Fractal structure of multipartite entanglement in monitored quantum circuits

TL;DR

This paper investigates the fractal structure of multipartite entanglement in monitored quantum circuits, revealing a measurement-induced phase transition and a self-similar, fractal nature of entangled clusters.

Contribution

It introduces the concept of entanglement depth and demonstrates its power law scaling and fractal properties across different phases in monitored quantum circuits.

Findings

Entanglement depth scales as a power law with system size in both phases.

Largest entangled clusters exhibit fractal dimensions between 0 and 1.

Fractal dimension correlates with entanglement depth exponent.

Abstract

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.