Kavli Affiliate: Matthew P. A. Fisher
| First 5 Authors: Maria Tikhanovskaya, Ali Lavasani, Matthew P. A. Fisher, Sagar Vijay,
| Summary:
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.
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