Information theoretical perspective on the method of Entanglement Witnesses

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 $\mathrm{sign}\left(\mathbb{E}\right [ W ])$.