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Register a new userBimodal behavior of post-measured entropy and one-way quantum deficit for two-qubit X states
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M.A. Yurischev
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A method for calculating the one-way quantum deficit is developed. It involves a careful study of post-measured entropy shapes. We discovered that in some regions of X-state space the post-measured entropy $tilde S$ as a function of measurement angle $thetain[0,pi/2]$ exhibits a bimodal behavior inside the open interval $(0,pi/2)$, i.e., it has two interior extrema: one minimum and one maximum. Furthermore, cases are found when the interior minimum of such a bimodal function $tilde S(theta)$ is less than that one at the endpoint $theta=0$ or $pi/2$. This leads to the formation of a boundary between the phases of one-way quantum deficit via {em finite} jumps of optimal measured angle from the endpoint to the interior minimum. Phase diagram is built up for a two-parameter family of X states. The subregions with variable optimal measured angle are around 1$%$ of the total region, with their relative linear sizes achieving $17.5%$, and the fidelity between the states of those subregions can be reduced to $F=0.968$. In addition, a correction to the one-way deficit due to the interior minimum can achieve $2.3%$. Such conditions are favorable to detect the subregions with variable optimal measured angle of one-way quantum deficit in an experiment.
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The one-way quantum deficit, a measure of quantum correlation, can exhibit for X quantum states the regions (subdomains) with the phases $Delta_0$ and $Delta_{pi/2}$ which are characterized by constant (i.e., universal) optimal measurement angles, correspondingly, zero and $pi/2$ with respect to the $z$-axis and a third phase $Delta_vartheta$ with the variable (state-dependent) optimal measurement angle $vartheta$. We build the complete phase diagram of one-way quantum deficit for the XXZ subclass of symmetric X states. In contrast to the quantum discord where the region for the phase with variable optimal measurement angle is very tiny (more exactly, it is a very thin layer), the similar region $Delta_vartheta$ is large and achieves the sizes comparable to those of regions $Delta_0$ and $Delta_{pi/2}$. This instils hope to detect the mysterious fraction of quantum correlation with the variable optimal measurement angle experimentally.
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