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arXiv:2009.11866 · Journal of Mathematical Physics

Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings

Marius Junge, Nicholas D. LaRacuente

Published Dec 1, 2022• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0066653

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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.
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.
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.
We prove several trace inequalities that extend the Golden–Thompson and the Araki–Lieb–Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden–Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.
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 prove several trace inequalities that extend the Golden–Thompson and the Araki–Lieb–Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden–Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.
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.