Measurement-induced criticality in random quantum circuits

TL;DR

This paper explores the critical behavior of entanglement transitions in random quantum circuits, mapping the problem to a statistical mechanics model and analyzing universal properties near the transition.

Contribution

It introduces a replica-based approach to analyze entanglement transitions, deriving scaling laws and exact results in the infinite-dimensional limit, and discusses universality classes.

Findings

Entanglement transition maps to an ordering transition in a 2D statistical mechanics model.

Derived universal scaling properties of entanglement entropies near the transition.

Computed exact universal coefficients in the infinite on-site Hilbert space dimension limit.

Abstract

We investigate the critical behavior of the entanglement transition induced by projective measurements in (Haar) random unitary quantum circuits. Using a replica approach, we map the calculation of the entanglement entropies in such circuits onto a two-dimensional statistical mechanics model. In this language, the area- to volume-law entanglement transition can be interpreted as an ordering transition in the statistical mechanics model. We derive the general scaling properties of the entanglement entropies and mutual information near the transition using conformal invariance. We analyze in detail the limit of infinite on-site Hilbert space dimension in which the statistical mechanics model maps onto percolation. In particular, we compute the exact value of the universal coefficient of the logarithm of subsystem size in the $n$th R\'enyi entropies for $n \geq 1$ in this limit using relatively recent results for conformal field theory describing the critical theory of 2D percolation, and we discuss how to access the generic transition at finite on-site Hilbert space dimension from this limit, which is in a universality class different from 2D percolation. We also comment on the relation to the entanglement transition in Random Tensor Networks, studied previously in Ref. 1.