Reflected entropy: Not a correlation measure

By explicit counterexample, we show that the “reflected entropy” defined by Dutta and Faulkner is not monotonically decreasing under partial trace, and so is not a measure of physical correlations. In fact, our counterexamples show that none of the Rényi reflected entropies $S_R^{\left(\alpha \right)}$ for $0<\alpha <2$ is a correlation measure; the usual reflected entropy is realized as the $\alpha =1$ member of this family. The counterexamples are given by quantum states that correspond to classical probability distributions, so reflected entropy fails to measure correlations even at the classical level.

Figure 1

A graph of the difference of the $\alpha \mathrm{th}$ Rényi reflected entropies $S_{R,\rho }^{\left(\alpha \right)}(A:BC)$ and $S_{R,\rho }^{\left(\alpha \right)}(A:B)$ for the $\beta =10$ case of the density matrix given in Eq. (8). It is negative for $\alpha =0$ up to around $\alpha =1.9,$ indicating that the Rényi reflected entropies are not correlation measures in this range.