Hierarchy of continuous-variable quantum resource theories

TL;DR

This paper extends the resource theories of coherence and non-uniformity to continuous-variable quantum systems, especially Gaussian states, establishing analytical measures and hierarchical relationships among various quantum resources.

Contribution

It introduces the concept of maximal coherence at fixed energy for Gaussian systems, and proves the equivalence of Gaussian non-uniformity and maximal Gaussian coherence.

Findings

Maximal Gaussian coherence can be quantified analytically by the relative entropy.

Gaussian non-uniformity equals maximal Gaussian coherence.

Hierarchy established among non-uniformity, coherence, discord, and entanglement.

Abstract

Connections between the resource theories of coherence and purity (or non-uniformity) are well known for discrete-variable, finite-dimensional, quantum systems. We establish analogous results for continuous-variable systems, in particular Gaussian systems. To this end, we define the concept of maximal coherence at fixed energy, which is achievable with energy-preserving unitaries. We show that the maximal Gaussian coherence (where states and operations are required to be Gaussian) can be quantified analytically by the relative entropy. We then propose a resource theory of non-uniformity, by considering the purity of a quantum state at fixed energy as resource, and by defining non-uniformity monotones. In the Gaussian case, we prove the equality of Gaussian non-uniformity and maximal Gaussian coherence. Finally, we show a hierarchy for non-uniformity, coherence, discord and entanglement in continuous-variable systems.