Measurement-Induced Phase Transitions in the Dynamics of Entanglement

TL;DR

This work analyzes how random projective measurements at rate $p$ affect entanglement dynamics in many-body quantum systems, revealing a robust measurement-induced entanglement transition between entangling (volume-law) and disentangling (area-law) phases with a finite critical rate $p_c$. It introduces a solvable toy model mapping $S_0$ to bond percolation, predicting linear growth, saturation, and logarithmic growth at criticality in 1+1D, and provides scaling forms with universal exponent $ u=4/3$ for the toy model. The authors then demonstrate that the generic transition for higher Rényi entropies $S_n$ with $n\ge1$ persists in realistic 1+1D quantum circuits under both random unitary and Floquet dynamics, featuring a finite $p_c$ (approximately $0.26$ for random circuits) and a correlated exponent $\nu\approx2$, with a dynamical exponent $z\approx1$ and clear scaling collapses. Spatial correlations, via mutual informations $I_n(x)$, exhibit power-law behavior at criticality and exponential decay off criticality, consistent with scale-invariant entanglement structures. The results have implications for simulating quantum dynamics with measurements and hint at deep connections to conformal field theories and tensor-network holography, while suggesting practical impacts on the computational hardness of trajectory-based simulations.

Abstract

We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate $p$ for each degree of freedom, we show that the system has two dynamical phases: `entangling' and `disentangling'. The former occurs for $p$ smaller than a critical rate $p_c$, and is characterized by volume-law entanglement in the steady-state and `ballistic' entanglement growth after a quench. By contrast, for $p > p_c$ the system can sustain only area-law entanglement. At $p = p_c$ the steady state is scale-invariant and, in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth Rényi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher Rényi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains, and show that the phenomenology of the two phases is similar to that of the toy model, but with distinct `quantum' critical exponents, which we calculate numerically in $1+1$D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.

Figures (16)

• Figure 1: Phase diagram as a function of $p$, the rate at which measurements are made for each degree of freedom. Arrows indicate renormalization group flow.
• Figure 2: Schematic illustration of entanglement production after a quench from a product state in 1+1D. The growth of bipartite entanglement entropy between the two semi-infinite halves of an infinite chain is shown. In the entangling phase ($p<p_c$; upper curve) the entanglement grows 'ballistically' with time. At the critical point ($p=p_c$; middle curve), the entanglement grows logarithmically. In the disentangling phase ($p>p_c$; lower curve), the entanglement saturates to a finite value. (Random fluctuations are averaged over.)
• Figure 3: Circuit representation for evolution of the quantum system. Bricks indicate unitary operators (specified in the text). Dots indicate spacetime locations where measurements may take place. Note that the full dynamics is nonlinear, since the state must be re-normalized after each projective measurement event.
• Figure 4: Mapping between circuit dynamics with measurement (Left) and bond percolation on the square lattice (Right). For the purposes of the minimal cut picture, a projective measurement breaks a bond. The minimal cut may pass through broken bonds at zero cost. For the figure on the right, which is topologically equivalent, we represent each unitary as the vertex of a square lattice. Broken bonds are dashed (unoccupied) and unbroken bonds are solid (occupied). This is bond percolation on the square lattice, with a probability $1-p$ for a bond to be occupied. The minimal cut lives on the dual square lattice.
• Figure 5: Example comparison between the unitary circuit and classical percolation results for the zeroth Rényi entropy, $S_0$, between two halves of a spin chain that has undergone unitary evolution. In this example, the circuit contains $L = 24$ spins and the evolution time is $t = 48$. The value of $p$ on the horizontal axis represents the probability of measurement for each spin after each time step. The magenta circles with error bars show the result of the full simulation of the unitary circuit, while the black line shows the result of the classical percolation simulation. The inset shows the same data on a logarithmic scale.