Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings

TL;DR

This paper extends trace inequalities and recovery map concepts from finite to type III von Neumann algebras, crucial for quantum field theory, by employing Haagerup and Kosaki $L_p$ spaces.

Contribution

It introduces a von Neumann algebra version of universal recovery maps and extends key inequalities to non-tracial settings using Haagerup approximation.

Findings

Generalized Araki-Lieb-Thirring and Golden-Thompson inequalities for type III algebras

Proved a von Neumann algebra version of universal recovery map corrections

Established subharmonicity of a logarithmic p-fidelity of recovery

Abstract

Trace inequalities are general techniques with many applications in quantum information theory, often replacing classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivate entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki $L_p$ 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 (Sutter, Berta \& Tomamichel 2017). Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of univeral 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 non-decrease of relative entropy is equivalent to existence of an $L_1$-isometry implementing the channel on both input states.