Information theoretical perspective on the method of Entanglement Witnesses

TL;DR

This paper applies information theory to analyze entanglement witnesses, revealing that they contain more information about quantum entanglement when considering the expected value rather than just its sign, and quantifies this information content.

Contribution

It introduces an information-theoretic framework for evaluating entanglement witnesses, highlighting the potential for more efficient entanglement detection methods.

Findings

More information is contained in the expected value of witnesses than in their sign.

Entanglement witnesses can provide additional insights when using alternative functionals of their expectation.

A quantitative measure of information content in entanglement detection is established.

Abstract

We frame entanglement detection as a problem of random variable inference to introduce a quantitative method to measure and understand whether entanglement witnesses lead to an efficient procedure for that task. Hence we quantify how many bits of information a family of entanglement witnesses can infer about the entanglement of a given quantum state sample. The bits are computed in terms of the mutual information and we unveil there exists hidden information not \emph{efficiently} processed. We show that there is more information in the expected value of the entanglement witnesses, i.e. $\mathbb{E}[W]=\langle W \rangle_\rho$ than in the sign of $\mathbb{E}[W]$. This suggests that an entanglement witness can provide more information about the entanglement if for our decision boundary we compute a different functional of its expectation value, rather than $\mathrm{sign}\left(\mathbb{E}\right [ W ])$.