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arXiv:1104.4043 · Physical Review A

Dynamics of geometric and entropic quantifiers of correlations in open quantum systems

Bruno Bellomo, Rosario Lo Franco, Giuseppe Compagno

Published Jul 13, 2012• 83 citations
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Abstract

We extend the Hilbert-Schmidt (square norm) distance, previously used to define the geometric quantum discord, to define also geometric quantifiers of total and classical correlations. We then compare the dynamics of geometric and entropic quantifiers of the different kinds of correlations in a non-Markovian open two-qubit system under local dephasing. We find that qualitative differences occur only for quantum discords. This is taken to imply that geometric and entropic discords are not, in general, equivalent in describing the dynamics of quantum correlations. We then show that geometric and entropic quantifiers of total correlations also present qualitative disagreements in the state space. This aspect indicates that the differences found for quantum discord are not attributable to a different separation, introduced by each measure, between the quantum and classical parts of correlations. Finally, we find that the Hilbert-Schmidt distance formally coincides with a symmetrized form of linear relative entropy.

DOI
10.1103/physreva.86.012312

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We extend the Hilbert-Schmidt (square norm) distance, previously used to define the geometric quantum discord, to define also geometric quantifiers of total and classical correlations. We then compare the dynamics of geometric and entropic quantifiers of the different kinds of correlations in a non-Markovian open two-qubit system under local dephasing. We find that qualitative differences occur only for quantum discords. This is taken to imply that geometric and entropic discords are not, in general, equivalent in describing the dynamics of quantum correlations. We then show that geometric and entropic quantifiers of total correlations also present qualitative disagreements in the state space. This aspect indicates that the differences found for quantum discord are not attributable to a different separation, introduced by each measure, between the quantum and classical parts of correlations. Finally, we find that the Hilbert-Schmidt distance formally coincides with a symmetrized form of linear relative entropy.
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We derive an exact formula of geometric measure of quantum discord (GMQD) for an arbitrary 3-qubit state. The dynamics of GMQD in a quantum spin system coupled to a thermal bath is studied by simulation and analytical approximation. It is found that tripartite quantum discord has distinct time behaviors within two different eigenspaces. Further, we extend the square norm distance to define tripartite geometric quantifiers of total and classical correlations. We obtain their explicit expressions and discuss the possibility of having a closed additive relation among various kinds of correlations during the dynamical evolution.
In this paper, we are concerned with the non-commutativity measure of quantum discord. We first present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. Then we compare the geometric quantum discords for two dynamic models with their non-commutativity measure of quantum discords. Furthermore, we show that the results conducted by the non-commutativity measure of quantum discord are different from those conducted by both or one of the Hilbert-Schmidt distance discord and trace distance discord. These intrinsic differences indicate that the non-commutativity measure of quantum discord is incompatible with at least one of the well-known geometric quantum discords in the quantitative and qualitative representation of quantum correlations.
The dynamics of three different entropic measures of quantum correlations in mixed bipartite qubit states in the presence of two different classical noises, the static noise (SN) and the random telegraph noise (RTN), are investigated. The three entropic measures of quantum correlations correspond to one-way information deficit, geometric quantum discord and the cubic information. General analytic relations for each quantifier in the two configurations are obtained. In both configurations, the minimized value of each measure of quantum correlations corresponds to the conditional entropy of the same projectors. It is shown that one-way information deficit captures more correlations in highly mixed initial states. On the contrary, in both configurations the cubic information reduces to the geometric quantum discord and captures more correlations for highly pure initial states. The periodic revival of each measure of quantum correlation is more prominent in the case of RTN.
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A symmetric measure of quantum correlation based on the Hilbert—Schmidt distance is presented in this paper. For two-qubit states, we considerably simplify the optimization procedure so that numerical evaluation can be performed efficiently. Analytical expressions for the quantum correlation are attained for some special states. We further investigate the dynamics of quantum correlation of the system qubits in the presence of independent dissipative environments. Several nontrivial aspects are demonstrated. We find that the quantum correlation can increase even if the system state is suffering from dissipative noise. Sudden changes occur, even twice, in the time evolution of quantum correlation. There exists a certain correspondence between the evolution of quantum correlation in the systems and that in the environments, and the quantum correlation in the systems will be transferred into the environments completely and asymptotically.