Optimal Extensions of Resource Measures and their Applications

TL;DR

This paper introduces a framework for extending resource measures to larger domains, establishing bounds and optimal extensions, with applications in quantum information theory, including entanglement measures.

Contribution

The paper presents a novel framework for resource measure extensions, proving bounds and identifying optimal extensions, with new techniques for entanglement measure extension.

Findings

All resource measure extensions are bounded by minimal and maximal extensions.

Relative entropies are bounded by min and max relative entropies.

Generalized trace distance, fidelity, and purified distance are optimal extensions.

Abstract

We develop a framework to extend resource measures from one domain to a larger one. We find that all extensions of resource measures are bounded between two quantities that we call the minimal and maximal extensions. We discuss various applications of our framework. We show that any relative entropy (i.e. an additive function on pairs of quantum states that satisfies the data processing inequality) must be bounded by the min and max relative entropies. We prove that the generalized trace distance, the generalized fidelity, and the purified distance are optimal extensions. And in entanglement theory we introduce a new technique to extend pure state entanglement measures to mixed bipartite states.