One-half reflected entropy is not a lower bound for entanglement of purification

TL;DR

This paper investigates whether the entanglement of purification is always bounded below by half of the reflected entropy at q=1, providing counterexamples that show it is not, thus clarifying the limitations of this inequality.

Contribution

The paper demonstrates that the proposed lower bound does not hold universally at q=1 by constructing explicit counterexamples through numerical optimization.

Findings

Counterexamples show the bound fails at q=1

The bound holds for certain random tensor network states

Potential for special states like CFTs to obey the bound

Abstract

In recent work, Akers et al. proved that the entanglement of purification $E_p(A:B)$ is bounded below by half of the $q$-R\'enyi reflected entropy $S_R^{(q)}(A:B)$ for all $q\geq2$, 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 counter-examples, which we arrive at through numerical optimization. Nevertheless, this result does not preclude the possibility that restricted sets of states, such as CFT states with semi-classical gravity duals, could obey the bound in question.