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Monotonicity of the quantum relative entropy under positive maps. (English) Zbl 1366.81045
Summary: 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 S. Beigi [J. Math. Phys. 54, No. 12, 122202, 11 p. (2013; Zbl 1315.81029)] 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.

MSC:
81P15 Quantum measurement theory, state operations, state preparations
94A17 Measures of information, entropy
47B65 Positive linear operators and order-bounded operators
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