Effects of measurements on entanglement dynamics for $1+1$D ${\mathbb{Z}}_2$ lattice gauge theory

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 the underlying non-Hermitian evolution and to model the effects of quantum measurements. This work focuses on measuring physical observables for a ${\mathbb{Z}}_2$ gauge theory. Tensor network calculations are performed to probe the effect of measurement 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 measurement rate and the coupling constant. We find that, under both local and non-local measurements, the late-time saturation value of the bipartite entanglement entropy remains independent of system size, indicating the absence of a measurement-induced phase transition in the no-click limit.