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arXiv:1111.3837 · Physical Review A

Necessary and sufficient condition for saturating the upper bound of quantum discord

Zhengjun Xi, Xiao-Ming Lu, Xiaoguang Wang, Yongming Li

Published Mar 6, 2012• 32 citations
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Abstract

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.

DOI
10.1103/physreva.85.032109

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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.
Abstract. In an initially uncorrelated mixed separable bi-partite system, quantum correlations can emerge under the action of a local measurement or local noise [1]. We analyze this counter-intuitive phenomenon using quantum discord as a quantifier. We then relate changes in quantum discord to system-environment correlations between the system in a mixed state and some purifying environmental mode using the Koashi-Winter inequality. On this basis, we suggest an interpretation of discord as a byproduct of transferring entanglement and correlations around the different subsystems of a global pure state.
In an initially uncorrelated mixed separable bi-partite system, quantum correlations can emerge under the action of a local measurement or local noise [ 1 ]. We analyse this counter-intuitive phenomenon using quantum discord as a quantifier. We then relate changes in quantum discord to system–environment correlations between the system in a mixed state and some purifying environmental mode using the Koashi–Winter inequality. On this basis, we suggest an interpretation of discord as a byproduct of transferring entanglement and correlations around the different subsystems of a global pure state.
We present an analytical solution for classical correlation, defined in terms of linear entropy, in an arbitrary system when the second subsystem is measured. We show that the optimal measurements used in the maximization of the classical correlation in terms of linear entropy, when used to calculate the quantum discord in terms of von Neumann entropy, result in a tight upper bound for arbitrary d⊗2 systems. This bound agrees with all known analytical results about quantum discord in terms of von Neumann entropy and, when comparing it with the numerical results for 10(6) two-qubit random density matrices, we obtain an average deviation of order 10(-4). Furthermore, our results give a way to calculate the quantum discord for arbitrary n-qubit GHZ and W states evolving under the action of the amplitude damping noisy channel.
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.
In contrast with entanglement, as measured by concurrence, in general, quantum discord does not possess the property of monogamy; that is, there is no tradeoff between the quantum discord shared by a pair of subsystems and the quantum discord that both of them can share with a third party. Here, we show that, as far as monogamy is considered, quantum discord of pure states is equivalent to the entanglement of formation. This result allows one to analytically prove that none of the pure three-qubit states belonging to the subclass of WWW states is monogamous. A suitable physical interpretation of the meaning of the correlation information as a quantifier of monogamy for the total information is also given. Finally, we prove that, for rank 2 two-qubit states, discord and classical correlations are bounded from above by single-qubit von Neumann entropies.