Nonlocal nonstabilizerness in free fermion models

TL;DR

This paper investigates nonlocal magic in fermionic Gaussian states, deriving bounds, analyzing typical behavior, and exploring dynamics, revealing how nonlocal nonstabilizerness varies across phases and under evolution.

Contribution

It provides a closed-form bound for nonlocal magic, benchmarks optimality, and studies its behavior in various models and dynamics, including phase transitions and random circuits.

Findings

Nonlocal magic is extensive in Gaussian Haar states and peaks near critical points.

Nonlocal magic grows diffusively in random circuits.

In the XY chain, nonlocal magic and entanglement show distinct behaviors.

Abstract

Nonlocal magic quantifies the irreducible nonstabilizerness of a bipartite quantum state after optimizing over local basis changes. We study nonlocal magic for pure fermionic Gaussian states, and derive a simple closed-form entanglement spectrum bound in terms of the singular values of the subsystem-restricted covariance matrix. We benchmark our result against simulated annealing over local Gaussian unitary transformations, which supports optimality along the full local Gaussian orbit. For states drawn from the Gaussian Haar ensemble, we show that the average nonlocal magic is extensive and determine its thermodynamic limit using random matrix theory for the appropriate circular unitary ensemble. We also study Gaussian ground states, focusing on the Kitaev chain, and find that nonlocal magic is suppressed deep in both trivial and topological phases and peaks near the critical points. Finally, we investigate Gaussian evolution via random circuits and in quenches with the XY chain. For random circuits, we find that nonlocal magic grows diffusively, while in the XY chain the XX limit reveals a striking separation between nonlocal magic and entanglement.