Dynamics of entanglement entropy for a locally monitored lattice gauge theory

Abstract

The $1+1$ dimensional $Z_2$ gauge theory is the simplest model that allows for quantum computation or 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 monitoring ultra-local physical observables for a $\mathbb Z_2$ gauge theory. Tensor network calculations are performed to dynamically probe entanglement entropy at larger lattice sizes. In this work, we report that continuously monitoring local and diagonal observables (electric and mass energy densities) in the computational basis demonstrates the absence of any measurement-induced phase transition, as indicated by the system-size independence of the late-time saturation value of the bipartite entanglement entropy.

Figures (6)

• Figure 1: A diagram denoting physical states on a 6-site lattice. The topmost panel denotes the lattice and the label for each site; the next one $|\Omega\rangle$ denotes the strong coupling vacuum - a global spin configuration on the lattice which corresponds to the presence of no particle, no anti-particle, and no gauge flux. The next two denote two global gauge invariant states $|\psi_1\rangle$ and $|\psi_2\rangle$, which contain 1 and 3 particle-antiparticle pairs, respectively, connected by gauge fluxes. Note that, for all the entanglement entropy configurations, the incoming boundary fluxes are chosen as zero ($s=-1/2$), and for both the outgoing ones are also zero ($s=-1/2$) as the difference between the total number of particles and anti-particles contained in the lattice is zero. All the entanglement entropy states correspond to a single global charge sector for the theory given by $s_{\textrm{global}}=-1/2$.
• Figure 2: A representative diagram for a physical state on a 8 state lattice. The dotted line corresponds to the position of the link through which we bi-partition the system. The left side, consisting of 4 sites, is the subsystem $\textrm{A}$, and the right side is the subsystem $\textrm{B}$. When measuring local observables (either localized on each sites or each link), we apply them to all sites of the system (in both the sub-systems A and B), but for non-local observables (which acts on both links and sites), the results presented in the manuscript is mainly for the case where we restrict our measurements to one subsystem to avoid cutting through the link on whic the measurement operators are acting.
• Figure 3: (a) The time evolution of entanglement entropy without any measurement cleary shows that entanglement entropy does not show any saturation even at late time, (b) time averaged entanglement entropy
• Figure 4: (a) Entanglement dynamics under the measurement of the electric flux operator for different measurement rates for $x = 0.5$ and $L = 64$ in (b) we show $\gamma$ as a function of saturated entanglement entropy. $(a=0.9735,~b = 0.0358,~c =0.0267,~d = -0.9447)$ and here $t_{sat}= 60$.
• Figure 5: (a) Entanglement dynamics under the measurement of the particle-anti operator for different measurement rates for $x = 0.5$ and $L = 64$ in (b) we show $\gamma$ as a function of saturated entanglement entropy. $(a=-0.0032,~b = 1.6452,~c =-0.0039,~d = 0.0301)$ and here $t_{sat}= 60$.