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

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.

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• Quantum algorithm for Petz recovery channels and pretty good measurements
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• A reduction method for noncommutative $L_p$-spaces and applications
• Equality conditions of Data Processing Inequality for $\alpha$-$z$ Rényi relative entropies
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Cited by in corpus (4)

• Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
• Monotonic multi-state quantum $f$-divergences
• Quantum Rényi divergences and the strong converse exponent of state discrimination in operator algebras
• Trace- and improved data processing inequalities for von Neumann algebras