Abstract
We study here the elementary properties of the relative entropy \({\mathcal{H}_\varphi(A, B) = {\rm Tr}[\varphi(A) - \varphi(B) - \varphi'(B)(A - B)]}\) for φ a convex function and A, B bounded self-adjoint operators. In particular, we prove that this relative entropy is monotone if and only if φ′ is operator monotone. We use this to appropriately define \({\mathcal{H}_\varphi(A, B)}\) in infinite dimension.
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Lewin, M., Sabin, J. A Family of Monotone Quantum Relative Entropies. Lett Math Phys 104, 691–705 (2014). https://doi.org/10.1007/s11005-014-0689-y
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DOI: https://doi.org/10.1007/s11005-014-0689-y