= Relative entropy of entanglement
{title2=$E_R(\rho)=\inf_{\sigma\in\mathrm{Sep}}D(\rho\|\sigma)$}
= Relative entropies of entanglement
{synonym}
The relative entropy of entanglement is the smallest <quantum relative entropy> from a <density operator> to a <separable quantum state>, with the separable set specified for the relevant division into parties. <Nonnegativity of quantum relative entropy> implies $E_R\geq0$. For a <separable quantum state>, choosing $\sigma=\rho$ gives $E_R=0$. For finite-dimensional bipartite states, compactness of the separable set and lower semicontinuity of the relative entropy give the converse. The support convention is $D(\rho\|\sigma)=+\infty$ unless $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$.
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