Parameterized entanglement measures based on Rényi-<i>α</i> entropy

Abstract

Parameterized entanglement measures have demonstrated their superiority compared with kinds of unparameterized entanglement measures. Entanglement concurrence has been widely used to describe entanglement in quantum experiments. As an entanglement measure it is related to specific quantum Rényi-<i>α</i> entropy. In the work, we propose a parameterized bipartite entanglement measure based on the general Rényi-<i>α</i> entropy, which is named <i>α</i>-logarithmic concurrence. This measure, different from existing parameterized measures, is defined first for pure states, then extended to the mixed states. Furthermore, we verify three necessary conditions for <i>α</i>-logarithmic concurrence to satisfy the entanglement measures. We show that this measure is easy to calculate for pure states. However, for mixed states, analytical calculations are only suitable for special two-qubit states or special higher-dimensional mixed states. Therefore, we devote our efforts to developing the analytical lower bound of the-logarithmic concurrence for general bipartite states. Surprisingly, this lower bound is a function on positive partial transposition criterion and realignment criterion of this mixed state. This shows the connection among the three entanglement measures. The interesting feature is that the lower bound depends on the entropy parameter associated with the detailed state. This allows us to choose appropriate parameter <i>α</i> such that <inline-formula><tex-math id="M3">\begin{document}$ G_\alpha({\boldsymbol{\rho}})\gg0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231503_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231503_M3.png"/></alternatives></inline-formula> for experimental entanglement detection of specific state <i>ρ</i>. Moreover, we calculate expressions of the <i>α</i>-logarithmic concurrence for isotropic states, and give a the analytic expressions for isotropic states with <inline-formula><tex-math id="M4">\begin{document}$ d = 2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231503_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231503_M4.png"/></alternatives></inline-formula>. Finally, the monogamy of the <i>α</i>-logarithmic concurrence is also discussed. We set up a mathematical formulation for the monogamous property in terms of <i>α</i>-logarithmic concurrence. Here we set up the functional relation between concurrence and <i>α</i>-logarithmic concurrence in two qubit systems. Then we obtain some useful properties of this function, and by combining the Coffman–Kundu–Wootters (CKW) inequality, we establish the monogamy inequality about <i>α</i>-logarithmic concurrence. We finally prove that the monogamy inequality holds true for <i>α</i>-logarithmic concurrence.