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Letters in Mathematical Physics

Approximate recoverability and relative entropy II: 2-positive channels of general von Neumann algebras

Thomas Faulkner, Stefan Hollands

Published Mar 15, 2022• 6 citations
Publisher

Abstract

Abstract We generalize our results in paper I in this series to quantum channels between general von Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two von Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.

DOI
10.1007/s11005-022-01510-9

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Abstract We generalize our results in paper I in this series to quantum channels between general von Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two von Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.
We generalize our results in paper I in this series to quantum channels between general v. Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two v. Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.
We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite-dimensional density matrices, yields a bound for how well a quantum state can be recovered after it has been passed through a channel. Some natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.
In this paper, we discuss a refinement of quantum data processing inequality for the sandwiched quasi-relative entropy S2\mathcal{S}_2S2​ on a tracial von-Neumann algebra. The main result is a universal recoverability bound with the Petz recovery map, which was previously obtained in the finite dimensional setup.
Abstract We prove the existence of a universal recovery channel that approximately recovers states on a von Neumann subalgebra when the change in relative entropy, with respect to a fixed reference state, is small. Our result is a generalization of previous results that applied to type-I von Neumann algebras by Junge at al. [ arXiv:1509.07127 ]. We broadly follow their proof strategy but consider here arbitrary von Neumann algebras, where qualitatively new issues arise. Our results hinge on the construction of certain analytic vectors and computations/estimations of their Araki–Masuda LpL_pLp​ L p norms. We comment on applications to the quantum null energy condition.
Scenarios ranging from quantum error correction to high energy physics use recovery maps, which try to reverse the effects of generally irreversible quantum channels. The decrease in quantum relative entropy between two states under the same channel quantifies information lost. A small decrease in relative entropy often implies recoverability via a universal map depending only the second argument to the relative entropy. We find such a universal recovery map for arbitrary channels on von Neumann algebras, and we generalize to p-fidelity via subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that non-decrease of relative entropy is equivalent to the existence of an L 1 -isometry implementing the channel on both input states. Our primary technique is a reduction method by Haagerup, approximating a non-tracial, type III von Neumann algebra by a finite algebra. This technique has many potential applications in porting results from quantum information theory to high energy settings.
The data processing inequality states that the quantum relative entropy between two states ρ\rho ρ and σ\sigma σ can never increase by applying the same quantum channel N\mathcal {N}N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ\rho ρ and the closest recovered state (R∘N)(ρ)(\mathcal {R} \circ \mathcal {N})(\rho )(R∘N)(ρ) , where R\mathcal {R}R is a recovery map with the property that σ=(R∘N)(σ)\sigma = (\mathcal {R} \circ \mathcal {N})(\sigma )σ=(R∘N)(σ) . We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ\sigma σ and the quantum channel N\mathcal {N}N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.