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arXiv · 2309.02506

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

Josiah Couch·Phuc Nguyen·Sarah Racz·Georgios Stratis·Yuxuan Zhang
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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.

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Josiah Couch, Phuc Nguyen, Sarah Racz, Georgios Stratis, Yuxuan Zhang. 2023-09-05. One-half reflected entropy is not a lower bound for entanglement of purification. https://doi.org/10.1103/physreva.109.022426

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