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arXiv · 1512.06117

Monotonicity of the Quantum Relative Entropy Under Positive Maps

Alexander Müller-Hermes·David Reeb
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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.

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Alexander Müller-Hermes, David Reeb. 2015-12-18. Monotonicity of the Quantum Relative Entropy Under Positive Maps. https://doi.org/10.1007/s00023-017-0550-9

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