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.

v3: published version; v2: 9 pages, extended results to infinite dimensions and lifted trace-preservation condition, added discussions and references; v1: 5 pages

References in corpus (6)

• Quantum Non-Markovianity: Characterization, Quantification and Detection
• Open system quantum dynamics with correlated initial states, not completely positive maps and non-Markovianity
• Multivariate Trace Inequalities
• Approximate reversibility in the context of entropy gain, information gain, and complete positivity
• Preservation of a quantum Renyi relative entropy implies existence of a recovery map
• Data processing for the sandwiched Rényi divergence: a condition for equality

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