Efficient computation of average subsystem Bures distance in transverse field Ising chain

TL;DR

This paper introduces an efficient method to compute the average subsystem Bures distance in the transverse field Ising chain, addressing computational challenges and eigenstate degeneracies, revealing distinct scaling behaviors in integrable systems.

Contribution

We develop a fast algorithm for Bures distance calculation between Gaussian states and systematically handle degeneracies using local conserved charges.

Findings

Bures distance scales linearly with subsystem size in integrable systems.

Distinct scaling behaviors are linked to degeneracies and conserved charges.

No linear increase observed in distances between random Gaussian states.

Abstract

The average subsystem trace distance has been proposed as an indicator of quantum many-body chaos and integrability. In integrable systems, evaluating the trace distance faces two challenges: the computational cost for large systems and ambiguities in defining and ordering eigenstates. In this paper, we calculate the average subsystem Bures distance in the spin-1/2 transverse-field Ising chain. We develop an efficient algorithm to evaluate the Bures distance between two Gaussian states, which allows us to access larger system sizes. To address the degeneracy issue, we consider simultaneous eigenstates of all local conserved charges and use these charges to systematically order degenerate states. The results align with the conjectured linear increase with subsystem size. We demonstrate that the distinct scaling behaviors of the average subsystem trace and Bures distances in chaotic versus integrable systems stem from discontinuities of local conserved charges across the spectrum in integrable systems. Additionally, we investigate the average subsystem distances between random pure Gaussian states but do not observe a linear increase.