# Strengthened monotonicity of relative entropy via pinched Petz recovery map

@article{Sutter2015StrengthenedMO, 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={2015}, 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…

## 64 Citations

### Monotonicity of quantum relative entropy and recoverability

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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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- 2015

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### The Fidelity of Recovery Is Multiplicative

- Computer ScienceIEEE Transactions on Information Theory
- 2016

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

- Computer ScienceProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2016

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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### On variational expressions for quantum relative entropies

- Computer ScienceArXiv
- 2015

A new variational expression is created for the measured Rényi relative entropy, which is exploited to show that certain lower bounds on quantum conditional mutual information are superadditive.

### Recoverability for optimized quantum f-divergences

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The refinements state that the absolute difference between the optimized $f-divergence and its channel-processed version is an upper bound on how well one can recover a quantum state acted upon by a quantum channel, whenever the recovery channel is taken to be a rotated Petz recovery channel.

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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…

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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.

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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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A new variational expression is created for the measured Rényi relative entropy, which is exploited to show that certain lower bounds on quantum conditional mutual information are superadditive.

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