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Bound entanglement in thermalized states and black hole radiation

We study the mixed-state entanglement structure of chaotic quantum many-body systems at late times using the recently developed $\textit{equilibrium approximation}$. A rich entanglement phase diagram emerges when we generalize this technique to evaluate the logarithmic negativity for various universality classes of macroscopically thermalized states. Unlike in the infinite temperature case, when we impose energy constraints at finite temperature, the phase diagrams for the logarithmic negativity and the mutual information become distinct. In particular, we identify a regime where the negativity is extensive but the mutual information is sub-extensive, indicating a large amount of $\textit{bound entanglement}$. When applied to evaporating black holes, these results imply that there is quantum entanglement within the Hawking radiation long before the Page time, although this entanglement may not be distillable into EPR pairs.