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 $\hat{I}{¾}_E$ 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.

8 pages, 6 figures. Identifies localizable entanglement as an operational order parameter for measurement-induced phase transitions in monitored quantum circuits. Reveals an emergent entanglement correlation length whose divergence allows extraction of the critical exponent from single-system-size data.A two-ancilla protocol provides an experimentally accessible probe of the transition