of the monotonicity of the quantum discord and the entropy reduction of a local measurement under quantum channels acting on an unmeasured subsystem.

We revisit the upper bound of quantum discord given by the von Neumann entropy of the measured subsystem. Using the Koashi-Winter relation, we obtain a tradeoff between the amount of classical correlation and quantum discord in the tripartite pure states. The difference between the quantum discord and its upper bound is interpreted as a measure on the classical correlative capacity. Further, we give the explicit characterization of the quantum states saturating the upper bound of quantum discord, through the equality condition for the Araki-Lieb inequality. We also demonstrate that the saturating of the upper bound of quantum discord precludes any further correlation between the measured subsystem and the environment.

How strongly can one system be correlated with another? In the classical world, this basic question concerning correlative capacity has a very satisfying answer: The ``effective size'' of the marginal system, as quantified by the Shannon entropy, sets a tight upper bound to the correlations, as quantified by the mutual information. Although in the quantum world bipartite correlations, like their classical counterparts, are also well quantified by mutual information, the similarity ends here: The correlations in a bipartite quantum system can be twice as large as the marginal entropy. In the paradigm of quantum discord, the correlations are split into classical and quantum components, and it was conjectured that both the classical and quantum correlations are (like the classical mutual information) bounded above by each subsystem's entropy. In this work, by exploiting the interplay between entanglement of formation, mutual information, and quantum discord, we disprove that conjecture. We further indicate a scheme to restore harmony between quantum and classical correlative capacities. The results illustrate dramatically the asymmetric nature of quantum discord and highlight some subtle and unusual features of quantum correlations.

Several important measures of quantum correlations of a state of a finite-dimensional composite system are defined as linear combinations of marginal entropies of this state. This paper is devoted to the infinite-dimensional generalizations of such quantities and to the analysis of their properties. We introduce the notion of faithful extension of a linear combination of marginal entropies and consider several concrete examples starting with the quantum mutual information and the quantum conditional entropy. Then we show that the conditional mutual information can be uniquely defined as a lower semicontinuous function on the set of all states of a tripartite infinite-dimensional system possessing all the basic properties valid in finite dimensions. Infinite-dimensional generalizations of some other measures of quantum correlations in multipartite quantum systems are also considered. It is shown that almost all of these generalized measures are globally lower semicontinuous and possess local continuity properties which essentially simplify their use in analysis of quantum systems. In the second part of the paper we consider applications of the general results of its first part, in particular, to the theory of infinite-dimensional quantum channels and their capacities. We also show the existence of the Fawzi-Renner recovery channel reproducing marginal states for all tripartite states (including states with infinite marginal entropies) starting with the corresponding finite-dimensional result.

The evolution of pairwise quantum correlations of Bell cat states under amplitude damping is examined using the concept of quantum discord which goes beyond entanglement. A closed expression of the quantum discord is explicitly derived. We used the Koashi–Winter relation, a relation which facilitates the optimization process of the conditional entropy. We also discuss the temporal evolution of bipartite quantum correlations under a dephasing channel and compare the behaviors of quantum discord and entanglement whose properties are characterized through the concurrence.

The evolution of pairwise quantum correlations of Bell cat-states under amplitude damping is examined using the concept of quantum discord which goes beyond entanglement. A closed expression of the quantum discord is explicitly derived. We used of the Koashi-Winter relation. A relation which facilitates the optimization process of the conditional entropy. We also discuss the temporal evolution of bipartite quantum correlations under a dephasing channel and compare the behaviors of quantum discord and entanglement whose properties are characterized through the concurrence.

This work presents a generalization of the Koashi-Winter relation for

α

-Renyi entropies. This result is based on the Renyi\apos s entropy version of quantum Jensen Shannon divergence. By means of this definition, a classical correlations quantifier

C_α(ρ_{AB}) = \sup_{ξ_{AB}^{M_B}} Q_α(ξ_{AB}^{M_B})

is proposed, where the optimization is taken over the ensembles

ξ_{AB}^{M_B}

created by the outputs of the local measurement process. The main result is applied to the capacity of a quantum classical channel over a tripartite pure state

ψ_{ABE}

, that is rated above in function of the probability of success to discriminate the states in the ensemble

ξ_{AE}^{M_E}

, created by the local dephasing over partition

E

, and the asymptotic log generalized robustness of partition

AB

. Some analytical results are calculated for classical correlations and entanglement of formation.