The quantum relative entropy is commonly used in quantum information theory as a benchmark for quantum resources like entanglement, discord and coherence. The relative entropy of entanglement for a density matrix $\rho$ for example is $$ E_r (\rho) = \min_{\sigma \subset \mathcal{S}} S(\rho \| \sigma) \equiv S(\rho \| \sigma_*) , \\ $$ where $S$ is the set of separable states, and the minimization state from $S$ is denoted $\sigma_*$.

Questions: How do the von Neumann entropy of the starting state $S(\rho)$ relate to that of the target state $S(\sigma_*)$, in other words, should they be equal or close in value (since their distance in terms of relative entropy is minimized it seems natural that they should be similar in some sense)? How does this differ across quantum resources?


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