Strengthened monotonicity of relative entropy via pinched Petz recovery map

@article{Sutter2016StrengthenedMO,
  title={Strengthened monotonicity of relative entropy via pinched Petz recovery map},
  author={David Sutter and Marco Tomamichel and Aram Wettroth Harrow},
  journal={2016 IEEE International Symposium on Information Theory (ISIT)},
  year={2016},
  pages={760-764}
}
The quantum relative entropy between two states satisfies a monotonicity property, meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a “recovery map” that exactly reverses the effects of the quantum channel on both states. In this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the… 

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A remainder term is established that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map, and it is shown that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities.

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Monotonicity of the Quantum Relative Entropy Under Positive Maps

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

The Fidelity of Recovery Is Multiplicative

The FoR is generalized and it is shown that the resulting measure is multiplicative by utilizing semi-definite programming duality and in contrast to the previous approaches, the proof does not rely on de Finetti reductions.

Universal recovery map for approximate Markov chains

It is proved that the conditional mutual information I(A:C|B) of a tripartite quantum state ρABC can be bounded from below by its distance to the closest recovered state RB→BC(ρAB), where the C-part is reconstructed from the B-part only and the recovery map RB→ BC merely depends on ρBC.

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