Fractal structure of multipartite entanglement in monitored quantum circuits

TL;DR

The paper analyzes multipartite entanglement in a one-dimensional random monitored Clifford circuit with single-site measurements at probability $p$, revealing a measurement-induced phase transition between volume-law and area-law entanglement. It introduces the entanglement depth, the size of the largest entangled cluster, and shows it scales as a power law with system size $L$ in both phases, with exponent $\gamma=1$ in the volume-law phase and at criticality, and $\gamma\to 0$ as $p\to 1$ in the area-law phase, around the critical probability $p_c \sim 0.16$. The largest entangled cluster exhibits fractal structure with fractal dimension $d \in (0,1)$, inferred from box-counting, and $d$ matches $\gamma$ away from criticality while near criticality there can be small deviations. Overall, the work demonstrates robust long-range multipartite entanglement in the area-law phase, introduces an operator-independent metric for entanglement structure, and showcases fractal, self-similar organization arising from the competition between unitary growth and measurements.

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.

Figures (6)

• Figure 1: (a) Entanglement structure schematic of a twelve-qubit state $|\psi\rangle$. The state is separable into a tensor product over clusters A (green),B (black) and C (orange) denoted as $|\psi\rangle = |\phi_A\rangle \otimes |\phi_B\rangle \otimes |\phi_C\rangle$. The colored bar shows the qubits arranged in their native spatial order. The entanglement depth of this state is 5, corresponding to the size of the largest cluster C. (b) Monitored circuit in brickwork form with two-qubit random Clifford unitaries and local projective $z$-basis measuremements with probability $p$.
• Figure 2: Log-log plot of average entanglement depth as a function of system size $L$ for the steady state ensemble of states formed by applying random nearest-neighbor two-site clifford gates, interspersed with random single site measurements. The probability of a local measurement is $p$. The curves are plotted in steps of $0.04$ for $p < p_c$ and in steps of $0.08$ for $p > p_c$. There is a phase transition at $p=p_c\sim 0.16$trans2trans3.
• Figure 3: Power law exponent of entanglement depth growth with system size, $\gamma$ (blue dots) and fractal dimension, $d$ of the largest cluster of entangled qubits as a function of measurement probability, $p$. The critical point for the volume law-area law phase transition is shown as $p_c = 0.16$.
• Figure 4: A typical 120 qubit state generated at measurement probability, $p=0.2$. The same state is shown with different coarse-graining scales, denoted by gridlines enclosing $b$ qubits. For a given coarse-grained picture, the orange squares denote qubits that are part of the largest entangled cluster while black squares denote the remaining qubits. The orange cluster spans the entire system, but is filled with 'holes' of multiple sizes that break it into discontinuous pieces.
• Figure 5: Log-log plot of number of boxes, $N_b$ as a function of box size, $b$ for various measurement probabilities, $p$. The curves are plotted in steps of $0.04$ for $p < p_c$ and in steps of $0.08$ for $p > p_c$. The inset separately shows the same data deep in the volume law phase for $p=0.04,p=0.08$.