Extremal properties of conditional entropy and quantum discord for XXZ symmetric quantum states

Abstract

For the XXZ subclass of symmetric two-qubit X states, we study the behavior of quantum conditional entropy S_{cond} as a function of measurement angle θ\in[0,π/2]. Numerical calculations show that the function S_{cond}(θ) for X states can have at most one local extremum in the open interval from zero to π/2 (unimodality property). If the extremum is a minimum the quantum discord displays region with variable (state-dependent) optimal measurement angle θ^*. Such θ-regions (phases, fractions) are very tiny in the space of X state parameters. We also discover the cases when the conditional entropy has a local maximum inside the interval (0,π/2). It is remarkable that the maxima exist in surprisingly wide regions and the boundaries for such regions are defined by the same bifurcation conditions as for those with a minimum. Moreover, the found maxima can exceed the conditional entropy values at the ends of interval [0,π/2] more than by 1%. This instils hope in the possibility to detect such maxima in experiment.

arXiv comment24 pages, 15 figures

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