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 Renyi reflected entropies SR(alpha) 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.