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

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.