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arXiv:1512.06117 · Annales Henri Poincaré

Monotonicity of the Quantum Relative Entropy Under Positive Maps

Alexander Müller‐Hermes, David Reeb

Published Jan 27, 2017• 110 citations
arXivHTMLPDF
DOI
10.1007/s00023-017-0550-9

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Itis well known that in quantum information quantum relative is monotonically decreasing under the completely positive and trace-preserving maps. For a new proposed sandwiched R\'enyi quantum relative, a map that is linear trace-preserving and whose Hilber-Schmidt adjoint map satisfies Schwarz inequality, monotonicity still holds. We give a new proof of monotonicity of sandwiched R\'enyi relative entropy for α∈[12,1)\alpha\in[\frac{1}{2},1)α∈[21​,1). The proof is based on complex interplotation techniques, which already has been used to prove monotonicity under trace-preserving and positive map for α∈(1,∞)\alpha\in(1,\infty)α∈(1,∞).
Consider the set $\mathcal{S}={{\ensuremath{\rho}}_{SE}}$ of possible initial states of the system-environment, steered from a tripartite reference state ${\ensuremath{\omega}}_{RSE}$. Buscemi [F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014)] showed that the reduced dynamics of the system, for each ${\ensuremath{\rho}}_{S}\ensuremath{\in}{\mathrm{Tr}}_{E}\mathcal{S}$, is always completely positive if and only if ${\ensuremath{\omega}}_{RSE}$ is a Markov state. There, during the proof, it has been assumed that the dimensions of the system and the environment can vary through the evolution. Here, we show that this assumption is necessary: we give an example for which, though ${\ensuremath{\omega}}_{RSE}$ is not a Markov state, the reduced dynamics of the system is completely positive, for any evolution of the system-environment during which the dimensions of the system and the environment remain unchanged. As our next result, we show that the result of Muller-Hermes and Reeb [A. Muller-Hermes and D. Reeb, Ann. Henri Poincare 18, 1777 (2017)], of monotonicity of the quantum relative entropy under positive maps, cannot be generalized to the Hermitian maps, even within their physical domains.