Monotonicity of the unified quantum (r, s)-entropy and (r, s)-mutual information

Abstract

Monotonicity of the unified quantum (r, s)-entropy $E_{r}^{s}(\rho )$ and the unified quantum (r, s)-mutual information $I_{r}^{s}(\rho )$ is discussed in this paper. Some basic properties of them are explored, and the following conclusions are established. (1) For any $0<r<1, E_{r}^{s}(\rho )$ is increasing with respect to $s\in (-\infty ,+\infty )$, and for any $r\ge 1, E_{r}^{s}(\rho )$ is decreasing with respect to $s\in (-\infty ,+\infty )$; (2) for any $s>0$, $E_{r}^{s}(\rho )$ is decreasing with respect to $r\in (0,+\infty )$; (3) for any $r>0, E_{r}^{s}(\rho )$ is convex with respect to $s\in (-\infty ,+\infty )$; (4) for a product state $\rho _{AB}$, there are two real numbers a and b such that $I_{r}^{s}(\rho _{AB})$ is increasing with respect to $s\in [0,a]$ when $r\ge 1$ and it is decreasing with respect to $s\in [b,0]$ when $0<r<1$; (5) for a product state $\rho _{AB}$, $I_{r}^{s}(\rho _{AB})$ is decreasing with respect to $r\in [r_s,+\infty )$ for each $s>0$, where $r_s={\mathrm {max}}\{a_s,b_s\}$, $m>2$ with $m-2\ln m=1$ and ${\mathrm {tr}}\rho _{A}^{a_s}={\mathrm {tr}}\rho _{B}^{b_s}=m^{-\frac{1}{s}}$.