Monotonicity of the Quantum Relative Entropy Under Positive Maps

• Published in: Annales Henri Poincaré Vol. 18; no. 5; pp. 1777 - 1788
• Main Authors: Müller-Hermes, Alexander, Reeb, David
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.05.2017; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Data processing; Dynamical Systems and Ergodic Theory; Elementary Particles; Entropy; Markov processes; Mathematical and Computational Physics; Mathematical Methods in Physics; Physics; Physics and Astronomy; Quantum Field Theory; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-017-0550-9

We prove that the quantum relative entropy decreases monotonically under the application of any positive trace-preserving linear map, for underlying separable Hilbert spaces. This answers in the affirmative a natural question that has been open for a long time, as monotonicity had previously only been shown to hold under additional assumptions, such as complete positivity or Schwarz-positivity of the adjoint map. The first step in our proof is to show monotonicity of the sandwiched Renyi divergences under positive trace-preserving maps, extending a proof of the data processing inequality by Beigi (J Math Phys 54:122202, 2013 ) that is based on complex interpolation techniques. Our result calls into question several measures of non-Markovianity that have been proposed, as these would assess all positive trace-preserving time evolutions as Markovian.