Erich J. Mueller

We analyze the distribution of multipartite entanglement in states produced in a one-dimensional random monitored quantum circuit where local Clifford unitaries are interspersed with single-site measurements performed with a probability $p$. This circuit has a measurement-induced phase transition at $p=p_c$, separating a phase in which the entanglement entropy scales with the system size (a volume law state) and one in which it scales with the boundary (an area law state). We calculate the entanglement depth, corresponding to the size of the largest cluster of entangled qubits, finding that it scales as a power law with system size in both the phases. The power law exponent is 1 in the volume law phase ($p < p_c$) and continuously decreases to 0 as $p \to 1$ in the area law phase. We explain this behavior by studying the spatial distribution of entangled clusters. We find that the largest cluster of entangled qubits in these states has a fractal dimension between 0 and 1 and appears to be self-similar. Away from the critical point, this fractal dimension matches the entanglement depth power law exponent.

We numerically calculate the dynamic structure factor of the simplest two dimensional $Z_2$ lattice gauge theory. This provides an important benchmark for future experiments which will explore the dynamics of such models. As would be expected, the spectrum is gapped away from the critical point, and can be understood in terms of the elementary excitations.