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Quantum Information Processing

On the Brodutch and Modi method of constructing geometric measures of classical and quantum correlations

Zbigniew Walczak, Iwona Wintrowicz

Published Feb 10, 2017• 3 citations
Publisher

Abstract

Recently, Brodutch and Modi proposed a general method of constructing meaningful measures of classical and quantum correlations. We systematically apply this method to obtain geometric classical and quantum correlations based on the Bures and the trace distances for two-qubit Bell diagonal states. Moreover, we argue that in general the Brodutch and Modi method may provide non-unique results, and we show how to modify this method to avoid this issue.

DOI
10.1007/s11128-017-1539-5

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Recently, Brodutch and Modi proposed a general method of constructing meaningful measures of classical and quantum correlations. We systematically apply this method to obtain geometric classical and quantum correlations based on the Bures and the trace distances for two-qubit Bell diagonal states. Moreover, we argue that in general the Brodutch and Modi method may provide non-unique results, and we show how to modify this method to avoid this issue.
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We apply the modified Brodutch and Modi method of constructing geometric measures of correlations to obtain analytical expressions for measurement-induced geometric classical and quantum correlations based on the Hellinger distance for two-qubit Bell diagonal states. Moreover, we study dynamics of the classical and quantum correlations for these states under decoherence induced by the action of bit flip, bit-phase flip, phase flip, depolarizing and generalized amplitude damping local quantum channels and identify all possible types of the decoherence dynamics. In particular, we show that these correlations exhibit freezing, sudden change and sudden transition between classical and quantum decoherence phenomena.
Abstract We investigate and compare three distinguished geometric measures of bipartite quantum correlations that have been recently introduced in the literature: the geometric discord, the measurement-induced geometric discord, and the discord of response, each one defined according to three contractive distances on the set of quantum states, namely the trace, Bures, and Hellinger distances. We establish a set of exact algebraic relations and inequalities between the different measures. In particular, we show that the geometric discord and the discord of response based on the Hellinger distance are easy to compute analytically for all quantum states whenever the reference subsystem is a qubit. These two measures thus provide the first instance of discords that are simultaneously fully computable, reliable (since they satisfy all the basic Axioms that must be obeyed by a proper measure of quantum correlations), and operationally viable (in terms of state distinguishability). We apply the general mathematical structure to determine the closest classical-quantum state of a given state and the maximally quantum-correlated states at fixed global state purity according to the different distances, as well as a necessary condition for a channel to be quantumness breaking.
We identify ambiguities in the available frameworks for defining quantum, classical, and total correlations as measured by discord-like quantifiers. More specifically, we determine situations for which either classical or quantum correlations are not uniquely defined due to degeneracies arising from the optimization procedure over the state space. In order to remove such degeneracies, we introduce a general approach where correlations are independently defined, escaping therefore from a degenerate subspace. As an illustration, we analyze the trace-norm geometric quantum discord for two-qubit Bell-diagonal states.
We introduce the concepts of geometric classical and total correlations through Schatten 1-norm (trace norm), which is the only Schatten p -norm able to ensure a well-defined geometric measure of correlations. In particular, we derive the analytical expressions for the case of two-qubit Bell-diagonal states, discussing the superadditivity of geometric correlations. As an illustration, we compare our results with the entropic correlations, discussing both their hierarchy and monotonicity properties. Moreover, we apply the geometric correlations to investigate the ground state of spin chains in the thermodynamic limit. In contrast to the entropic quantifiers, we show that the classical correlation is the only source of 1-norm geometric correlation that is able to signal an infinite-order quantum phase transition.