Localizable Entanglement as an Order Parameter for Measurement-Induced Phase Transitions

Abstract

We identify localizable entanglement (LE) as an order parameter for measurement-induced phase transitions (MIPT). LE exhibits universal finite-size scaling with critical exponents that match previous MIPT results and gives a nice operational interpretation connecting MIPTs to classical percolation. Remarkably, we find that LE decays exponentially with distance in the area-law phase as opposed to being essentially constant for the volume-law phase thereby, discover an intrinsic length scale $ξ_E$ that diverges at the critical measurement probability $p_c$. While classical percolation transition captures successful transport across a network, MIPT as characterized by LE can be interpreted as quantifying the amount of quantum teleportation between two given nodes in a quantum circuit. Building on this insight, we propose a two-ancilla protocol that provides an experimentally accessible readout of entanglement redistribution across the transition.

Figures (6)

• Figure 1: (a) Schematic of a $(1+1)$-dimensional monitored brickwall quantum circuit composed of discrete time layers of random unitary gates and projective measurements, exhibiting a measurement-induced transition. All quantities in the main text are obtained by averaging independent realizations over such circuits. (b) Graph state representations of stabilizer states generated from a 20-qubit MIPT circuit with Clifford gates evolved for a sufficiently long time at three representative measurement probabilities, arranged from left to right with increasing measurement rate: volume-law phase $(p = 0.14)$, critical point $(p=0.16)$, and area-law phase $(p=0.18)$. The graphs show the connectivity at each phase. The schematic shown below the graph state depicts the corresponding behavior of the averaged correlation function $\left<C_{\text{LE}} \right>$: distance-independent in the volume-law phase, intermediate at criticality, and exponentially decaying in the area-law phase.
• Figure 2: Order parameter $\left\langle R \right\rangle$ as a function of measurement probability $p$ for random Clifford brickwall circuits of different system sizes $L$. $\left\langle R \right\rangle$ quantifies the maximal normalized separation between qubits with nonvanishing LE, serving as an operational order parameter for the measurement-induced phase transition. The crossing of curves at $p_c \approx 0.16$ signals the transition from the entangling (volume-law) to the disentangling (area-law) phase.
• Figure 3: Semilog plot of the entanglement correlation function $\left<C_{\text{LE}}(r;p)\right>$. In the area-law phase $(p>p_c)$, the curves are exponential in $r$, enabling extraction of the entanglement correlation length $\xi_E(p)$. At $p<p_c$, the curves approach a constant, reflecting long-range entanglement. The inset shows the same data on linear–linear axes to clearly demonstrate the saturation of $\left<C_{\text{LE}}(r;p)\right>$ for $p<p_c$.
• Figure 4: Extraction of the entanglement correlation length in the area-law phase. Log–log plot of the entanglement correlation length $\xi_E(p)$ as a function of the distance from the critical measurement rate, $|p-p_c|$, in the area-law regime $p>p_c$. The data (points) follow a clear power-law divergence, $|p-p_c|^\nu$, and the linear fit (solid line) yields an exponent $\nu=1.31$ consistent with the MIPT.
• Figure 5: Average LE between a reference qubit and the rest of the system $\left<\text{LE}_{\text{ref}} \right>$, shown as a function of measurement probability $p$ for system sizes $L=20,40,80,160$. For small measurement rates, $\left<\text{LE}_{\text{ref}} \right> \approx 1$, reflecting the volume-law entangled phase in which the reference qubit shares finite distillable entanglement with the many-body state. As $p$ increases, the curves sharpen with increasing $L$