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Guide

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"The fact that the quantum relative entropy $D(\cdot \| \cdot)$ is non-increasing with respect to quantum channels lies at the core of many optimality theorems in quantum information theory. In 1986 Petz showed that for every quantum channel $\mathcal{M}(\cdot)$, the relative entropy is constant for a particular $\rho$ and $\sigma$, $D(\mathcal{M}(\rho) \| \mathcal{M}(\sigma))= D(\rho \| \sigma)$, iff both
$\rho$ and $\sigma$
can be recovered perfectly; namely iff $\tilde{\mathcal{M}}(\mathcal{M}(\rho))=\rho$ and $\tilde {\mathcal{M}}(\mathcal{M}(\sigma))=\sigma$, where $\tilde{\mathcal{M}} (\cdot):=\sigma^{1/2} \mathcal{M}^\dagger \left(\mathcal{M}(\sigma)^{-1/2} (\cdot) \mathcal{M}(\sigma)^{-1/2}\right) \sigma^{1/2}$
is now known as the Petz recovery map (a completely positive trace non-increasing map). Later, Li and Winter conjectured that this could be strengthened to include approximate recovery: $D(\rho \| \sigma)- D(\mathcal{M}(\rho) \| \mathcal{M}(\sigma)) \geq D(\rho\|\bar{\mathcal{M}} ( \mathcal{M}(\sigma)))$, for a recovery map $\bar{\mathcal{M}}$, which enjoys the nice properties that the Petz recovery map possesses \cite{winter-li}.
Li and Winter knew that such a bound is false with $\bar{\mathcal{M}}$ taken as the Petz recovery map, but they conjectured that it might hold for some other natural recovery map.
However, for particular applications for which the map $\mathcal{M}(\cdot)$ has more structure, we have proven that the bound, or variations of it, hold. As a consequence, these bounds have now found applications in a variety of physical contexts in quantum information theory and physics, including thermodynamics, uncertainty relations, resource theories of asymmetry, approximate cloning, and dynamical maps."