Possibility of entanglement of purification to be less than half of the reflected entropy

Josiah Couch, Phuc Nguyen, Sarah Racz, Georgios Stratis, Yuxuan Zhang

Abstract: In recent work, Akers et al. [arXiv:2306.06163] proved that the entanglement of purification $E_{p}(A:B)$ is bounded below by half of the $q$-Rényi reflected entropy $S_{R}^{q}(A:B)$ for all $q\geq 2$, showing that $E_{p}(A:B) = \frac{1}{2} S_{R}^{q}(A:B)$ for a class of random tensor-network states. Naturally, the authors raise the question of whether a similar bound holds at $q=1$. Our work answers that question in the negative by finding explicit counterexamples, which we arrive at through numerical optimization. Nevertheless, this result does not preclude the possibility that restricted sets of states, such as conformal field theory states with semiclassical gravity duals, could obey the bound in question.