Hilbert's projective metric in quantum information theory

TL;DR

This paper explores the application of Hilbert's projective metric in quantum information theory, providing bounds on state distinguishability, entanglement measures, and quantum channel contractivity, with new duality insights and inequalities.

Contribution

It introduces Hilbert's projective metric into quantum information theory, establishing bounds, dualities, and inequalities that enhance understanding of quantum state discrimination and channel behavior.

Findings

Bounds on quantum state distinguishability measures

Contractivity bounds for quantum channels, improving existing inequalities

Duality between distinguishability measures and base norms

Abstract

We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or PPT operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under LOCC protocols or other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.