Extremal quantum states

TL;DR

This paper explores the extremal properties of quantum states using phase-space tools like the Husimi Q function, identifying states with maximal or minimal quantumness and analyzing their properties across different quantum systems.

Contribution

It introduces extremal principles based on phase-space quantities to classify quantum states' extremality, applicable beyond specific symmetry groups.

Findings

Extremal quantum states are characterized using Wehrl entropy and related measures.

The extrema largely coincide in continuous-variable systems.

Caution is needed when applying extremal principles to spin systems.

Abstract

The striking differences between quantum and classical systems predicate disruptive quantum technologies. We peruse quantumness from a variety of viewpoints, concentrating on phase-space formulations because they can be applied beyond particular symmetry groups. The symmetry-transcending properties of the Husimi $Q$ function make it our basic tool. In terms of the latter, we examine quantities such as the Wehrl entropy, inverse participation ratio, cumulative multipolar distribution, and metrological power, which are linked to intrinsic properties of any quantum state. We use these quantities to formulate extremal principles and determine in this way which states are the most and least "quantum;" the corresponding properties and potential usefulness of each extremal principle are explored in detail. While the extrema largely coincide for continuous-variable systems, our analysis of spin systems shows that care must be taken when applying an extremal principle to new contexts.