A nonstabilizerness monotone from stabilizerness asymmetry

TL;DR

This paper introduces the basis-minimized stabilizerness asymmetry (BMSA), a new measure of nonstabilizerness that is a strong monotone for magic-state resource theory and relates to other measures, with practical computation methods.

Contribution

It proposes the BMSA as a novel nonstabilizerness monotone, establishes its relation to existing measures, and develops numerical methods for its computation.

Findings

BMSA is a strong monotone for pure states.

BMSA coincides with basis-minimized measurement entropy.

Numerical methods highlight BMSA's advantages and limitations.

Abstract

We introduce a nonstabilizerness monotone which we name basis-minimised stabilizerness asymmetry (BMSA). It is based on the notion of $G$-asymmetry, a measure of how much a certain state deviates from being symmetric with respect to a symmetry group $G$. For pure states, we show that the BMSA is a strong monotone for magic-state resource theory, while it can be extended to mixed states via the convex roof construction. We discuss its relation with other magic monotones, first showing that the BMSA coincides with the recently introduced basis-minimized measurement entropy, thereby establishing the strong monotonicity of the latter. Next, we provide inequalities between the BMSA and other nonstabilizerness measures known in the literature. Finally, we present numerical methods to compute the BMSA, highlighting its advantages and drawbacks compared to other nonstabilizerness measures in the context of pure many-body quantum states. We also discuss the importance of additivity and strong monotonicity for measures of nonstabilizerness in many-body physics, motivating the search for additional computable nonstabilizerness monotones.