Monotonicity of the Quantum Relative Entropy Under Positive Maps

TL;DR

This paper proves that quantum relative entropy decreases monotonically under any positive trace-preserving linear map, extending previous results and challenging existing measures of non-Markovianity in quantum systems.

Contribution

It establishes the monotonicity of quantum relative entropy under all positive trace-preserving maps, a long-standing open problem in quantum information theory.

Findings

Monotonicity of sandwiched Renyi divergences under positive maps

Quantum relative entropy decreases under positive trace-preserving maps

Implications for measures of non-Markovianity in quantum dynamics

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.