Effects of monitoring on entanglement dynamics for $1+1$D $\mathbb Z_2$ lattice gauge theory

Abstract

The $(1+1)$-dimensional $\mathbb Z_2$ gauge theory is the simplest model that allows for quantum simulation to probe the fundamental aspects of a gauge theory coupled with dynamical fermions. To reliably benchmark such a system, it is crucial to understand the non-unitary quantum dynamics arising from effective non-Hermitian evolution and post-selected monitoring protocols. This work focuses on the post-selected non-Hermitian filtering dynamics of a $\mathbb Z_2$ gauge theory, where the non-Hermitian terms are associated with local and non-local gauge-invariant operators naturally present in the theory. We interpret the resulting dynamics as post-selected filtering, where different operator sectors are coupled to loss channels with different rates. This gives a unified framework for both the local electric flux and particle-number terms and the non-local mesonic hopping term. Tensor network calculations are performed to probe the effect of the filtering for larger lattice sizes (up to 256-site systems). Using Matrix Product State calculations, the dynamics of entanglement entropy are studied as a function of the filtering rate and the coupling constant. We find that, under both local and non-local filtering, the late-time saturation value of the bipartite entanglement entropy remains independent of system size, providing no evidence of a measurement-induced phase transition-like phenomenon in the post-selected dynamics across the range of filtering strengths, evolution times, and system sizes considered here.