Monotonicity of the Quantum Relative Entropy Under Positive Maps

authored by
Alexander Müller-Hermes, David Reeb
Abstract

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.

Organisation(s)
Institute of Theoretical Physics
CRC 1227 Designed Quantum States of Matter (DQ-mat)
External Organisation(s)
University of Copenhagen
Type
Article
Journal
Annales Henri Poincare
Volume
18
Pages
1777-1788
No. of pages
12
ISSN
1424-0637
Publication date
01.05.2017
Publication status
Published
Peer reviewed
Yes
ASJC Scopus subject areas
Statistical and Nonlinear Physics, Nuclear and High Energy Physics, Mathematical Physics
Electronic version(s)
https://doi.org/10.1007/s00023-017-0550-9 (Access: Closed)