Letters in Mathematical Physics

, Volume 104, Issue 6, pp 691–705 | Cite as

A Family of Monotone Quantum Relative Entropies

  • Mathieu LewinEmail author
  • Julien Sabin


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.

Mathematics Subject Classification (2010)

47A63 81Q99 46N50 


relative entropy strong subadditivity matrix inequalities Klein inequality 


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Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Mathematics Department (UMR 8088)CNRS and Université de Cergy-PontoiseCergy-PontoiseFrance
  2. 2.Mathematics Department (UMR 8088)Université de Cergy-PontoiseCergy-PontoiseFrance

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