Universality of the cross entropy in $\mathbb{Z}_2$ symmetric monitored quantum circuits

TL;DR

This paper demonstrates that the linear cross-entropy (LXE) can distinguish different entangled phases and reveal universal critical behavior in symmetry-protected monitored quantum circuits, especially relating to percolation theory.

Contribution

It shows that LXE can identify distinct phases and universal behavior in monitored circuits with $ ext{Z}_2$ symmetry, connecting to percolation theory and boundary conditions.

Findings

LXE distinguishes spin glass and paramagnetic phases.

LXE at criticality relates to crossing probabilities in percolation.

Numerical results agree with universal functions of aspect ratio.

Abstract

The linear cross-entropy (LXE) has been recently proposed as a scalable probe of the measurement-driven phase transition between volume- and area-law-entangled phases of pure-state trajectories in certain monitored quantum circuits. Here, we demonstrate that the LXE can distinguish distinct area-law-entangled phases of monitored circuits with symmetries, and extract universal behavior at the critical points separating these phases. We focus on (1+1)-dimensional monitored circuits with an on-site $\mathbb{Z}_{2}$ symmetry. For an appropriate choice of initial states, the LXE distinguishes the area-law-entangled spin glass and paramagnetic phases of the monitored trajectories. At the critical point, described by two-dimensional percolation, the LXE exhibits universal behavior which depends sensitively on boundary conditions, and the choice of initial states. With open boundary conditions, we show that the LXE relates to crossing probabilities in critical percolation, and is thus given by a known universal function of the aspect ratio of the dynamics, which quantitatively agrees with numerical studies of the LXE at criticality. The LXE probes correlations of other operators in percolation with periodic boundary conditions. We show that the LXE is sensitive to the richer phase diagram of the circuit model in the presence of symmmetric unitary gates. Lastly, we consider the effect of noise during the circuit evolution, and propose potential solutions to counter it.