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Universal recovery and P-Fidelity in von neumann algebras

Marius Junge, Nicholas D. LaRacuente

Published Sep 24, 2020• 4 citations
OpenAlex

Abstract

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.

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

Thomas Faulkner, Stefan Hollands

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.

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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.
Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states.
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.
Let (M,τ)(\mathcal{M},τ)(M,τ) and (N,τ′)(\mathcal{N},τ^{\prime})(N,τ′) be tracial von-Neumann algebras and let φ:M→Nφ:\mathcal{M}\to\mathcal{N}φ:M→N be a strictly completely positive, trace preserving map. Given a positive, invertible B∈MB\in\mathcal{M}B∈M with τ(B)=1τ(B)=1τ(B)=1, a state on M\mathcal{M}M given by a positive A∈L1(M,τ)A\in L^1(\mathcal{M}, τ)A∈L1(M,τ) is said to be recoverable if R(φ(A))=A\mathcal{R}(φ(A))=AR(φ(A))=A where R\mathcal{R}R is the Petz recovery map corresponding to BBB and φφφ. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of R∘φ\mathcal{R}\circφR∘φ. We show that there exists a completely positive, trace preserving map ψ:M→Mψ:\mathcal{M}\to\mathcal{M}ψ:M→M such that ψ(A)ψ(A)ψ(A) is recoverable for all AAA and (R∘φ)n→ψ(\mathcal{R}\circφ)^n\toψ(R∘φ)n→ψ in norm as operators on Lp(M,τ)L^p(\mathcal{M},τ)Lp(M,τ) for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in L1L^1L1. Finally, we prove an interesting decomposition theorem for normal states on M\mathcal{M}M.
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.
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.
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.