Measurement-Induced Phase Transitions in the Dynamics of Entanglement

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 $1+1\mathrm{D}$, 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 $1+1\mathrm{D}$. 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.

Popular Summary

The laws of quantum mechanics dictate that interacting systems become “entangled” as time passes, which means that the system settles into a state that cannot be described as the sum of its individual components. The degree of entanglement generally increases the longer the system is allowed to evolve. But when some component of the system is measured, the state of that component becomes well defined without reference to the rest of the system, and thus the total amount of entanglement is reduced. Here, we mathematically examine what happens to the entanglement of a quantum system as measurements are made.

Our model is a 1D chain of quantum-mechanical spins that undergoes unitary evolution in discrete time. The spin chain is subjected to sporadic measurements of the $z$ component of a randomly chosen spin. Using theoretical arguments and numerical simulations, we find evidence for a critical rate of measurement that separates two distinct phases of entanglement. When the measurement rate is below the critical value, the entanglement grows over time and saturates only at a value proportional to the system size. At rates higher than the critical value, the entanglement growth truncates at a value that is independent of the system size.

Our findings have direct bearing on the question of how difficult it is to simulate quantum systems using classical computers and may have larger implications for quantum computing schemes that rely on maintaining long-range entanglement.

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 $1+1\mathrm{D}$. 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 renormalized after each projective measurement event.

Figure 4

Mapping between circuit dynamics with measurement (left panel) and bond percolation on the square lattice (right panel). 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 to the one on the left, we represent each unitary as the vertex of a square lattice. Broken bonds are dashed (unoccupied), and unbroken bonds are solid (occupied). This configuration corresponds to 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.

Figure 6

Cartoon of the minimal cut in the entangling phase for $S_0$ (at $L=\infty$). The small white domains are clusters of broken bonds: We only show those clusters that the minimal cut passes through. A minimal cut is shown traversing the spacetime patch. The red sections have nonzero cost per unit length, while the green regions have zero cost. In the entangling phase, the cost of the minimal cut scales like $t$ (with subleading KPZ fluctuations), so $S_0(t)\sim t$.

Figure 7

Cartoon of the minimal cut in the disentangling phase for $S_0$ (cf. Fig. 6). The clusters of broken bonds now percolate, forming the infinite white region. The cost of the minimal cut remains finite as $t\to \infty$: Only a finite number of unbroken bonds need to be traversed to reach the infinite white cluster. Therefore, $S_0(t)\sim t^0$.

Figure 8

Cartoon for the scaling argument showing $S_0(t)\sim \log t$ at the percolation critical point (cf. Figs. 6 and 7). The minimal cut passes through the sequence of white domains shown in blue and white. Writing the linear sizes of consecutive domains in this sequence as $R_1,R_2,\dots$, the ratio $R_{i+1}/R_i$ is typically larger than 1 (see text), so for an $i$ of order $\log t$, the minimal cut reaches a cluster of $\mathcal{O}(t)$ size that borders the boundary and the sequence ends. Domains $i$ and $i+1$ typically approach each other to within one lattice spacing, so the cost scales as the number of domains in the sequence.

Figure 9

Growth of the minimal-cut cost $S_0$ separating two halves of the classical percolation network, as measured by numerical simulations. (a) Dependence of $S_0$ on time $t$ for a network with aspect ratio $L=4t$ (note the double logarithmic scale). Different curves correspond to different values of $p$, ranging from $p=0$ (topmost curve) to $p=0.9$ (bottom curve) in steps of 0.1. The dashed lines show, respectively, the dependence $S_0\propto t$ and $S_0=\text{const}$. The inset shows $S_0$ at $p=0.5$ on a semilogarithmic scale, and the thin solid line shows a fit to the form $S_0=A\ln t+B$, which gives $A=0.27\pm 0.01$ and $B=0.063$. (b) The rate of entanglement growth $v_0{s}_0$ as a function of $p$, extracted by measuring the linear slope of the data in panel (a). The red line shows $v_0{s}_0\propto (p-p_c{)}^{4/3}$, as suggested by Eq. (10). (c) The coefficient of the volume-law entanglement, $s_0$, as a function of $p$, which is extracted by measuring the linear slope of the dependence $S_0(L)\propto s_{0}L$ for networks with $t=4L$. The red line is identical to the one in (b). All data in this figure are averaged over 4000 realizations of the random network.

Figure 10

Demonstration of critical behavior in the zeroth Rényi entropy near the percolation threshold $p_c=1/2$. (a) $S_0$ as a function of $p$ for different values of the system size $L$, with $t=4L$. (b) Critical scaling of this same data, with $\nu =4/3$ and $p_c=1/2$ taken from the known values for two-dimensional percolation on the square lattice. The vertical axis is $S_0(p,L)-S_0(p_c,L)$, where $S_0(p_c,L)$ is the measured value of $S_0$ at $p=1/2$ for a given value of $L$.

Figure 11

Decay of the mutual information $I_0$ between two spins as a function of their separation $x$, as determined by the classical percolation picture. The main panel shows $I_0(x)$ for three different values of $p$. The dashed line indicates the dependence $I_0\propto 1/x^4$ expected for $p=p_c$ at large $x$. The inset shows the same data plotted on a semilogarithmic scale. The dashed straight lines in the inset indicate exponential decay, which describes the curves at $p\ne p_c$. All data correspond to system size $L=3x$, with the two spins located at positions $L/3$ and $2L/3$. Data are averaged over $3\times {10}^6$ random realizations, and the evolution time is $t=L$. Error bars indicate 1 standard deviation divided by the square root of the number of realizations.

Figure 12

Examples of minimal-cut configurations for $S_0(\mathbf{a}\cup \mathbf{b})$ in cases where $I_0=2$ (upper panel) and $I_0=1$ (lower panel). In the former case, the cost of the minimal cut $S_0(\mathbf{a}\cup \mathbf{b})=0$, and in the latter case, it is $S_0(\mathbf{a}\cup \mathbf{b})=1$; i.e., the obstacle marked in the upper figure is of minimal width. In both cases, $S_0(\mathbf{a})=S_0(\mathbf{b})=1$.

Figure 13

Critical behavior of the von Neumann entanglement entropy $S_1$ for the random unitary circuit. (a) $S_1$ as a function of $p$ for different values of the system size $L$, with $t=2L$. (b) Critical scaling of these same data. The inset shows a zoomed-in view of the same data near the origin (with the solid lines suppressed).

Figure 14

Critical scaling for the bipartite entanglement $S_n$ as measured by numerical simulations of the random unitary circuit (left column of plots) and the Floquet dynamics (right column). Each plot is labeled by the corresponding value of $n$ and shows curves for seven different system sizes, ranging from $L=6$ to $L=24$ [the same as in the legend of Fig. 13]. The simulation time was $t=2L$ for the simulations of random unitaries and $t=8L$ for the Floquet dynamics. The corresponding values of $p_c$ and $\nu$ for each plot are listed in Table 1.

Figure 15

Dependence of the mutual information $I_n$ between two spins on their separation $x$. The main figure shows $I_n(x)$ for values of $p$ that are close to the critical value $p_c$. The plotted data are taken from simulations of the random unitary dynamics, for which $p_c=0.267\pm 0.027$. Blue, red, and green symbols correspond, respectively, to $n=1$, $n=2$, and $n\to \infty$, while upward- and downward-facing triangles correspond, respectively, to $p=0.26$ and 0.27. The dashed line shows the relation $I\propto 1/x^4$, as in the classical problem. The inset shows these same data on a semilogarithmic scale, along with results for $I_1(x)$ at $p=0.4$, which is away from the critical point. These latter results are well fit by an exponential dependence (red dashed line).

Figure 16

A $2+1\mathrm{D}$ analogue of the correspondence in Fig. 4. Left diagram: A simple choice of circuit geometry in $2+1\mathrm{D}$. Some unitaries (bricks) are hidden in the figure, so as to expose just six in each layer. Right diagram: Each unitary corresponds to a node of the four-coordinated lattice shown. Measurements break bonds of this lattice. The minimal cut (not shown) is a two-dimensional sheet whose cost is the number of unbroken bonds it bisects.