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)

BibTeX

@article{d7b516e08b5f494c98a4e9ed5245afa1,
title = "Monotonicity of the Quantum Relative Entropy Under Positive Maps",
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.",
author = "Alexander M{\"u}ller-Hermes and David Reeb",
note = "Publisher Copyright: {\textcopyright} 2017, Springer International Publishing.",
year = "2017",
month = may,
day = "1",
doi = "10.1007/s00023-017-0550-9",
language = "English",
volume = "18",
pages = "1777--1788",
journal = "Annales Henri Poincare",
issn = "1424-0637",
publisher = "Birkhauser Verlag Basel",
number = "5",
}