Letters in Mathematical Physics

, Volume 107, Issue 12, pp 2239–2265 | Cite as

On variational expressions for quantum relative entropies

  • Mario BertaEmail author
  • Omar Fawzi
  • Marco Tomamichel


Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback–Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki’s quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz’ conclusion remains true if we allow general positive operator-valued measures. Second, we extend the result to Rényi relative entropies and show that for non-commuting states the sandwiched Rényi relative entropy is strictly larger than the measured Rényi relative entropy for \(\alpha \in (\frac{1}{2}, \infty )\) and strictly smaller for \(\alpha \in [0,\frac{1}{2})\). The latter statement provides counterexamples for the data processing inequality of the sandwiched Rényi relative entropy for \(\alpha < \frac{1}{2}\). Our main tool is a new variational expression for the measured Rényi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.


Quantum entropy Measured relative entropy Relative entropy of recovery Additivity in quantum information theory Operator Jensen inequality Convex optimization 

Mathematics Subject Classification

94A17 81Q99 15A45 



We acknowledge discussions with Fernando Brandão, Douglas Farenick and Hamza Fawzi. MB acknowledges funding by the SNSF through a fellowship, funding by the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NFS Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028), and funding support form the ARO grant for Research on Quantum Algorithms at the IQIM (W911NF-12-1-0521). Most of this work was done while OF was also with the Department of Computing and Mathematical Sciences, California Institute of Technology. MT would like to thank the IQIM at CalTech and John Preskill for his hospitality during the time most of the technical aspects of this project were completed. He is funded by an ARC Discovery Early Career Researcher Award fellowship (Grant No. DE160100821).


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Institute for Quantum Information and MatterCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of ComputingImperial College LondonKensington, LondonUK
  3. 3.Laboratoire de l’Informatique du ParallélismeÉcole Normale Supérieure de LyonLyonFrance
  4. 4.Centre for Quantum Software and InformationUniversity of Technology SydneySydneyAustralia

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