# Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory

Adam Nahum, Shitadhi Roy, Brian Skinner, Jonathan Ruhman

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Adam Nahum, Shitadhi Roy, Brian Skinner, Jonathan Ruhman. Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory. PRX Quantum, 2021, 2 (1), pp.010352. 10.1103/prxquantum.2.010352. hal-03456318

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# Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory

Adam Nahum<sup>*</sup>,<sup>1,1</sup> Shitadhi Roy<sup>1,2</sup> Brian Skinner<sup>3</sup> and Jonathan Ruhman<sup>4,5,†</sup>

<sup>1</sup>Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, United Kingdom

<sup>2</sup>Physical and Theoretical Chemistry, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom

<sup>3</sup>Department of Physics, Ohio State University, Columbus, Ohio 43210, USA

<sup>4</sup>Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel

<sup>5</sup>Center for Quantum Entanglement Science and Technology, Bar-Ilan University, Ramat Gan 52900, Israel

<figure><img src="image_2.png" /></figure>

(Received 10 October 2020; accepted 19 January 2021; published 30 March 2021)

A quantum many-body system whose dynamics includes local measurements at a nonzero rate can be in distinct dynamical phases, with differing entanglement properties. We introduce theoretical approaches to measurement-induced phase transitions (MPTs) and also to entanglement transitions in random tensor networks. Many of our results are for “all-to-all” quantum circuits with unitaries and measurements, in which any qubit can couple to any other, and related settings where some of the complications of low-dimensional models are reduced. We also propose field-theory descriptions for spatially local systems of any finite dimensionality. To build intuition, we first solve the simplest “minimal cut” toy model for entanglement dynamics in all-to-all circuits, finding scaling forms and exponents within this approximation. We then show that certain all-to-all measurement circuits allow exact results by exploiting local treelike structure in the circuit geometry. For this reason, we make a detour to give general universal results for entanglement phase transitions in a class of random tree tensor networks with bond dimension 2, making a connection with the classical theory of directed polymers on a tree. We then compare these results with numerics in all-to-all circuits, both for the MPT and for the simpler “forced-measurement phase transition” (FMPT). We characterize the two different phases in all-to-all circuits using observables that are sensitive to the amount of information that is propagated between the initial and final time. We demonstrate signatures of the two phases that can be understood from simple models. Finally we propose Landau-Ginsburg-Wilson-like field theories for the measurement phase transition, the forced-measurement phase transition, and for entanglement transitions in random tensor networks. This analysis shows a surprising difference between the measurement phase transition and the other cases. We discuss variants of the measurement problem with additional structure (for example free-fermion structure), and questions for the future.

DOI: 10.1103/PRXQuantum.2.010352

## I. INTRODUCTION

A quantum system whose unitary dynamics is inter-
spersed with repeated measurements follows a random
trajectory through Hilbert space [1–5], determined both
by the unitary part of the dynamics and by the sequence
of measurement outcomes. In the many-body case this

random dynamics admits a “measurement phase transition” (MPT) between two qualitatively different, stable dynamical phases, with distinct entanglement properties [6–27]. For definiteness, consider a system of many spins in a pure state, evolving under a quantum circuit that includes both entangling two-spin unitary gates and measurements, which are made at random times at a finite rate per spin. Informally, sufficiently frequent measurements yield a “disentangling” phase: in this phase, the state at a given time is weakly entangled, and is fully specified by the outcomes of a relatively recent set of measurements. (The limiting case of this disentangling dynamics is where all the spins are measured simultaneously, leaving the system in a product state that can be read off from the measurement outcomes.) But when the frequency of measurements falls below a critical threshold,

<footnote>* adam.nahum@physics.ox.ac.uk</footnote>

<footnote>† jonathan.ruhman@biu.ac.il</footnote>

<footnote>Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.</footnote>
the dynamics enters an entangling phase [6,7]. In this phase the dynamics produces states with extensive entanglement, which retain quantum information from much earlier times. If the initial state is mixed, rather than pure, then it will rapidly be purified [12] by the repeated measurements in the disentangling phase, but not in the entangling phase.

The simplest toy model for the MPT arises from thinking about the connectivity of the space-time diagram of the quantum circuit, viewed as a tensor network [6]. In this representation a measurement event is a break in the worldline of a spin, across which quantum information cannot be transmitted. When measurements become sufficiently frequent, the circuit falls apart into disconnected pieces, implying that entanglement in the final state is short ranged and there is no transmission of quantum information, from the initial to the final state, over long timescales.

The existence of the MPT poses several types of questions. Viewing the circuit as a quantum information processor, the MPT is a transition in the properties of a randomly generated error-correcting code [11,24,28], the structure of which can be optimal in a certain sense [12]; understanding the MPT may lead to useful insights into fault-tolerant quantum computation [29].

The transition also has consequences for the computational difficulty, for a *classical* computer, of simulating various types of open or monitored quantum systems [6,30–33]. As a simplified thought experiment, we may imagine that we are given the sequence of measurement outcomes obtained in an experimental run (as well as the information about the Hamiltonian, and the initial state), and asked to calculate the quantum state of the system following these measurements. If the dynamics is in the disentangling phase, an efficient matrix-product or tensor-network representation of the evolving state may be possible, while if the dynamics is in the entangling phase, the computation may be intractable. In this sense the MPT can function as an “epistemological” phase transition in which the quantum wave function becomes essentially unknowable.

Philosophically, we may also wonder what the existence of two phases implies about how to distinguish dynamical processes that are intrinsically quantum from those that are effectively classical. For example, in both of the phases separated by the MPT, quantum correlations between local observables are “weak,” but for different reasons. In the disentangled phase, a local operator is correlated only with a few others nearby. In the entangled phase, it may have nontrivial correlations, but these are detectable only by highly nonlocal “scrambled” operators, and hidden from local ones. Only close to the transition point does the system escape both mechanisms, allowing nontrivial correlations for local operators [6,9,13–17,24]. Yet another key question is how to probe the MPT experimentally [13,17].

This question is nontrivial: for example, a naive approach leads to a severe sampling problem (due to the need to compare measurements in distinct experimental runs that have the same measurement outcomes).

Another way of looking at the MPT is as a problem in statistical mechanics and critical phenomena [5, 7, 8, 12, 13, 21, 24, 25, 27, 34]. Open questions abound, both about the nature of the phases and about the critical point separating them. Many variants of the measurement transition can be imagined; how do we sort them into universality classes? Are there simplifying limits where exact results are possible? Are there useful continuum field theories for the MPT and related problems, that allow us to apply the tools of the renormalization group?

This statistical mechanics problem is closely connected to an entanglement transition that takes place in random tensor networks [14,35,36] (we explore the similarities and differences further here) and the same questions apply in that setting. These problems are challenging partly because of the need to average over randomness: either intrinsic randomness in the definition of the dynamics (for example, if we consider dynamics using a random quantum circuit) or simply the inevitable quantum-mechanical randomness in measurement outcomes.

Our focus in this paper is on circuits built from generic unitary gates (for example, Haar-random gates). An alternative profitable direction is to study circuits made from Clifford unitaries [6, 11, 15, 16, 22, 23, 26, 37–41]. Clifford circuits are efficiently classically simulable, which has allowed direct tests of conformal invariance at the MPT in 1 + 1D [9,15] and simulations in 2 + 1D [22]. In general, the universality class of the MPT is expected to differ for Clifford versus generic unitaries (see, e.g., Ref. [16]), though many features of the stable phases are similar.

This paper is a journey through several approaches to the MPT, and also to the closely related “forced” measurement phase transition (FMPT, defined in Sec. II A below), and to the entanglement transition in various types of random tensor networks (RTNs). Our aim is to find settings in which exact results can be obtained for the transition, as well as to clarify the properties of the two phases. We examine several different tools and settings, but the unifying feature is that we consider measurement and entanglement transitions in situations where the complications arising from low-dimensional spatial structure are reduced.

Much of this paper is concerned with circuits with all-to-all couplings between qubits, i.e., with no fixed spatial geometry, which we study using analytical arguments and numeric simulations. (Various types of all-to-all circuit have also been discussed recently in Refs. [12,26].) These circuits are in turn closely related to tree tensor networks, for which we give exact results, including the first exact identification of an entanglement transition in a generic system with finite bond dimension.
Turning to models with a finite number of spatial dimensions, we discuss and extend tools based on mappings to effective “lattice magnets” [13,14,34–36,42–45], involving a replica limit [13,14,34,36], which capture the properties of the two phases, and in principle the critical point. We suggest alternative ways of thinking about these effective models, making connections with ideas from disordered magnetism: in particular, we suggest a construction of order parameters for the MPT and for entanglement transitions in random tensor networks, based on overlap of Feynman trajectories. A key outstanding question is the existence of effective field theories for the MPT. Here we propose—speculatively—two Landau-Ginsburg theories, one for the MPT and one for both the FMPT and the RTN.

The sketch in Fig. 1 contrasts an all-to-all measurement circuit and a circuit with a fixed spatial geometry. In this figure, time runs vertically, and each worldline represents a spin or qubit. Unitary gates are applied between randomly chosen spins at random times, and projective measurements are applied to randomly chosen spins. All-to-all coupling is perhaps the simplest setting for the MPT. Since the distinction between area and volume law breaks down in the all-to-all case (as also in the limit of infinite dimensions), it is natural to focus instead on the transmission of information between initial and final times. Here we characterize this transmission via the operator entanglement [46–51] of the nonunitary time-evolution operator, defined below. This quantity has a simple interpretation in terms of the surface tension of the “entanglement membrane” in the effective replica description, which we discuss. An even simpler heuristic picture for it comes from the classical toy model, in terms of the minimal cut that separates the top of the circuit from the bottom.

<figure><img src="image_3.png" /><figcaption>FIG. 1. Random circuits with measurements. Vertical lines are world lines of individual spins, with time running vertically. The blocks connecting different world lines are independently random unitary gates. These are interspersed by projective measurements, represented by the red blobs. The left panel shows an all-to-all two-local circuit whereas the right panel shows a regular 2 + 1D circuit.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 2. Some of the approaches in this paper. To gain intuition, we start by solving a classical toy model for the transition in the all-to-all circuit, which gives scaling forms for the “minimal-cut” cost that determines the zeroth Rényi entropy (Sec. III). In Sec. IV we turn to the truly “quantum” problem, obtaining exact results for random tree tensor networks. These can be applied to the true quantum transition in all-to-all circuits with “forced” measurements. Section V simulates all-to-all measurement circuits, using operator entanglement of the time-evolution operator, and the convergence with time of two initially orthogonal states, to diagnose the preservation of information over an exponentially long timescale that is a hallmark of the entangling phase. Section VI develops analytical approaches to the MPT and to entanglement transitions based on the replica trick, clarifying the properties of the two phases and suggesting candidate field theories for the critical points in various settings.</figcaption></figure>

We apply all the approaches mentioned above (tree approximations, simulations, replica field theories) in the setting of generic quantum circuits for spin 1/2, as well as related random tensor networks, giving results for scaling properties in the entangled phase and close to the critical point. We also study a solvable “classical” limit of the problem. Our main approaches are illustrated in Fig. 2, and the ensuing section, Sec. II, gives an overview of our results. In closing this Introduction, however, let us briefly clarify the logic of our four-pronged approach to understanding the MPT and its relatives.

Before tackling the “true” quantum circuit problem, we find it instructive to first solve the classical toy model mentioned above, in the particular setting of all-to-all circuits (Sec. III). In this model the entanglement is described in terms of a “minimal cut” through a circuit in which worldlines have been broken by measurement. The minimal cut becomes an exact description of the MPT in certain limits, but in general it does *not* capture either the location of the critical point or the true critical scaling of the quantum problem. Nonetheless, the minimal cut problem yields some useful lessons for the full quantum problem. Most prominently, it captures key qualitative
features of the two phases, including the appearance of an exponentially long timescale for survival of quantum information within the entangled phase. Solving the minimal cut problem also makes clear certain crucial concepts for understanding the MPT in all-to-all circuits, including the local tree structure of the circuit and the relevance of crossover scaling phenomena.

The fact that all-to-all circuits have a local tree structure then motivates us to study entanglement transitions in quantum trees (Sec. IV). In this setting we are able to obtain the exact location of the entanglement transition (and exact critical properties) for a tree tensor network that is relevant to dynamics with Haar-random gates. This result may be useful for further investigations: studies of the MPT in systems with generic unitaries are often hampered by the restriction of numerics to small sizes, which make it difficult to accurately pinpoint entanglement transitions. Moreover, we argue that the critical measurement rate that we identify in the quantum tree is also the exact result for the full all-to-all quantum circuit with forced (postselected) measurements.

Armed with the understanding gained from the minimal-cut and quantum tree problems, we turn our attention to direct numerical simulations of the quantum circuit (Sec. V). The results we obtain are consistent with the critical scaling forms suggested by the previous approaches, and highlight the emergence of an exponentially long timescale associated with information transmission through the circuit in the entangling phase.

Finally, in Sec. VI we discuss mappings of the MPT and of random tensor networks to effective lattice models for a “pairing field,” and we discuss how to coarse grain such models. We construct the simplest candidate Lagrangians that are consistent with the replica symmetry and describe some of their features. We also touch on free fermions subject to measurement [52], which do not show the same kind of transition between weakly and strongly entangled phases but do show transitions of a different type [37,53,54]. We contrast these free systems (which have a continuous, rather than discrete, replica symmetry) with generic models, and we discuss some other variants of the MPT.

## II. OVERVIEW

### A. Models

Our starting point is a dynamical process in which a large number $N$ of spin 1/2s undergoes unitary evolution punctuated by projective single-spin measurements: Fig. 1, left. (Circuits with both unitaries and measurements are referred to as “monitored” or “hybrid” quantum circuits.) The spins are “all-to-all” coupled, meaning that unitary gates may be applied between any two spins in the system. These gates are applied at a uniform rate between randomly chosen pairs of spins, and are themselves drawn

independently from a random ensemble (e.g., the Haar ensemble). Measurements, which are made in the $Z$ basis, are also applied at a uniform rate to randomly chosen spins. The only parameter is $r \in [0, 1]$, which determines the relative rate of measurements and unitaries: in a unit interval of time there are on average $rN$ measurements and $(1-r)N$ unitary operations.

We distinguish between two possibilities for the projective measurements, which we refer to as “measurements” and “forced measurements,” respectively. (Correspondingly we refer to the “measurement phase transition,” or MPT, and “forced-measurement phase transition,” or FMPT.) The outcomes of “measurements” are determined as usual by the Born rule, based on the state of the system at the time of measurement. By contrast the probability of a given outcome for a “forced measurement” is independent of the state. We take it to be 1/2 for both of the two possible outcomes, $\uparrow$ and $\downarrow$—but in fact, for the ensembles of random unitaries we consider, it is completely equivalent to take all the measurement outcomes to be $\uparrow$. We can think of the FMPT as pertaining to a protocol in which we run (exponentially) many samples, discarding all those except those that yield the desired (“postselected”) sequence of outcomes.

To formalize the distinction between MPT and FMPT, define $V_m$ to be the nonunitary time evolution operator represented by a given realization of the circuit. This operator is the product of unitaries and projection operators: we label it by a given sequence $\mathbf{m}$ of outcomes for the measurement events: for example $\mathbf{m} = (\uparrow, \downarrow, \dots, \uparrow)$. ($V_m$ also depends on the total time $t$, locations and times of the unitaries and measurements, and the specific random unitaries in the circuit realization, but we leave these dependencies implicit.) For the MPT, and for a given sequence of unitaries and measurement locations, the probability of a sequence of measurement outcomes is

$$P(\mathbf{m}) = \langle\psi(0)|V_m^\dagger V_m|\psi(0)\rangle, \quad (1)$$

where $|\psi(0)\rangle$ is the initial state. For the FMPT it is

$$P(\mathbf{m}) = 2^{-|\mathbf{m}|}, \quad (2)$$

where $|\mathbf{m}|$ is the number of measurements in a given realization of the circuit. In both cases, the time evolution of a pure state is

$$|\psi(t)\rangle = \frac{V_m |\psi(0)\rangle}{|V_m| |\psi(0)\rangle}. \quad (3)$$

It is occasionally useful to generalize the circuit to a variable number of spin states $q$ for each site. In particular, the limit of large $q$ is one way to motivate the classical problem we describe below.

Having started with the models above, we are led to consider some other related problems. These models are
<figure><img src="image_5.png" /><figcaption>FIG. 3. The entanglement entropy of states and operators can be described in terms of the surface tension of an effective membrane or domain wall. Left: interpretation of entanglement entropy of a spatial subregion, in the entangling phase, as the free energy of an anchored membrane. Right: operator entanglement of the nonunitary circuit (in the entangling phase) in terms of a “horizontal” membrane (see also Ref. [13]).</figcaption></figure>

introduced as we need them. Section IV considers a class of tree tensor networks, one example of which is closely related to the FMPT case above. Section VI addresses both circuits and tensor network models in a finite number of dimensions, in which we do have a sense of spatial locality.

## B. Detecting the entangling phase

Before turning to the critical properties, we discuss the more basic issue of how to distinguish the two phases.

The entanglement transition can be identified with the vanishing of an effective surface tension for a membrane-like object in space time, as we discuss below. In the classical toy model, this membrane is a minimal cut through the circuit [6]. In a more precise picture, it is a domain wall in an effective statistical mechanics problem (see following sections). The surface tension of this membrane or domain wall is positive in the entangling phase [6].

In finite dimensions, the vanishing of this surface tension, which we denote $s_n(r)$ [55], implies a vanishing of the entanglement entropy density of the states produced by the dynamics at late time. This density is the coefficient of the volume law for the entanglement entropy of a spatial subregion, and is given by the surface tension $s_n(r)$. This is because the subregion’s entropy maps to the free energy of a membrane that is anchored on the boundary of region A on the final time surface: see Fig. 3 for a schematic in 1 + 1D.

In the all-to-all circuit there is no distinction between areas and volumes (as in the limit of high dimensions), so the naive attempt to define an entropy density using the entropy of a spatial subregion is contaminated by trivial short-range entanglement (i.e. entanglement that could be removed with a shallow-depth circuit). Instead it is simpler to consider the entanglement properties of the operator that implements the time evolution itself. The operator entanglement [46–50], defined below, is a measure of the amount of quantum information transmitted from the initial to the final time by the nonunitary evolution operator $V_m$. In the membrane picture, this operator entanglement

is equal to the free energy of a “horizontal” membrane that completely traverses the system [13], as shown in Fig. 3. This observable also detects the vanishing surface tension $s_n(r)$ for the domain wall, as detailed below, but it does not require us to specify a spatial subregion.

Gullans and Huse proposed in Ref. [12] to think about dynamics with measurements in terms of the entropy of a state that starts out as maximally mixed, and is gradually purified by the dynamics. (The entangling phase is then a “mixed” phase, where the state remains mixed for a long time, and the disentangling phase is a “pure” phase where the state is rapidly purified.) This mixed state entropy is in fact equal to the operator entanglement of the nonunitary evolution operator. Reference [12] noted the exponentially long timescale for the survival of quantum information in the entangled phase (and plateaus in various observables), which will play an important role below. See also the recent Refs. [27,56].

Formally, the nth operator entanglement entropy of the circuit, denoted $S_n$ throughout this paper, may be defined via the singular value decomposition of the nonunitary time-evolution operator $V_m$:

$$V_m \propto \sum_{j=1}^{2^N} \lambda_j |j\rangle_t \langle j|_0, \quad (4)$$

where $\{|j\rangle_0\}$ and $\{|j\rangle_t\}$ are bases corresponding to the initial and final time. Normalizing the $\lambda_j$ so $\sum_j \lambda_j^2 = 1$,

$$S_n = \frac{1}{1-n} \ln \left( \sum_j \lambda_j^{2n} \right). \quad (5)$$

For the unitary case ($r=0$), $S_n = N \ln 2$ is maximal at all times. For positive $r$, and for asymptotically late times, a single term dominates Eq. (4), meaning essentially that all initial states are projected onto the same final state—i.e., the final state can be read off from measurement outcomes $m$ (and the structure of the circuit) without knowledge of which initial state was fed in.

We also discuss another observable for quantifying the transmission of quantum information from initial to final times, which is more numerically tractable: this is the overlap between two initially orthogonal states, both subjected to the same $V_m$. (In the entangling phase, initially orthogonal states remain orthogonal for a long time.)

We characterize the operator entanglement in the classical toy model (Sec. III), in numerical simulations (Sec. V), using the replica trick (Sec. I), and with a toy model based on multiplying random matrices (Appendix E). The following basic points hold in all of these approaches.
First, in the entangling phase a nonzero density can be associated with the operator entanglement:

$$s_n(r) = \lim_{t \to \infty} \lim_{N \to \infty} \frac{S_n(r, N, t)}{N}. \quad (6)$$

We think of this quantity as the information transmitted per spin, or in the membrane picture as the surface tension for a “horizontal” membrane. $s_n(r)$ is positive in the entangling phase, and vanishes continuously, for all $n \ge 1$, as the critical measurement rate $r_c$ is approached from below.

As for almost any product of many random matrices, we expect that if $N$ and $r$ are fixed, then at sufficiently late times one of the singular values dominates the others and $S_n$ decays exponentially in time. But if $s_n(r)$ is positive, this exponential decay does not set in until a time $\tau(r, N)$ that is exponentially large in $N$. We may define

$$a(r) = \lim_{N \to \infty} \frac{\ln \tau(r, N)}{N}. \quad (7)$$

Close to the transition, at $r \lesssim r_c$ where $s_n(r)$ is small,

$$a(r) \sim s_n(r) \quad (8)$$

(up to an order-1 constant of proportionality). On times $t$ satisfying $\ln t \ll \ln \tau$, the entanglement deviates logarithmically from the “plateau” value dictated by Eq. (6). For example in one regime,

$$S_n(r, N, t) \approx s_n(r)N - \ln t. \quad (9)$$

This formula has also been obtained in various limits in Refs. [12,27]: in particular Li and Fisher in Ref. [27] give a discussion very similar to that in Sec. I, in terms of domain walls in an effective quasi-1D model. In this interpretation $\ln t$ is the translational entropy of a domain wall. More generally, randomness and other effects can modify the nature of the subleading term above slightly, depending on the time regime.

In contrast to the above, the information transmitted per spin, $\lim_{N \to \infty} S_n(r, N, t)/N$, decays exponentially with $t$ in the disentangled phase.

We now give an overview of our approaches to critical properties of these circuits and related models, considering each approach in turn (summarized in Fig. 2). The reader may obtain the key points of each approach from the corresponding Overview section. We also highlight some points that are not yet resolved, and places where our arguments rely on conjectures that could be tested further.

### C. Min-cut toy model

Before attempting an exact treatment of the true quantum transition in the spin-1/2 circuit, we consider a limit (we sometimes refer to this as the “classical” limit) in

which the entanglement transition becomes a simple geometric problem involving a random graph. This graph is defined such that its edges represent the time evolution of each spin, which can be severed by measurement, and its nodes represent interactions (applied gates) between spins. The analog of the operator entanglement entropy is the cost of a “minimal cut” that disconnects the initial-time and final-time nodes: see Sec. III for a detailed definition.

Determining the scaling of this min-cut cost is a toy problem that provides intuition for the generic “quantum” problem. The minimal cut becomes an exact description of the operator entanglement only in special limits, as described in Sec. III (specifically, for projective measurements in the case where the local Hilbert space dimension $q$ goes to infinity, and for a generic local Hilbert space dimension if we consider the somewhat unphysical zeroth Rényi entropy, $S_0$). The “classical” problem has its transition at a measurement rate $r_c^{\text{cl}}$ that is, for spin 1/2, strictly larger than the critical measurement rate $r_c$ for the true quantum transition, as diagnosed, for example, by all the $S_n$ with $n \ge 1$.

We first identify the critical point $r_c^{\text{cl}}$ associated with percolation on the graph, which illustrates the importance of local tree structure in all-to-all circuits. We then present an effective continuum field theory for percolation on this graph, which gives the relevant scaling forms near $r_c^{\text{cl}}$. We demonstrate this critical scaling using extensive numerical simulations for percolation observables and correlation functions. This demonstration is possible despite significant finite-time corrections, which arise because the critical timescale scales as $N^{1/5}$ and is modest even for simulations with very large $N$.

We demonstrate the plateau in the cost of the minimal cut that was described above, $S_0 \sim s_0(r)N$ over a long timescale. Close to criticality at $r \lesssim r_c^{\text{cl}}$ we find the entanglement density (min-cut tension)

$$s_0(r) \sim (r_c^{\text{cl}} - r)^{5/2}, \quad (10)$$

which is an appropriate limit of a general scaling form $S_0 = H(t/N^{1/5}, \delta r N^{2/5})$, and the corresponding long timescale

$$\tau \sim \exp[a(r) \times N], \quad a(r) \sim (r_c^{\text{cl}} - r)^{5/2}. \quad (11)$$

The scaling we identify applies not only for the all-to-all problem, but also for spatially local circuits with spatial dimension $d \ge 5$, as follows from a standard crossover scaling argument.

The all-to-all percolation model has also been analyzed independently in Ref. [57], using a different method in which rate equations for the number of percolation clusters of a given size are solved. This analysis also implies that the scaling variables are $t/N^{1/5}$ and $\delta r N^{2/5}$, in agreement with what we find.
## D. Tree tensor networks: exact results

When the system size $N$ is large, the structure of the quantum circuit in the vicinity of a given unitary is treelike (the smallest loops involve a parametrically large number of unitaries). This means that it is trivial to locate the classical critical point mentioned above. But in some cases (forced measurements) it also allows exact results for the quantum problem. This motivates us to study entanglement transitions on “quantum trees,” i.e., tree tensor networks, in Sec. IV.

While our approach could be generalized, we focus on trees with bond dimension 2, where each node is a random tensor whose probability distribution is invariant under U(2) rotations on its legs. This includes trees that appear spontaneously in the spin-1/2 FMPT circuit for unitaries drawn from the Haar measure, for example.

Formally we can think of an (upside-down) tree like that shown in Fig. 2 as a tensor-network wave function for a single spin at the apex (root) and many spins (leaves) at the base. Our starting point is to characterize the entanglement between apex and base, which for a bond dimension 2 tree is characterized by a single number, $Z$. For an asymptotically large tree, $Z$ has a critical vanishing at a particular measurement rate $r_c$. (In more general trees, $r$ can be thought of as a parameter in the node tensors’ distribution.)

We write a random recursion relation for $Z$ as a function of the generation number $k$ of the tree. This recursion relation allows us to derive the location of the critical point $r_c$ analytically for the case with Haar-random unitaries (we also study a slightly broader class of distributions):

$$r_c = \frac{212 + 75\pi}{362 + 75\pi}. \qquad (12)$$

This critical point $r_c$ is detected by any Rényi entropy $S_n$ with $n > 0$; $S_0$ instead detects the classical transition, at the strictly larger value $r_c^{cl}$, discussed in the previous section. This is the difference between the existence of a percolating path connecting the root of the tree to infinity (for $r < r_c^{cl}$) and the ability of the tree to broadcast a nonzero amount of quantum information from the root of the tree to infinity, rather than an amount that decays exponentially with the distance from the root.

Assuming a plausible conjecture, Eq. (12) is also the value of $r_c$ for the FMPT in the all-to-all Haar circuit, and yields a bound on the critical scaling of the entanglement density $s_2(r)$ [Eq. (6)]. While the treatment of the tree may hold lessons for the MPT in addition to the FMPT, we do not discuss the MPT from this perspective: the measurement correlations encoded in Eq. (1) hamper our approach.

We also obtain the critical scaling of $Z$ for $r \lesssim r_c$. Since the full nonlinear recursion relation for $Z$ is complicated, this requires us to make a conjecture, which is that the universal features of the scaling are faithfully retained

in a simplified nonlinear recursion relation. We can then write a continuum description that describes a Fisher-Kolmogorov-Petrovsky-Piskunov- (FKPP) like traveling wave [58]. This is a standard description for the partition function of a directed polymer on a tree [58], with the addition of a diffusion constant that varies with the (fictitious) spatial coordinate, reflecting the nonlinearity of the recursion. For the parameters of the trees we treat, there is a surprisingly rapid scaling close to the critical point: the entanglement between apex and base of an infinite tree scales as ($S_2^{\text{tree}} \sim 2Z$)

$$S_2^{\text{tree}} \sim \exp \left( - \frac{\text{const}}{\sqrt{r_c - r}} \right). \qquad (13)$$

We also address the scaling of $S_n$ as a function of tree size exactly at $r_c$.

Using a nonrigorous argument, we extend these formulas to give the entanglement of a subset of the spins, in a tree tensor-network wave function for a spin chain (whose spins are the leaves of the tree). These results are not relevant to the all-to-all circuit, but are interesting in the context of tree tensor-network states, which are toy models for some features of scale invariance in 1+1D, and are also useful numerical tools [59–68]. We obtain a “modified minimal cut” formula for the tree, in which the cost of cutting a bond in the tree is loosely speaking weighted by appropriate factors of the quantity $Z$, which is parametrically small close to $r_c$. This gives a quantitative picture of how the entanglement of $\ell$ consecutive spins in a tree tensor-network state goes from the well-known logarithmic scaling, $S_\ell \sim c(r) \ln \ell$, suggested by its hierarchical structure [67–69], to an area law state, $S_\ell = O(\ell^0)$. We find that $c(r)$ vanishes exponentially as $r \to r_c$ and that the state is area law even at $r_c$.

Recently, Ref. [68] studied the entanglement transition in a quantum tree state, with bond dimension 3, using a different approach. The authors conjectured that the scaling was the same as in a statistical mechanics model that shares some of the features of a replica formulation derived from the tree (the exact replica formulation was not tractable). Surprisingly, the findings in Ref. [68] are quite different from those we obtain (assuming the conjecture mentioned in the previous paragraph) in the trees studied here. For example, Ref. [68] finds that the coefficient $c$ in $S \sim c \ln \ell$ is a power law in the control variable close to the transition, and that entanglement is super-area-law at $r_c$. The reason for the different results in these two models remains to be understood.

Our conjectured continuum description for scaling in the tree has a parameter $\Delta$ that describes the degree of disorder in the tensor network, and which determines the critical exponents. For the trees we study, whose node tensors have a distribution with a U(2) invariance property on the legs, this parameter is fixed to $\Delta = 1/4$ at $r_c$. This corresponds
to a “strong disorder” regime [58]. We raise the question of whether general distributions of tensors allow us to explore the phase transition at other values of $\Delta$. If so, it is possible to obtain a range of universality classes for the tree transition, analogous to a renormalization group fixed line. However, we do not determine whether this is possible.

## E. Direct simulations of quantum circuits

We perform direct simulations of the all-to-all measurement circuit and forced-measurement circuit, and interpret the results in the light of the tree calculation and the replica approach described below. These simulations are computationally demanding: we are limited to system sizes $N \le 20$ for quantities involving states and to smaller sizes for the operator entanglement. Determining $r_c$ accurately (the value of which is expected to differ for measurements and forced measurements) is not possible, but we are able to confirm many of the key features of the entangled phase in Sec. II B.

We give evidence for the plateau, Eq. (6), in the operator entanglement, with a nonzero information transmission per spin $s(r)$ that is asymptotically time independent, and for a positive exponential growth coefficient $a(r) > 0$ for the characteristic timescale within the entangled phase.

It is convenient to define this timescale $\tau$ via the late-time convergence of two distinct, initially orthogonal, states $|\psi_1(t)\rangle$ and $|\psi_2(t)\rangle$ that are postselected to undergo the same sequence of measurement outcomes, so that they are evolved with the same $V_m$. These states remain approximately orthogonal for a long time in the entangled phase: a kind of effective unitarity of the nonlinear, nonunitary time evolution Eq. (3) for a given $m$. (This orthogonality is related to the error-correction property of the dynamics [11,12].) The two states collapse at late times. We show that $a(r)$ is positive at small $r$ and vanishes at large $r$.

For forced measurements our expectation is that $r_c$ is given by the result of the tree calculation. Numerically, it in fact becomes unmeasurably small at a significantly smaller value of $r$. Our interpretation of this is that, because of exponential scaling in Eq. (13), the quantities $s(r)$ and $a(r)$ vanish extremely fast as $r \to r_c$. A more stringent test of the identity of the two transition points would be valuable.

We have also examined the observables discussed here in 1+1D circuits, motivated by the fact that, since they do not require us to introduce a spatial bipartition of the system, they avoid introducing a length scale that is smaller than the system size. We will report on these findings elsewhere.

## F. Replicas and field theories

A key question is whether useful continuum field theories can be written for the MPT and FMPT, and also for entanglement transitions in (reasonably generic [70]) random tensor networks. This question has not been resolved,

despite progress on mapping the quantum problems to effective “classical” lattice models [13,14,34–36,42,43]. A basic issue is the need to handle disorder. The most familiar approach to this is to use the replica trick [13,14,34,36]. (In this section we use $N$ to denote the number of replicas: this should not be confused with the number of physical spins in the previous sections.) However the complicated $N$ dependence of the interactions makes it unclear a priori how to coarse grain these effective lattice models.

In Sec. VI we start by reviewing the approach of mapping circuits and tensor networks to effective lattice models for permutations. We discuss coarse graining of such models in a heuristic way. We then suggest an alternative way of thinking about effective statistical mechanics models for circuits (motivated by a physical picture for the emergence of permutations, in terms of phase cancellation in sums over Feynman histories [34,43]). This picture connects entanglement transitions to approaches familiar from disordered magnetism, the random-field Ising model, spin glasses, etc. [71].

With this motivation, we construct the simplest Lagrangians that capture the global symmetry associated with the replica formulation [34,36,43], which we denote

$$G_N = (S_N \times S_N) \times \mathbb{Z}_2, \quad (14)$$

and which pass some basic consistency tests.

The limiting number of replicas $N$ is distinct for the case of (i) the MPT and (ii) both the FMPT and the RTN [13,14,36]. For the FMPT and RTN we need to take $N \to 0$, as in standard quenched disorder problems. For the MPT, realizations are weighted by the additional Born-rule factor, which increases the number of replicas: we need to take $N \to 1$ [13,14]. Previously, properties in the vicinity of a fine-tuned point have been used to motivate the suggestion that all of these problems may have similar universal properties, despite the differing numbers of replicas [14]. However, we find that the simplest field theory candidates (which may of course be too simple) are strikingly different in the two different cases.

The Lagrangians we propose have the schematic forms

$$\begin{aligned} \mathcal{L}_X &= \sum_{ab} [(\partial X_{ab})^2 + \mu X_{ab}^2 + X_{ab}^3], \\ \mathcal{L}_Y &= \sum_{ab} [(\partial Y_{ab})^2 + \nu Y_{ab} + Y_{ab}^3] + \sum_{abcd} Y_{ab} F_{ab,cd} Y_{cd}. \end{aligned} \quad (15)$$

$F$ is the tensor $F_{ab,cd} = \delta_{bd} + \delta_{ac}$. In these equations we suppress all coupling constants except the crucial one that drives the transition, denoted $\mu$ or $\nu$. Both space and time derivatives are grouped together in the derivative term: in the case of the circuit there will in general be a nonuniversal speed $\nu$ appearing, so that the derivative terms have the
form $(\partial_t X)^2 + v^2(\nabla X)^2$. The plus sign means that there is an emergent Euclidean, rather than Lorentzian, space-time symmetry [6,9,15] (see endnote [72]).

In the Lagrangian $\mathcal{L}_X$, the field $X_{ab}$ is a real $N \times N$ matrix satisfying $\sum_a X_{ab} = 0$ and $\sum_b X_{ab} = 0$. It may be thought of (modulo a constant shift) as a coarse-grained permutation matrix. This Lagrangian is appropriate for the replica limit $N \to 1$. It has upper critical dimension $D = 6$ (this is the space-time dimension in the case of the circuit). This is a candidate Lagrangian for describing the MPT.

At first we might assume that the same Lagrangian $\mathcal{L}_X$ for the measurement transition could be continued to the distinct limit $N \to 0$ in order to describe the random tensor network and the forced-measurement transition. We argue in Sec. VI that this is not the case. Instead, the simplest candidate for the FMPT and RTN is the Lagrangian $\mathcal{L}_Y$. Here, the field $Y$ is a real $N \times N$ matrix, with $N \to 0$, that does *not* satisfy any constraints on its row and column sums. The upper critical dimension for this theory is the unexpectedly large value $D = 10$. See Sec. VI for further discussion.

We caution that these theories are conjectures based on symmetry considerations and certain limited consistency checks. Further investigation is required to determine whether they are in fact sufficient to describe the problems of interest. It is possible that more elaborate continuum descriptions are required, either for a particular microscopic model or in general.

Indeed, the trees described in Sec. II D, which have exponential order parameter scaling close to the critical point, appear to be one case that is not captured by $\mathcal{L}_Y$ (Contrary to the naive guess that the high-dimensional limit of the field theory and the tree would show similar “mean-field” critical scaling.) We defer an examination of the reason for this to a future work.

In Sec. VI we also present some results that are independent of the speculative field theories above. In particular, we use effective domain wall pictures to obtain the scaling within the phases (mentioned above in Sec. II B).

We also briefly discuss the use of Ising toy models for the properties of the second Rényi entropy in measurement dynamics, pointing out that the formalisms of Ref. [34] or Ref. [43] allow these to be justified in certain strongly entangled regimes, rather than being regarded simply as toy models as in previous work. However, quenched disorder must be taken into account in the resulting Ising model. Additionally, the Ising picture breaks down close to the critical point (or in the disentangling phase) and also at long times.

Finally we discuss variants of the MPT, FMPT, and RTN phase transitions. We point out that quite different scaling obtains for models of free fermions subjected to measurements, as a result of continuous rather than discrete replica symmetry.

### III. MINIMAL CUT PROBLEM

A natural starting point for understanding the MPT is to map the quantum circuit to a classical graph on which one can study a classical “minimal cut” optimization problem [6]. In this mapping there is a phase transition at the point where the graph percolates.

We think of this classical min-cut problem as a toy model for the generic quantum transition. In the circuits we study, the cost of the minimal cut gives the exact value of the (somewhat unphysical) zeroth Rényi entropy [73], $S_0$. It also gives exact results for the other Rényi entropies in the limit of large local Hilbert space dimension (e.g., a large value of each spin), with Haar-random gates. But in general, the minimal cut is only an upper bound on the entanglement entropies $S_n$ with $n \ge 1$. (There can be no quantum information propagated from the initial to the final time if the associated classical graph is disconnected; in this regime, the “cost” of the minimal cut vanishes.) The true “quantum” transition in general occurs at a smaller value of $r$ than the classical transition discussed in this section (and in general has distinct universal properties). Despite this, the classical problem conveys some useful lessons.

Viewed as a graph, the circuit is a bond percolation configuration, as described below. The frequency of projective measurements determines the fraction of broken bonds in this percolation configuration. The minimal cut is the minimal number of additional bonds that must be severed in order that two parts of the boundary of the circuit, $A$ and $\bar{A}$, no longer have any percolating path between them. This minimal cut is a unifying heuristic [35,51,67,74–76] for the entanglement of various objects, depending on how we choose $A$ and $\bar{A}$. If these are taken to be two complementary subsets of the legs of the circuit at the final time, then the minimal cut gives the entanglement $S_0$ of a subset $A$ of the spins in the final state quantum state, assuming the initial state is a product state. Here we are more interested in a minimal cut separating the top boundary of the circuit from the bottom. That is, $A$ contains all the circuits “legs” at the final time, and $\bar{A}$ all those at the initial time. This “horizontal” minimal cut is a measure of information transmitted from the initial to the final time, equal to the operator entanglement $S_0$ for the nonunitary time evolution operator $V$ (Sec. II).

In the percolating regime, this horizontal minimal cut must sever a number of bonds that is extensive in the number of spins $N$, so that $S_0 \simeq sN$. The coefficient $s$ is a “surface tension” [51] for the minimal cut, which vanishes continuously at the percolation threshold. In 1+1D this transition is conformally invariant. Many of the critical exponents, such as the correlation length exponent $\nu$, are standard percolation exponents, while others are less familiar, since the minimal cut is an additional optimization problem built on top of the percolation configuration [6,77].
<figure><img src="image_6.png" /><figcaption>FIG. 4. (a) Example of a small unitary circuit with $N = 6$ spins. Black worldlines represent the evolution of a particular spin, with time proceeding vertically. Colored blocks indicate two-spin unitaries, and broken lines (marked with red crosses) indicate single-spin measurements. (b) The equivalent graph, with nodes representing unitaries [node of a particular color corresponds to the unitary of the same color in (a)] and edges representing unbroken segments of worldline. Small red (gray) circles denote the initial (final) time for a given spin. A possible minimal cut for this graph is shown by the dashed line: removing the two indicated edges disconnects the initial and final times. (c) The classical graph arranged as a tree, with the purple node used as a seed and generation number $k$ proceeding downward. (This illustrative circuit forms a tree; in general the structure of a large circuit is only locally treelike.).</figcaption></figure>

In the circuit without fixed spatial structure, where any qubit can couple randomly to any other, the location of the critical point, and the basic critical exponents, can be determined exactly, as we show in this section. These exponents also apply to the finite-dimensional minimal cut problem when the spatial dimensionality $d$ is greater than or equal to 5 (Sec. III E). Interestingly, there is also reason to speculate that the exponents apply for some versions of the quantum measurement transition in high dimensions, even without the minimal cut approximation (see Sec. VI, where we discuss Landau theory for the measurement transition and entanglement transitions).

For all-to-all circuits, the classical percolation problem is defined as follows. The circuit defines a random graph, in which the nodes (vertices) correspond to unitaries and the edges are the sections of spin worldline that are not broken by measurements. In other words, an edge connects two nodes whenever (i) the two nodes correspond to successive unitaries in the time evolution of a particular spin; and (ii) that spin is not measured during the time in between the two unitaries. Figure 4(a) shows an example circuit, and Fig. 4(b) shows the corresponding graph. Each node has at most four edges connected to it, corresponding to the four legs of each unitary in Fig. 4(a). The minimal cut in the figure indicates an operator entanglement $S_0 = 2$ for this small circuit.

We take the number of spins to be very large, $N \gg 1$, while by definition the degree (connectivity) of each node

is only of order 1. In this situation, standard considerations [78] imply that the local structure of the graph is treelike on both sides of the percolation transition. Above the percolation transition closed loops do exist, but their length is of order $\ln N$.

### A. Local tree structure and percolation

To relate the classical graph to a tree, imagine starting at an arbitrarily chosen “seed” unitary in the bulk of the circuit (far from the initial and final time boundaries) and tracing out its cluster: finding the nodes connected to the seed by an edge, then those connected to the seed by a path of length 2 edges, etc. In this way the cluster containing the seed may be arranged in a tree, with the seed at the top and subsequent generations of connected nodes below: see Fig. 4(c). We denote the generation number by $k$, with $k = 0$ being the seed.

The probability $p$ that a given one of a unitary’s four possible edges is absent is equal to the probability that (as we travel along that segment of worldline) the spin undergoes a measurement before it is involved in another unitary. This probability is given by

$$p = \frac{r}{2-r} \qquad (16)$$

The small circuit shown in Fig. 4 contains no loops. In general the circuit can contain loops. However, a subcluster of any finite size is guaranteed to be free of loops in the limit $N \to \infty$ (since the probability that two unitaries in generation $k$ both connect to the same unitary in generation $k+1$ is of order $1/N$).

To understand the location of the critical point, note that the average branching number of the tree (the average number of descendants of a given node with $k > 0$) is $3 \times (1 - p)$. The percolation transition in the graph occurs when the branching number is 1, i.e., at $p_c = 2/3$ (as also noted in Ref. [12]), or

$$r_c = \frac{4}{5}. \qquad (17)$$

(In this section only, $r_c$ denotes the classical transition point, $r_c = r_c^{cl}$.) When $r$ is greater than $r_c$, all trees are finite even in the limit $N, t \to \infty$, where the graph itself is infinite: starting at a seed node, the tree inevitably dies out after a finite number of generations. Therefore, at $r > r_c$ all unitaries are in finite clusters; this is the nonpercolating phase. When $r < r_c$, however, there is a nonzero probability $f_\infty$ that a tree continues forever, or rather until it includes a number of nodes proportional to $N$. In the percolation problem, $f_\infty$ is the order parameter—the probability that the unitary lies in the infinite cluster. The critical exponent $\beta$ for this order parameter is 1, which is
the mean-field value for percolation. A simple recursive treatment (Appendix A 1) shows that, close to $r_c$,

$$f_{\infty} \approx \frac{50}{3}(r_c - r)^{\beta}, \quad \beta = 1. \qquad (18)$$

Note that the window of $r$ in which we can hope to observe critical scaling, corresponding to $0 < f_{\infty} \ll 1$, is rather narrow as a result of the large (nonuniversal) prefactor in Eq. (18).

## B. Effective 1D continuum theory

We now show that near the critical point the basic scaling variables for the percolation and minimal cut problems are

$$\frac{t}{N^{1/5}}, \quad \delta r N^{2/5}, \qquad (19)$$

where $\delta r = r - r_c$ and $t$ is, say, the temporal duration of the evolution. For example, the characteristic timescale for a large system at its critical point scales as $N^{1/5}$. In Secs. III C, III D and Appendices A 2–A 4 we show how these variables appear in scaling forms for the minimal cut and other observables. The critical exponents in Eq. (19) have also been obtained in independent work [57], by an approach that is complementary to the one below (Sec. C).

This problem is similar to one of crossover scaling, in which a system that is effectively very high dimensional on short timescales crosses over to one that is one-dimensional on long scales. This analogy can be used to obtain the above exponents, as we discuss in Sec. III E. This approach also sheds light on the quantum problem (Sec. VI). Here, however, we solve the classical problem directly.

To simplify the discussion, let us consider a percolation problem with the same basic features as the circuit, but with a simpler connectivity rule inspired by the Erdős-Rényi random graph [78]. This simplification does not change the universality class, as we show numerically in Appendix 3. The random graph we consider has a layered structure, with one layer for each timestep. This graph may be contrasted with one studied in Ref. [26], which maps a measurement transition in a class of “instantaneous quantum polynomial time” circuits to the percolation transition in an Erdős-Rényi graph without a time dimension.

We discretize the time $t$ in integer steps. At each $t$ we have $N$ nodes, labeled $(i, t)$ with $i = 1, \dots, N$. We allow edges only between sites in adjacent $t$ layers, each edge being present with probability $b'/2N$, independently of the others. This scaling with $N$ ensures that the average degree of a site, $b'$, is $O(1)$, as in the circuit. It is easy to see by thinking about the local tree structure that the phase transition is at $b'_c = 1$. As in the circuit, connectivity is local in time, but there is no notion of spatial structure with a layer at a fixed time.

Classical percolation can be mapped to the $Q$-state Potts model in the limit $Q \to 1$ [79–82]. For our problem, the fact that each site couples to all the sites on the adjacent layers means that the Potts partition function simplifies after a Hubbard-Stratonovich transformation with a field $\Phi(t)$ that depends only on time. This transformation is shown in detail in Appendix A 2. The field $\Phi$ may be taken to be a $Q \times Q$ traceless diagonal matrix, on which Potts symmetry acts by permuting the diagonal components.

It is possible to take the continuum limit in a controlled way, to give an effective one-dimensional field theory. Close to the critical point, such that $b' - 1 = \delta b' \ll 1$, the partition function for this field theory is

$$Z = \int \mathcal{D}\Phi \exp\left(-\int dt \mathcal{L}\right), \qquad (20)$$

with

$$\mathcal{L} = \frac{1}{4} \mathrm{tr}(\partial_t \Phi)^2 - \frac{\delta b'}{2} \mathrm{tr} \Phi^2 - \frac{1}{6\sqrt{N}} \mathrm{tr} \Phi^3. \qquad (21)$$

Modulo the values of the order-1 constants, we expect the same field theory to apply to the percolation model arising from the circuit.

The factor of $1/\sqrt{N}$ in Eq. (21) allows a long timescale and nontrivial scaling forms to emerge at the critical point $\delta b' = 0$, despite the fact that the effective field theory is one-dimensional. One-dimensionality implies that for any fixed $N$, correlations decay exponentially at sufficiently large $t$, but the timescale diverges with $N$.

The critical exponents for the minimal cut problem in the all-to-all circuit follow from the observation that the change of variables

$$\tilde{t} = \frac{t}{N^{1/5}}, \quad u = \delta r N^{2/5}, \quad \tilde{\Phi} = \frac{\Phi}{N^{1/10}}, \qquad (22)$$

eliminates $N$ from the action. Scaling forms for correlation functions follow from this fact together with the corresponding scalings for operators. We discuss some examples in the following subsection and in Appendix A 4.

We may also obtain these exponents from a crossover scaling argument if we assume that they are the same as those in a system that does have spatial structure, but with a very high spatial dimensionality $d$. This crossover is described in Sec. III E.

In fact, the exponents in Eq. (19) apply for any $d > 5$ (in an appropriate regime of timescales) with logarithmic corrections in $d = 5$. This is because $d = 5$ gives a total space-time dimension of 6, which is the upper critical dimension for percolation. This fact allows an even simpler mnemonic for the above exponents. Suppose for a moment that we are considering a graph with a regular lattice in spatial dimension $d = 5$, with $N = L^d = L^5$,
where L is the system size. d = 5 is the lowest dimension in which mean-field exponents apply (up to logarithms). In this picture, the first scaling variable above is simply t/L, corresponding to the dynamical exponent z = 1 in the five-dimensional theory, and the second scaling variable is u = δrL¹/², with the mean-field correlation length exponent v = 1/2. In d > 5 we must also consider the dangerous irrelevance of the interaction term in the field theory [83] (which means that the relevant timescale is no longer t/L), but this term can be treated using a standard coarse-graining argument (Sec. III E).

### C. The percolation probability

Before describing the minimal cut itself (Sec. III D), we first consider an observable that is simpler to study both analytically and numerically—namely, the probability $P_{\text{perc}}$ of percolation between initial and final times in the classical graph. The value of $1 - P_{\text{perc}}$ is equivalent to the probability that the operator entanglement is exactly zero, since nonpercolation of the classical graph implies that the initial and final times are causally disconnected.

$P_{\text{perc}}$ has scaling dimension zero, i.e., it has no power-law prefactor in $N$, so it is useful for numerical tests of the scaling defined by Eq. (19). In Appendix A4 we present numerical results for two observables with nontrivial scaling dimension: namely, the probability of two nodes on either the same or opposite time boundaries being connected to the same cluster. We show that these observables are also described by the scaling variables in Eq. (19).

In the Potts language, $P_{\text{perc}}$ is expressed in terms of the free-energy cost of twisted boundary conditions [84]. (In the 1D field theory this free energy involves boundary magnetic fields that are parametrically large in $N$; this is discussed in Appendix A2.) We obtain the scaling form:

$$P_{\text{perc}} = F\left(\frac{t}{N^{1/5}}, N^{2/5}\delta r\right). \qquad (23)$$

(Here $t$ denotes the full temporal duration of the dynamics.) First consider the critical point $r = r_c$, for which

$$P_{\text{perc}} = F_{\text{crit}}\left(\frac{t}{N^{1/5}}\right). \qquad (24)$$

In principle we should obtain a scaling collapse simply by plotting $P_{\text{perc}}$ as a function of the scaling argument. Practically speaking, however, the characteristic timescale $N^{1/5}$ is modest for the values of $N$ we can access numerically, and it appears to be necessary to include a subleading correction. This correction is of a type that is generically present for nonperiodic boundary conditions, and corresponds to replacing the scaling variable with $(t - c_0)/N^{1/5}$, for a nonuniversal $O(1)$ constant $c_0$.

Figure 5 (inset) shows raw data for the percolation probability $P_{\text{perc}}$ of the classical graph (for the all-to-all circuit)

<figure><img src="image_7.png" /><figcaption>FIG. 5. The probability of percolation, $P_{\text{perc}}$, as a function of time for the classical circuit with the critical measurement rate $r = r_c = 4/5$. Different curves correspond to different system sizes $N$. The inset shows the raw data. In the main panel the time is rescaled by $N^{1/5}$ and a shift $c_0$ is introduced, with $c_0 \approx 1.3$. All data is averaged over 40 000 realizations.</figcaption></figure>

as a function of time and $N$. As can be seen in the main panel, this data collapses onto a single curve when $P_{\text{perc}}$ is plotted against $(t + c_0)/N^{1/5}$, where $c_0 \approx 1.3$.

At any fixed values of $r$ and $N$, the probability $P_{\text{perc}}$ decays exponentially with time $t$ at large enough values of $t$. One can extract the associated decay time $\tau(r, N)$, which according to Eq. (23) has the scaling form

$$\tau(r, N) = N^{1/5} W(N^{2/5} \delta r). \qquad (25)$$

This scaling is confirmed in Fig. 6. Figure 6 comprises a check of off-critical scaling close to $r_c$ as well as the scaling at $r_c$ that is shown in Fig. 5.

The decay time $\tau(r, N)$ of the percolation probability constitutes one way of defining a characteristic timescale over which information is able to propagate between the initial and final times. A key feature of the classical graph, which carries over to the quantum case, is that the timescale $\tau$ grows very rapidly with $N$ within the entangling phase. At any fixed $r < r_c$, we can argue that as $N \to \infty$ the timescale $\tau$ grows as

$$\tau_{r<r_c} \sim \exp[a(r) \times N], \qquad (26)$$

neglecting power-law prefactors. In the present classical problem, this exponential growth can be understood in terms of rare events that disconnect the cluster. Close to the transition we must have

$$a(r) \sim (\delta r)^{5/2} \qquad (27)$$

in order to match the scaling form [85]. We expect that $P_{\text{perc}}$ is close to 1 for $t \ll \tau$. This exponentially long
timescale can also be seen directly from the field theory in Eq. (21), in terms of “instantons” in the field theory (domain walls in time); see Appendix A 2.

As mentioned in the previous subsection, one can model the minimal cut problem using the simpler setup of a sequence of layered Erdős-Rényi random graphs. Within this model one can calculate the percolation probability $P_{\text{perc}}$ and characteristic decay time $\tau$. In Appendix 3 we show that this layered Erdős-Rényi model gives the same scaling behavior as in Figs. 5 and 6.

Percolation two-point functions give further information on the connectivity of the circuit. These are analyzed in Appendix A 4.

### D. Scaling for the minimal cut

Because of the lack of spatial structure in the all-to-all model, it is natural to focus on the transmission of information between the initial and final times. One measure of this transmission is the operator entanglement of the linear, but nonunitary, operator $V$ that defines the time evolution for a particular sequence of measurement outcomes (Sec. II).

In the minimal cut picture, the operator entanglement between initial and final times is the cost of the minimal cut through the circuit that separates the initial and final times [as illustrated in Fig. 4(b)]. We refer to this cost as $S_0$ (the Hartley entropy), although in some cases (including the limit of infinite local Hilbert space dimension, mentioned above) it is equal to the other Rényi entropies as well.

The behavior of $S_0$ is most interesting within the entangled phase, so let us consider some fixed $r < r_c$. As illustrated in the previous subsection, in this phase there is an

<figure><img src="image_8.png" /><figcaption>FIG. 6. The characteristic decay time $\tau$ for the percolation probability. The inset shows $\tau$ as a function of the measurement rate $r$ for different values of the system size $N$. The main figure shows this same data plotted as a function of the scaling variables in Eq. (19).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIG. 7. The cost of the minimal cut, $S_0$, divided by the system size $N$, extrapolated to the limit of $N \to \infty$. The main figure shows $S_0/N$ as a function of $r$, while the inset shows $S_0/N$ as a function of $r_c - r$ in double-logarithmic scale. The dashed red line shows the dependence $S_0/N \propto (r_c - r)^{5/2}$. Details of the extrapolation procedure are discussed in the text. Error bars are smaller than the symbol size.</figcaption></figure>

exponentially large (in $N$) timescale over which the percolation probability is close to 1. Correspondingly, there is a parametrically large time range, corresponding to [86] $1 \ll \ln t \ll N$, over which $S_0/N$ is approximately constant. The crudest picture for the subleading corrections gives (see endnote [87])

$$S_0/N = s - \mathcal{O}(\sqrt{N^{-1} \ln t}). \qquad (28)$$

We refer to the range of times where subleading terms are negligible as the “plateau” in the entanglement. Over this large time window, the horizontal minimal cut has a well-defined cost per spin, $s$. This cost per spin $s$ is the infinite-dimensional version of the line tension for the minimal cut in the 1 + 1D case or the surface tension in the 2 + 1D case [6]. These quantities all vanish at the critical point. In the plateau regime, the information per spin transmitted by the circuit is nonzero, up to an exponentially large time.

The inset of Fig. 7 shows this cost per spin in the entangled phase, as measured by a numerical simulation using the Ford-Fulkerson method [88]. The details of the extrapolation to large $N$ are described below and in Appendix A 5.

Let us relate this minimal cut to the scaling theory close to the critical point. We expect the scaling form

$$S_0 = H\left(\frac{t}{N^{1/5}}, N^{2/5} \delta r\right). \qquad (29)$$

(see Sec. III B). If we assume that within the entangled phase there is a time regime during which $S_0$ is extensive
in N and independent of time (i.e., independent of the first scaling variable above), then we obtain in this regime

$$S_0 = s(r)N, \qquad (30)$$

with the entropy per spin $s(r)$ scaling as

$$s(r) \sim (r_c - r)^{5/2} \quad (r \lesssim r_c). \qquad (31)$$

The main panel of Fig. 7 shows $s(r)$ close to the critical point on a double logarithmic scale. Though we cannot extract a clear power law from the data, it seems roughly consistent with the prediction, Eq. (31).

In order to numerically obtain the value of $s(r)$ for the plots above, we measure $S_0$ as a function of the time $t$ and the system size $N$ from simulations. For a fixed $t$, we find that $S_0(t, N)/N$ has a linear dependence on $1/\sqrt{N}$ at large $N$ [in line with the simple picture in Eq. (28)]. This dependence allows us to estimate a value of $S_0(r)/N$ in the limit of $N \to \infty$ by extrapolating the linear relationship to $1/\sqrt{N} = 0$. Further details of this extrapolation procedure are presented in Appendix A 5.

### E. Finite dimensions with $d \ge 5$

The scaling exponents identified in Secs. III B and III C also apply to the classical problem in a system with a regular spatial lattice (and unitaries applied only between nearest neighbors) in a large enough number of spatial dimensions $d$, as we now discuss. The total space-time dimension, $d+1$, should be greater than 6, which is the upper critical dimension for percolation (in $d=5$ we have the same exponents with additional logarithms).

We start with the standard Potts representation of percolation [79–82] in $d+1$ dimensions. Suppressing all $O(1)$ constants, as well as a nonuniversal velocity scale, a continuum action is

$$S = \int dt d^d x \operatorname{tr}[(\partial_t \phi)^2 + (\nabla \phi)^2 + \delta r \phi^2 + \phi^3]. \qquad (32)$$

Here $\phi$ is a traceless diagonal $Q \times Q$ matrix, as in Sec. III B. Our system is of extent $L$ in each of the spatial dimensions, with

$$N = L^d, \qquad (33)$$

and extent $t \gg L$ in the time direction. We take the UV cutoff (“lattice spacing”) to be 1.

We coarse grain the system by a factor of order $L$, so that the spatial system size becomes comparable with the UV cutoff, and we have an effective 1D theory as far as correlations on scales $\gg L$ are concerned. Since the cubic coupling is irrelevant, with renormalization group (RG)

eigenvalue $y_3 = -(d-5)/2$, it decreases during the flow, leading to (again we suppress order-1 constants):

$$S_{\text{eff}} = \int ds \operatorname{tr}[(\partial_s \phi')^2 + \delta r L^2 \phi'^2 + L^{-(d-5)/2} \phi'^3]. \qquad (34)$$

Here $s$, the coarse-grained time coordinate, is equal to $t/L$, and $\phi' \sim L^{(d-1)/2}\phi$ from the scaling dimension of the field in $d+1$ dimensions. If we now write the action in terms of $t$ and $\Phi \equiv L^{1/2}\phi'$, we recover the form of the action in Eq. (21) with $N = L^d$.

Because of the dangerous irrelevance of $\phi^3$ [83], a finite-dimensional model with $d > 5$ has two distinct large timescales,

$$L = N^{1/d} \quad \text{and} \quad L^{d/5} = N^{1/5}. \qquad (35)$$

The shorter timescale (which is compressed to order 1 in the all-to-all model) marks the crossover between $(d+1)$-dimensional and one-dimensional scaling for correlation functions. The longer timescale is the one of more interest to us, and indicates the time at which the percolation probability starts to vary away from unity. This longer time becomes the characteristic critical timescale in the all-to-all model.

The scaling forms discussed above carry over to the present case $(5 < d < \infty)$ with $N \to L^d$.

### F. Lessons for the full quantum problem

So far we have discussed the (classical) minimal cut problem in all-to-all and high-dimensional circuits. A priori, one can expect the universal properties of the generic measurement transition to be different from those for the minimal cut transition: the minimal cut is only an exact representation of the entanglement in certain special cases (as described at the beginning of Sec. III). Nevertheless, as in $1+1$D, the solution of the minimal cut problem provides more general lessons.

First, there are qualitative features that carry over to the generic problem. The most basic feature is the existence of a transition between a phase in which the operator entanglement $S(t)$—the information propagated from the initial to the final time—decays quickly with time, and a phase in which an extensive value of entanglement, $S(t) \sim sN$, persists over a time that grows exponentially with the number $N$ of spins. In Secs. V and I we demonstrate that these features carry over to the operator entanglement (as measured by the von Neumann or Rényi entropies $S_{n \ge 1}$), and related observables, in spin-1/2 circuits with measurements or forced measurements. Another generic feature is that close to the critical point, the scaling of the exponential timescale is tied to that of the plateau entanglement: $\ln \tau(r) \sim Ns(r)$ (Sec. I).

The minimal cut model also illustrates a possible relationship between the all-to-all case and the case of a
high-dimensional regular lattice. In the classical problem, the exponents of the all-to-all model are those of finite but high dimensions, once we take account of the dangerous irrelevance of interactions in high dimensions, which leads to a critical timescale $L^{\text{const}}$ that is parametrically larger than the linear system size $L$ (a timescale $\tau \sim L$ is what one would naively expect from $z=1$ scaling). In Sec. VI we discuss similar crossovers in field theories for generic quantum models. However, we caution that our results in Sec. IV suggest more complex possibilities in the all-to-all systems.

Finally, we saw that in the classical problem, the percolation order parameter and the value of $r_c$ could be obtained exactly by studying a simpler problem on a tree. In the next section, we propose that exact results for the full quantum version of the FMPT can also be obtained by studying trees: not only their classical connectivity, as here, but their “quantum” connectivity as defined by entanglement measures for tree tensor networks.

# IV. ENTANGLEMENT TRANSITIONS IN QUANTUM TREES

## A. Motivation for studying quantum trees

Locally, the all-to-all circuit has the structure of a tree (Sec. A). Viewing the circuit as a graph whose nodes are unitaries and whose edges are segments of spin worldline, the size of the smallest loops diverges when $N \to \infty$. This is true for all values of the measurement or projection rate, including deep in the entangled phase. We propose that this allows some exact results for the phase transition in the circuit, in certain cases (the FMPT), by studying the entanglement transition in a tree tensor network. As a by-product, we give exact results for general tree tensor networks.

Figure 8(left) is a schematic of the first $k=3$ generations of the tree that is connected to one end of a link somewhere in the bulk of the circuit. For later convenience we use a slightly different convention for drawing the tree to that in Sec. A. Previously we “pruned off” all the branches below a projection operator, while in Fig. 8(left) we leave them in place, so that the number of descendants after $k$ generations (the number of links at the base of the tree) is always $3^k$. Each four-coordinated node in this figure, such as the one denoted $t$, includes a unitary, together possibly with projectors on its legs—we describe this below.

This tree is a tensor network. It has one free tensor index at the top and $3^k$ free indices at the bottom, and tensors $t$ in the interior (built from a unitary and projectors). A basic way to characterize such a tensor network is via the amount of quantum information shared between apex and base. We can quantify this by the entanglement entropy between apex and base (Sec. C below). This language suggests analogous, but distinct, criteria for the classical and quantum transitions.

<figure><img src="image_10.png" /><figcaption>FIG. 8. A tree tensor network with $k=3$ generations of nodes, and its singular value decomposition between apex and base. The individual node tensors are denoted by $t$ [see, for example, Eq. (36)].</figcaption></figure>

The classical percolation transition at $r_c^{\text{cl}}$ (Sec. III above) has a simple interpretation in terms of the tree tensor network. For an asymptotically large tree, $r_c^{\text{cl}}$ is the projection rate beyond which the apex and the base are guaranteed to be strictly disconnected by projectors. That is, once we go beyond the classical transition, the quantum information shared between apex and base vanishes for simple geometric reasons [89].

This suggests that we can also diagnose the quantum transition in the circuit, occurring at a value $r_c^{\text{qm}}$ (we see below that $r_c^{\text{qm}} < r_c^{\text{cl}}$) using the properties of the tree. We show that the tree has a transition at a critical value $r_c$, which we conjecture is also the location of the critical point for the circuit ($r_c = r_c^{\text{qm}}$). For $r > r_c$ the amount of information shared between apex and base decreases exponentially with the number of tree generations, even though the apex and the base may not be disconnected in the trivial geometrical sense. For $r < r_c$, the von Neumann entanglement entropy between apex and base instead remains positive: $\lim_{k \to \infty} \langle S_1 \rangle > 0$.

Motivated by this connection between the circuit and trees, in this section we derive some universal results for entanglement transitions of tree tensor networks. We argue that the tree structure allows us to find the exact location of the critical point for the simplest version of the all-to-all circuit model exactly. In the language of Sec. II, this is the FMPT rather than the MPT. We explain in Sec. B immediately below why it is necessary for us to restrict to the FMPT in this section.

Tree tensor networks are also interesting quite apart from the connection to the all-to-all circuit [59–68]. They are instructive toy models for 1D wave functions with a scale-invariant entanglement structure [66,67], and they also allow efficient numerical tensor contraction algorithms [59]. Many of the results of the following subsections apply to more general disordered tree tensor networks that are unrelated to the circuit (see the discussion in Sec. I).

We obtain specific universal results for a broad class of trees that includes those arising in the FMPT circuit. These trees have bond dimension 2, and the probability distribution of the local tensors has a simple invariance property.
We also discuss, speculatively, what happens for trees with more general disorder distributions. Our conjectured continuum theory allows, a priori, for the entanglement transition to be in distinct universality classes—a phenomenon analogous to a line of fixed points (there is an overview in Sec. D). Strikingly, for the class of trees that we study here, the transition is constrained to lie on a specific point on this line. It remains to be seen whether other points on the line can be obtained by varying the model.

Heuristically, these different possibilities for the tree transition can be related to different possibilities for the disentangled phase close to the transition. In the disentangled phase the entanglement between apex and base is exponentially small in $k$. But we can distinguish, in principle, between a “strong disorder” regime where this small amount of entanglement is (loosely speaking) dominated by a single path from apex to base, and a “weak disorder regime” where exponentially many paths through the tree contribute. For the tree tensor networks we study here, we show that the former (strong disorder) case applies. The possibility of these two regimes is due to the existence of a glass transition in the classical problem of a directed polymer on a tree [58]. We rely heavily on the methods developed in Ref. [58] for the directed polymer problem, which relate a linear recursion relation for the polymer’s partition function to a traveling wave equation.

## B. Structure of tree tensor network

### 1. Generalities

The trees we consider have branching number three and bond dimension 2 for each bond (these are not essential restrictions). The four-index tensor $t^a_{bcd}$ at a given node has bond index $a = 1, 2$ for the upper bond and $b, c, d$ for the lower bonds. Below we describe the structure of $t$ for the circuits we consider. We note that they fall within a special class of tree tensor networks with a simplifying feature, for which we be able to make strong statements.

Let us first consider trees such as Fig. 8(left) in general terms, without assuming that they arise from a circuit problem.

First, our analytical treatment will assume that the individual random tensors $t^a_{bcd}$ for the nodes are statistically uncorrelated. This is important as it allows a simple recursive equation for the entanglement between top and bottom. We also take them to be identically distributed.

Second, for most of this section we assume that the probability distribution of the local tensors $t^a_{bcd}$ has a simple invariance property. Namely, the distribution is invariant under multiplying an arbitrary U(2) matrix $u$ on any index: for example under

$$t^a_{bcd} \rightarrow t^a_{bcd'} u_{d'd}. \qquad (36)$$

This feature simplifies the recursive equation: it means we can write a recursion for singular values alone, without having to keep track of singular vectors. In fact we need only a weaker condition: below the invariance property will hold for only the lower indices of $t^a_{bcd}$, which is sufficient.

The two assumptions above will be satisfied naturally for the circuit ensembles we consider, for example those built from Haar-random two-site gates. They turn out to lead to surprisingly strong constraints on the structure of the recursion.

Towards the end of our discussion of trees (Sec. I) we speculate about what happens when we relax the second condition.

### 2. Application to FMPT in circuit

In applying our results on trees to the transition in a circuit, the first assumption above (on the statistical independence of the node tensors) restricts us to the FMPT rather than the MPT.

Recall that for the MPT the measurement outcomes in the circuit are determined by Born's rule. That means they have nontrivial statistics that depend on the random unitaries, violating the first assumption above. But in the FMPT the local projection operators are fixed independently of the choice of unitaries, not with Born's rule. This means all the nodes of the tree are statistically independent, allowing a recursive statistical treatment. The nodes are described explicitly below. For the ensembles of two-site unitaries we study, it does not in fact matter how the directions of the local projections are fixed, so long as this is done independently of the realization of unitaries. For definiteness, we take all the local projectors to be onto the spin-up state.

To complete the specification of the circuit model, we just need to fix the distribution from which each two-site unitary $U$ is drawn. (As in Sec. III, the rate at which projection operators is applied is $r$.)

### 3. Choice of ensemble of unitaries

The simplest choice is to take each $U$ independently Haar-random in U(4), i.e., drawn from the circular unitary ensemble.

$$\text{Ensemble 1: } U \sim \text{Haar}. \qquad (37)$$

For this ensemble we find a “quantum” transition point $r_c = 0.749$ that is quite close to the classical one, $r_c^{cl} = 0.8$. In order to increase the separation between these transitions, we also consider a second ensemble with more weakly entangling gates. Unitaries in this second ensemble, referred to below as the “$\Delta t$ ensemble,” are of the
form

$$ \text{Ensemble 2: } U = (V_1 \otimes V_2) U_{\text{fixed}}(W_1 \otimes W_2), \quad (38) $$

where $V_1$, $V_2$, $W_1$, and $W_2$ are Haar-random one-site unitaries (ensuring the invariance property mentioned above), and $U_{\text{fixed}}$ is a nonrandom, fixed unitary:

$$ U_{\text{fixed}} = \exp(-i\Delta t H). \quad (39) $$

The parameter $\Delta t$ controls the strength of the unitary, and consequently the position of the quantum transition (see Fig. 11). However, for concreteness we mostly refer to the fixed value $\Delta t = 0.3$. For $H$ we use [6]

$$ H = 0.3X_1X_2 + 0.2(X_1 + X_2) + 0.4Z_1Z_2 + 0.5(Z_1 + Z_2), \quad (40) $$

where $X_i$, $Y_i$, $Z_i$ are the Pauli matrices on site $i$. These parameter values are not fine tuned, and we do not expect results to depend qualitatively on the precise values.

The recursive treatment for $Z_k$ below applies to a more general family of distributions for the two-site unitaries for which the assumptions in Sec. I are obeyed. First, the distribution of $U$ should be invariant under left or right multiplication by single-site unitaries [90]. Second, it should have the property of being statistically invariant under exchange of the two spins acted on by the unitary, and under transposition of the unitary. (This is less crucial, but simplifies the recursion [91]).

### 4. Node tensor in tree

Recall from Sec. III that we can “grow” the tree by starting at some seed location in the circuit and following links (segments of spin worldline) to form a cluster of unitaries at greater and greater distance from the seed. In fact, if we start on a link, we can think of it as a seed for two trees, one attached to each end of the link. It suffices to consider the properties of one of these trees separately. Truncating the tree at $k$ generations gives a tensor network with a single bond at its apex and $3^k$ bonds at the base (we follow the convention in Sec. A where all branches are kept, even if they contain projections).

First consider a tree with no projections, where each node is a unitary. Now, when we include projections, each link of the tree has a probability $p$ to contain a projection [92]. If a projection is present, we choose to incorporate it

into the node below the link. The node tensor is therefore

<figure><img src="image_11.png" /></figure>

or in components (we write the row index of $U$ as a superscript, and the column index as a subscript; both are multi-indices, since the unitary acts on two spins):

$$ t_{bcd}^a = Q_a^a U_{bc}^{a'd}. \quad (42) $$

The matrix $Q$, shown as a circle in the picture, is either the identity, or the projector onto up, with probabilities $1-p$ and $p$ for each of the options. Recall that, in terms of the measurement rate [Eq. (16)],

$$ p = \frac{r}{2-r}. \quad (43) $$

In the case where the projector is present, we could simply prune off all the branches below it, but it is simpler to treat the geometry of the tree as fixed. Note that the distribution of Eq. (42) is invariant under multiplication of $U(2)$ matrices on any of the lower indices, as required in Sec. I.

### C. Entanglement between apex and base

We characterize the phase that the tree is in by the amount of quantum information shared between its apex and its base. Depending on the phase, this can either be exponentially small in the number $k$ of generations of the tree, or it can be order 1 even for asymptotically large trees.

We can always think of such a tree tensor network as a wave function for a single spin at the apex and multiple spins at the base. The information shared between apex and base is then quantified by the entanglement entropy between top and base, or more formally, by the singular value decomposition when we partition the tensor network between the top and the base: see Fig. 8(right) [93].

Since the bond at the apex has a bond dimension of 2, there are only two singular values. After normalizing the tree, their squares sum to one, so we are in fact characterizing the tree by just a single number. We take this to be the square of the smaller singular value, and will denote it by $Z_k$ for a tree with $k$ generations of nodes:

$$ Z_k = \lambda_{\min}^2. \quad (44) $$
The entanglement mentioned above is quantified by the Rényi entropies,

$$S_n = \frac{1}{1-n} \ln[Z_k^n + (1-Z_k)^n], \quad (45)$$

which at small $Z_k$ are approximately

$$S_{n>1} \approx \frac{n}{n-1} Z_k, \quad S_1 \approx Z_k \left( \ln \frac{1}{Z_k} + 1 \right). \quad (46)$$

In the random tensor network $Z_k$ is of course random. Its distribution can be obtained recursively, using the fact that a larger tree can be built up by combining subtrees.

### D. Overview: classes of quantum tree

Let us summarize our basic conclusions for $Z_k$ before getting into calculations. Depending on the location in the phase diagram, the random variable $Z_k$ may have a broad distribution, and it will be vital to define its typical value using the average of $\ln Z_k$:

$$\ln Z_k^{\text{typ}} = \langle \ln Z_k \rangle \neq 0. \quad (47)$$

We must condition on $Z_k$ not being strictly zero in order to define the typical value [94].

For the quantum circuit with two-site unitaries and projections, the tree undergoes an entanglement transition at a critical value $r_c$. The value of $r_c$ depends on the ensemble of unitaries, but a basic point is that it is strictly below the classical transition point for any ensemble satisfying our assumptions:

$$r_c < r_c^{\text{cl}}. \quad (48)$$

We compute the value of $r_c$ analytically for the Haar circuit:

$$r_c = \frac{212 + 75\pi}{362 + 75\pi}. \quad (49)$$

This critical point at $r_c \approx 0.749004$ lies not that far from the classical transition at $r_c^{\text{cl}} = 0.8$. For the $\Delta t = 0.3$ ensemble the spacing is increased,

$$r_c = 0.621(3). \quad (50)$$

In the disentangling phase the information shared between apex and base tends to zero exponentially with the size of the tree:

$$Z_k^{\text{typ}} \sim \exp(-|c_r|k) \quad \text{for } r > r_c, \quad (51)$$

with the “speed” $|c_r|$ vanishing linearly as $r \to r_c$.

In the entangling phase $Z_k^{\text{typ}}$ is instead nonzero as $k \to \infty$, so that information is shared between apex and

base even in the limit of an infinitely large tree. This information becomes small as we approach the transition from the entangled side. The scaling is very rapid:

$$Z_{\infty}^{\text{typ}} \sim \exp\left(-\frac{C}{\sqrt{r_c - r}}\right) \quad \text{for } r \lesssim r_c, \quad (52)$$

with $C$ a nonuniversal constant. The distribution of $Z$ is also very broad when $r_c - r$ is small. For example $\langle Z_{\infty} \rangle \sim \sqrt{Z_{\infty}^{\text{typ}}}$, so that the mean is parametrically larger than the typical.

If we are right at the critical point, the value of $Z$ decays more slowly with $k$ than in the disentangled phase. A somewhat heuristic argument in Sec. 4 suggests

$$\ln Z_k^{\text{typ}} \sim -k^{1/3}. \quad (53)$$

The results above rely on an exact treatment of the linearized form of the recursion relation for $Z_k$, together with the conjecture that the effect of nonlinearity is captured by a simplified, analytically tractable model. Making this assumption (for which we provide numerical evidence), the results above hold for the trees derived from any forced-measurement circuit ensemble with the structure described in Sec. 3 (recall that we assumed invariance of the distribution of $U$ under rotations on each leg).

In fact, they apply for the entanglement transition in any tree tensor network that obeys the two assumptions described in Sec. 1, in particular the U(2) invariance property of the node tensor. In this context $r$ is no longer interpreted as a measurement rate: instead it is any parameter characterizing $t_{bcd}^a$ that can be used to drive the entanglement transition. However for the purposes of the discussion we use the notation appropriate to the forced-measurement circuit.

An analysis of random tree tensor networks outside the above class is a task for the future (see also Ref. [68]). However, our conjectured effective description suggests the interesting possibility that there may be multiple universality classes for the tree entanglement transition. The effective description includes a parameter $\Delta > 0$, which controls the scaling of $Z_{\infty}^{\text{typ}}$ near $r = r_c$. At first glance $\Delta$ is a nonuniversal parameter that will depend on the model. But, surprisingly, the U(2) invariance property fixes $\Delta = 1/4$ at the entanglement transition. Equation (52) applies for $\Delta < 1$ in the effective description, but for $\Delta > 1$ there is instead power-law scaling of the “order parameter” $Z$ close to the transition. For completeness, we solve the effective model in this $\Delta > 1$ regime also. We find a regime with a variable exponent, and a regime where this exponent is pinned to 1:

$$Z_{\infty}^{\text{typ}} \sim (r_c - r)^{\frac{1}{\Delta-1}} \quad \text{for } 1 < \Delta < 2. \quad (54)$$

$$Z_{\infty}^{\text{typ}} \sim (r_c - r), \quad \text{for } \Delta > 2. \quad (55)$$
However, at present it is unclear whether these regimes of the effective model can be accessed by any tensor network, or whether they exist only in the effective model (Sec. I).

### E. Recursion relation for singular values

Let us think of the tree as a quantum state for a spin at the top and 3k spins at the base: this is just to fix notation for bras and kets. We may write its Schmidt (singular value) decomposition:

$$T = \sum_{i=1,2} \lambda_i |i\rangle_{\text{top}} |i\rangle_{\text{bottom}}. \quad (56)$$

The states are Schmidt states in the appropriate Hilbert spaces (the second ket lives in the $2^{(3^k)}$-dimensional Hilbert space associated with the base). In the problem we are studying, the overall normalization of the tree is not important, so we always take the Schmidt singular values to be normalized: $\lambda_1^2 + \lambda_2^2 = 1$.

Given three trees $T_k$, $T'_k$, and $T''_k$, each of k generations, we may form a tree $T_{k+1}$ of k + 1 generations by attaching $T_k$, $T'_k$, and $T''_k$ to the base of the t node shown in Eq. (41). The statistical invariance of U under single-site rotations means that we are free to take the Schmidt states $|i\rangle_{\text{top}}$ [Eq. (56)] for $T_k$, $T'_k$, and $T''_k$ to be simply the two basis states (up and down spin states), which we denote $|1\rangle_{\text{top}}$ and $|2\rangle_{\text{top}}$. Then

$$T_{k+1} = \sum_{a,b,c,d=1,2} t_{bcd}^a \lambda_b \lambda_c' \lambda_d'' |a\rangle_{\text{top}} |bcd\rangle_{\text{bottom}}. \quad (57)$$

Here $\lambda$, $\lambda'$, and $\lambda''$ are singular values for $T_k$, $T'_k$, and $T''_k$, $\{|a\rangle\}_{\text{top}}$ are computational basis states, and $\{|bcd\rangle\}_{\text{bottom}}$ is a set of eight orthonormal states associated with the base of the full tree, formed from the Schmidt states of the three subtrees. Equivalently, in this basis,

$$(T_{k+1})_{bcd}^{a} = t_{bcd}^{a} \lambda_{b} \lambda_{c}' \lambda_{d}'' . \quad (58)$$

It is straightforward to compute the normalized singular values of $T_{k+1}$ [95]. Let us denote the smaller singular value squared of $T_{k+1}$ by $Z_{k+1}$. If t includes the projector, then trivially

$$Z_{k+1} = 0. \quad (59)$$

We write the other case explicitly for completeness, though we need only a simple limit of it:

$$Z_{k+1}^2 + (1 - Z_{k+1})^2 = \frac{\sum U_{bc}^{ad} (U_{bc}^{a'd}) * U_{ef}^{a'g} (U_{ef}^{ag}) * \lambda_b^2 \lambda_c^2 \lambda_e^{1/2} \lambda_f^{1/2} \lambda_d^{1/2} \lambda_g^{1/2}}{(\sum |U_{bc}^{ad}|^2 \lambda_b^2 \lambda_c^2 \lambda_d^{1/2})^2}. \quad (60)$$

We are interested in a transition between a phase where $Z_k$ vanishes as $k \to \infty$, and a phase where the typical value

of $Z_k$ remains positive in this limit. Even in this phase, if we are close to the phase transition, this typical value of $Z_k$ is small. Therefore, to understand the critical properties we can study the recursion relation in the regime where the minimal singular values are close to 0 for all the trees. We order the singular values of any tree such that $\lambda_2 \le \lambda_1$, and define $Z = \lambda_{\min}^2 = \lambda_2^2$ for each tree.

The first step is to examine the linearized recursion relation. Taking Eqs. (59) and (60) to order Z,

$$Z_{k+1} = \begin{cases} A_1 Z_k + A_2 Z'_k + A_3 Z''_k & \text{with probability } 1-p \\ 0 & \text{with probability } p. \end{cases} \quad (61)$$

Here $A_i$ are three positive constants that depend on the random unitary:

$$\begin{align}
A_1 &= \frac{|U_{11}^{11} U_{21}^{21} - U_{21}^{11} U_{11}^{21}|^2}{(|U_{11}^{11}|^2 + |U_{11}^{21}|^2)^2}, \nonumber \\
A_2 &= \frac{|U_{11}^{11} U_{12}^{21} - U_{12}^{11} U_{11}^{21}|^2}{(|U_{11}^{11}|^2 + |U_{11}^{21}|^2)^2}, \tag{62} \\
A_3 &= \frac{|U_{11}^{11} U_{11}^{22} - U_{11}^{12} U_{11}^{21}|^2}{(|U_{11}^{11}|^2 + |U_{11}^{21}|^2)^2}. \nonumber
\end{align}$$

Analogous formulas hold for more general choices of the node tensor t, see Sec. I.

Let us consider the meaning of this equation. $Z_k$, $Z'_k$, and $Z''_k$ refer to trees of the same size, so they are drawn from the same probability distribution. The recursion relation then defines the probability distribution for a variable $Z_{k+1}$ at the next level in the hierarchy. This defines a sequence of probability distributions $P_k(Z)$ for increasing k. The initial condition at the lowest level of the hierarchy is $Z_0 = 1/2$ (for a single bond, the two singular values are equal), i.e., $P_0(Z) = \delta(Z - 1/2)$. (This initial condition is far outside the linear regime, but close to the transition, $Z_k$ becomes small at large k. The specific choice of the initial condition in the linear tree is unimportant as long as it is nonzero and positive.)

The linearized recursion relation, Eq. (61), is crucial. It is sufficient to obtain the exact location of the entanglement transition, although we need to add nonlinearity to understand what happens close to this transition in the entangled phase.

Let us collect here some properties of $A_i$ that will be useful below. It turns out that the statistical invariance of U under single-site rotations allows some exact statements, regardless of the precise choice of distribution for U. We demonstrate these in Appendix 2. In particular, we need the following identities, which hold for all $i = 1, 2, 3$ (so
<table><caption>TABLE I. Some averages of the $A_i$ that are used in the text.</caption><thead><tr><th>Quantity</th><th>Haar</th><th>Δt = 0.3</th></tr></thead><tbody><tr><td>⟨A<sub>i</sub>⟩</td><td>1</td><td>1</td></tr><tr><td>⟨√A<sub>i</sub> log A<sub>i</sub>⟩</td><td>0</td><td>0</td></tr><tr><td>⟨√A<sub>1</sub>⟩</td><td>64/75</td><td>0.981</td></tr><tr><td>⟨√A<sub>2</sub>⟩</td><td>64/75</td><td>0.420</td></tr><tr><td>⟨√A<sub>3</sub>⟩</td><td>π/4</td><td>0.419</td></tr><tr><td>⟨√A<sub>1</sub>(log A<sub>1</sub>)<sup>2</sup>⟩</td><td>17408/16875</td><td>0.227</td></tr><tr><td>⟨√A<sub>2</sub>(log A<sub>2</sub>)<sup>2</sup>⟩</td><td>17408/16875</td><td>1.429</td></tr><tr><td>⟨√A<sub>3</sub>(log A<sub>3</sub>)<sup>2</sup>⟩</td><td>π<sup>3</sup>/4 - 2π</td><td>1.428</td></tr></tbody></table>

long as $U$ is nontrivially entangling with probability 1):

$$ \langle A_i \rangle = 1, \quad \langle A_i^{1/2} \ln A_i \rangle = 0. \qquad (63) $$

The first of these is at first sight surprising, since if the unitary $U$ is the identity [for example, if $\Delta t \to 0$ for the distribution in Eqs. (38) and (39)] then $A_2$ and $A_3$ are exactly equal to zero [96]. However, this is a singular limit for $\langle A_i \rangle$, see below.

In the case where $U$ is a Haar-random U(4) matrix, we can obtain more general analytic results (Appendix 2):

$$ \langle A_1^\lambda \rangle = \langle A_2^\lambda \rangle = \frac{12}{12 + 8\lambda - 7\lambda^2 - 2\lambda^3 + \lambda^4}, \qquad (64) $$

$$ \langle A_3^\lambda \rangle = \frac{\pi \lambda (1 - \lambda)}{\sin(\pi \lambda)}. \qquad (65) $$

To determine the location of the phase transition we need the special cases $\langle A_i^{1/2} \rangle$. These are given in Table I (Appendix 2) for both Haar and $\Delta t = 0.3$ ensembles.

The asymptotics of the probability distributions of $A_i$ are also obtained in Appendix 2. These three variables are cor- related, but here we discuss only the marginal distribution of a given one. Let us define

$$ V_i \equiv \ln A_i. \qquad (66) $$

For any generic distribution of $U$, the tails of the $V_i$ distribution are exponential:

$$ P(V_i) dV_i \sim \begin{cases} e^{-2V_i} dV_i & V_i \gg 0, \\ e^{-|V_i|} dV_i & V_i \ll 0. \end{cases} \qquad (67) $$

If the distribution of unitaries is taken to be weakly entangling, for example if $\Delta t \ll 1$ in Eq. (39), then the right-hand tail of the distribution has an intermediate part, extending over the range $\ln \Delta t^2 \ll V_i \ll \ln \Delta t^{-2}$, that decays with the smaller exponent $-1/2$ (Appendix 2). This slowly decaying tail, cut off at a parametrically large $V_i$, is responsible for the failure of the limit $\Delta t \to 0$ to commute with the average in Eq. (63) that was mentioned above.

<figure><img src="image_12.png" /><figcaption>FIG. 9. Evolution of the typical value of $Z$ with the depth $k$ of the tree, for the linear recursion (left) and the nonlinear recursion (right), for various values of $p$. The evolution is similar in the two cases for $p > p_c$, but for $p < p_c$, the nonlinearity causes $Z^{\text{typ}}$ to saturate. The initial value $Z_0 = 10^{-6}$ is used.</figcaption></figure>

## F. Linearized recursion relation

The most basic question about the linearized recursion, Eq. (61), is whether the typical value of $Z$ is exponentially growing or exponentially shrinking at large $k$ [58]. We may define an exponential growth speed $c_p$:

$$ \ln Z_k^{\text{typ}} \sim c_p k. \qquad (68) $$

In this section we usually use $p$ as the parameter, rather than the equivalent $r$, Eq. (43), since $p$ has a more direct interpretation in terms of the tree.

Define the point $p_c$ to separate a regime of exponential growth, which we see occurs for $p < p_c$, from a regime of exponential decay at larger $p$:

$$ c_{p_c} = 0 \quad (\text{definition of } p_c). \qquad (69) $$

We see that $p_c$ is precisely the location of the entanglement phase transition for the tree. When the linear recursion predicts that $Z^{\text{typ}} \to 0$ at large $k$, this remains true when higher powers of $Z$ are included in the recursion. On the other hand, when the linear recursion predicts that $Z^{\text{typ}} \to \infty$ at large $k$, then the nonlinear terms in the recursion replace “$\infty$” with a finite value, in a universal manner that we discuss in Sec. H.

This is illustrated using simulations for the case of Haar-random unitaries in Fig. 9. This compares $Z_k^{\text{typ}}$ for the linear and nonlinear recursion relations. In the linear case, evolution follows $\ln Z_k^{\text{typ}} \propto k$ at large $k$, for all $p$. In the nonlinear case, this is only true for $p > p_c$. Details of these simulations are described in Appendix C 1.

The speed $c_p$ can be extracted using the method of Ref. [58], which relates the linear recursion to a traveling wave
problem. Define the generating function

$$G_k(x) = \langle \exp(-e^{-x} Z_k) \rangle, \quad (70)$$

where the average is over $Z_k$. The recursion relation, Eq. (61), then becomes

$$G_{k+1}(x) = p + (1-p) \langle G_k(x - V_1) \\ G_k(x - V_2)G_k(x - V_3) \rangle, \quad (71)$$

where the remaining average is only over the $V_i = \ln A_i$ defined in Eq. (66). [The fact that the $V_i$ appear additively in the arguments of the generating functions here is the reason why the generating function in Eq. (70) is usually written with the double exponential.]

It may be helpful to think of $G_k(x)$, defined in Eq. (70), as a smeared version of the cumulative probability distribution for $\ln Z$. This definition shows that for $x \gg \ln Z^{\text{typ}}$, $G_k$ plateaus at the value 1, while for $x \ll \ln Z^{\text{typ}}$, $G_k$ plateaus at the probability of $Z_k$ being exactly zero [97]. $G_k$ has a “front” at

$$x_{\text{front}}(k) = \ln Z_k^{\text{typ}} + o(k) \quad (72)$$

that interpolates between these two plateaus. It is useful to think of $x$ as a fictitious spatial coordinate, and of $k$ as fictitious time coordinate [58]. Then, at late time, this front propagates as a traveling wave with speed $c_p$, and obeys the traveling-wave ansatz:

$$G_k(x) = G^{(\lambda)}[x - v_p(\lambda)k]. \quad (73)$$

The wave speed $v_p(\lambda)$ depends on a parameter $\lambda$ of the solution $G^{(\lambda)}$. This parameter, which must be determined, is the exponential decay constant of $G^{(\lambda)}$ at large argument [58]:

$$G^{(\lambda)}(u) \sim 1 - \alpha e^{-\lambda u}. \quad (74)$$

Substituting this form into Eq. (71) gives an explicit formula for the speed $v(\lambda)$ of the traveling-wave solution with a given $\lambda$:

$$v_p(\lambda) = \frac{1}{\lambda} \ln \left[ (1-p) \left( \langle A_1^{\lambda} \rangle + \langle A_2^{\lambda} \rangle + \langle A_3^{\lambda} \rangle \right) \right]. \quad (75)$$

We must then determine the correct value of $\lambda$, i.e., which traveling-wave solution the initial condition converges to. This is done by standard considerations of velocity selection for traveling waves [58,98].

In outline, there is a privileged minimal speed traveling wave defined by the parameter value $\lambda = \lambda_*$ where $v_p(\lambda)$ is minimal:

$$v_p^{\min} \equiv v_p(\lambda_*) \quad v'_p(\lambda_*) = 0. \quad (76)$$

$G_k$ will converge to this minimal speed solution if $\lambda_*$ is less than 1 [99], while it will converge to the solution

<figure><img src="image_13.png" /><figcaption>FIG. 10. Velocity $v_p(\lambda)$ vs $\lambda$ for three values of $p$ (above, at and below the critical point). Left panel is for Haar evolution and right is for $\Delta t = 0.3$.</figcaption></figure>

with $\lambda = 1$ if $\lambda_* > 1$. In the latter case the speed is $v_p(1)$, which we refer to as the “annealed” value of the speed (for reasons described in Sec. G):

$$v_p^{\text{ann}} \equiv v_p(1). \quad (77)$$

Therefore, the desired exponential growth rate is given for any $p$ by

$$c_p = \begin{cases} v_p^{\min} & \text{if } \lambda_* < 1, \\ v_p^{\text{ann}} & \text{if } \lambda_* > 1. \end{cases} \quad (78)$$

Recall that $\lambda_*$ is determined using Eq. (75), via $v'_p(\lambda_*) = 0$, so it depends on $p$.

The above Eq. (78) can lead to a nonanalyticity in $c_p$ as $p$ is varied. This has a meaning in terms of the statistical mechanics of the linearized recursion relation [58], which we review in Sec. G. For now we simply note that, for the present class of circuits [100] the first line in Eq. (78) is always the one that applies for $p$ close to $p_c$. This is shown in Sec. G. Given this, $p_c$ is determined by solving

$$v_{p_c}(\lambda_*) = 0, \quad v'_{p_c}(\lambda_*) = 0, \quad (79)$$

for $\lambda_*$ and $p_c$.

Figure 10 shows $v_p(\lambda)$, defined in Eq. (75), for the Haar tree and the $\Delta t = 0.3$ tree, in the vicinity of their respective $p_c$ values. [Numerically, these are obtained by simple averages using a single tensor. In the Haar case, Eqs. (64) and (65) also give the exact form.] $c_p$ is given by the minimal value of the curve, $c_p = v_p(\lambda_*)$, which passes through zero at $p = p_c$.

This can be used to determine $p_c$ numerically, but in fact further analytical progress is possible. Using the definition of $v_p(\lambda)$ in Eq. (75), Eqs. (79) reduce to

$$\sum_{i=1}^{3} \langle A_i^{\lambda_*} \rangle = \frac{1}{1 - p_c}, \quad \sum_{i=1}^{3} \langle A_i^{\lambda_*} \ln A_i \rangle = 0. \quad (80)$$

Remarkably, the second identity in Eq. (63) shows that the solution is always at $\lambda_* = 1/2$, for any ensemble of
<figure><img src="image_14.png" /><figcaption>FIG. 11. Phase diagram of the quantum tree generated by the unitary gate in Eq. (39), in the space of $\Delta t$ and $r$.</figcaption></figure>

unitaries satisfying our assumptions. This fact gives an explicit expression for $p_c$ as a simple average for the local node tensor,

$$p_c = 1 - \frac{1}{\sum_{i=1}^{3} \langle A_i^{1/2} \rangle} \quad (81)$$

This may be evaluated analytically for the Haar case [Eqs. (64) and (65)], giving $p_c = (212 + 75\pi)/(512 + 75\pi)$ (equivalent to the $r_c$ value quoted in Sec. D) and numerically for the $\Delta t$ ensemble. The location of the critical point in the $\Delta t$ ensemble is shown for various values of $\Delta t$ in Fig. 11.

## G. Aside: glass transition in linear recursion

The canonical example of linear recursion relations like Eq. (61) is the problem of the directed polymer on a tree [58]: see Fig. 12. In the disentangled phase, where the linear treatment is valid at large $k$, this gives another interpretation of the singular-value-squared $Z_k$ as a sum over paths through the tensor network. Here we briefly review this mapping and use it to clarify which of the regimes in Eq. (78) is relevant. This subsection is not essential to the subsequent development.

Within the linear approximation Eq. (61), $Z_k$ is exactly equal to the partition function of a polymer that lies along a path from the top to the bottom of a tree of depth $k$, as in Fig. 12. We view $-V_1, -V_2, -V_3$ in Eq. (66) as random potentials on the three bonds below a given node. The energy of the polymer is the sum of the potentials for the bonds it visits:

$$Z_k = \sum_{\text{paths}} e^{\sum_{\text{on path}} \nu_{\text{bond}}} \quad (82)$$

This is easily seen to satisfy the recursive Eq. (61). There are minor differences from the standard polymer model.

<figure><img src="image_15.png" /><figcaption>FIG. 12. Schematic of a directed polymer on a tree.</figcaption></figure>

First, if $p > 0$, there are some bonds that the polymer cannot visit, where $A = 0$ or $V = -\infty$ (these bonds and the subtrees below them can simply be removed). Second, the $V$s have a nontrivial distribution, with links that share the same parent node having correlated potentials.

The polymer can be in either a glass phase or a paramagnetic phase [58]. These are distinct thermodynamic phases in the polymer problem, but to avoid confusion we refer to them as “regimes,” because they do not correspond to distinct phases of the entanglement problem. (The distinction between the glass and paramagnet is a feature of the linearized problem only, and is unrelated to the distinction between entangled and disentangled phases.)

The glass obtains when the pinning effect of disorder on the polymer defeats the depinning effect of entropy. Usually the glass would be entered by decreasing the temperature (increasing the scale of $V$). Here we increase the strength of disorder by increasing $p$. In the paramagnetic regime the polymer has extensive entropy (propotional to $k$) while in the glass the entropy per unit length vanishes.

The glass and paramagnet regimes have a simple translation to the language of the traveling wave (Sec. IV F), which we state only [58]. The polymer is in the glass regime if $\lambda_* < 1$, and in the paramagnetic regime if $\lambda_* > 1$ [58]. These correspond to the two lines in Eq. (78) for the growth rate $c_p$, which is simply (minus) the free energy per unit length of the polymer.

In our problem, the entanglement transition necessarily takes place in the glass regime of the linear recursion, essentially because of the fact that $\langle A_i \rangle = 1$. Let us give an intuitive picture.

To begin with, imagine that the polymer is in the paramagnetic regime. In this regime (but not in the glass [101]) the “annealed” expression for the free energy growth rate $c_p$ applies [the second line of Eq. (78)]. This expression is in fact just the annealed approximation to the free energy, in which we average the partition function of the polymer, $\langle Z_k \rangle$, instead of averaging its logarithm. In the present linearized problem this gives

$$v_p^{\text{ann}} = \ln(3[1-p]), \quad (83)$$

using $\langle A_i \rangle = 1$.
Recall that the entanglement transition is at the value of $p$ where $c_p = 0$. We see from Eq. (83) that if the polymer is in the paramagnetic regime in the vicinity of $p_c$, then the entanglement transition would coincide with the classical percolation transition at $p_c^{\text{classical}} = 2/3$!

We can see that this is inconsistent as follows. Consider the structure of large trees when we approach the classical percolation transition at $p_c^{\text{classical}} = 2/3$ from below. After deleting subtrees that terminate before reaching the base [102], a large tree with $Z \neq 0$ is made up of one-dimensional chains connected by branching events. Close to the classical transition, the typical length of one of these 1D chains grows like $(2/3 - p)^{-1}$ [103]. Treating them as renormalized bonds in the polymer problem, one may check that the effective disorder strength on these renormalized bonds grows without bound as they get longer. This increasing disorder strength implies that we must enter the glass regime before we get to the classical transition. That is, either the linear recursion relation is in the glass regime for all $p$, or it is in the glass regime for all $p > p_{\text{glass}}$ for some $p_{\text{glass}} < 2/3$.

When the polymer is in the glass phase, $c_p$ is strictly smaller than the annealed approximation above [Eq. (78)]. Therefore, $c_p$ in fact hits zero at a smaller value of $p$ than $v_p^{\text{ann}}$ does. In other words, $p_c$ is strictly smaller than $p_c^{\text{cl}}$.

The value of $p_{\text{glass}}$ is determined by the equation $v'_p(1) = 0$. For the $\Delta t = 0.3$ ensemble the value of $p_{\text{glass}}$ is evaluated numerically and found to be negative, indicating that this ensemble is always in the glassy phase. For the Haar ensemble, $p_{\text{glass}} = (3 - e^{7/9})/3 \approx 0.274$ [from Eqs. (64), (65), and (75)]. But since this value lies inside the entangled phase, where the linearized recursion is not valid, we do not expect that the glass transition is physically significant for the tensor network.

The arguments here, showing that the entanglement transition must take place within the glass regime of the linear recursion, extend to the class of tree tensor networks described in Sec. I. The possibility of other universality classes of entanglement phase transition for other kinds of quantum trees is discussed in Sec. I.

## H. Including the nonlinearity

Having understood the linear approximation to the recursion relation for the singular value squared, Eq. (61), we must now consider the effect of nonlinearity. The nonlinearity is necessary to make sense of the entangled phase, where $Z_k$ is of order 1, rather than being exponentially large in $k$ as the linear equation would predict. Our aim in this section is to determine the scaling of $Z$ close to the transition, on the entangled side. Our basic conclusions have already been summarized in Sec. D.

<figure><img src="image_16.png" /><figcaption>FIG. 13. Distribution of $\ln Z$ in a tree of $k = 150$ generations (for the full nonlinear problem). Here we remove instances where $Z$ is exactly zero (which have a finite support due to the forced measurements).</figcaption></figure>

## 1. Numerical results

Let us show numerical results before turning to an analytical treatment.

First, Fig. 13 shows the probability distribution of $\ln Z_k$ for the Haar ensemble (Sec. 3) in a tree of $k = 150$ generations, where we remove instances where $Z$ is exactly zero [104]. Various values of $p$ less than or equal to $p_c$ are shown. The maximal possible value of $Z_k$ is 1/2: deep in the entangled phase the distribution is concentrated near this upper limit, but as we approach the critical point $\ln Z_{\text{typ}}$ moves to the left. The shape of the distribution also stabilizes. (In fact it approaches the shape for the linear problem, except on the right where $Z$ is of order 1.)

Next, in Fig. 14 we show the scaling of $Z_{\text{typ}}$ for both choices of the ensemble of unitaries (Sec. 3), close to the critical point. The analytic treatment below gives $\ln Z_{\text{typ}} \simeq -D/\sqrt{r_c - r}$, which corresponds to a straight line with slope $-1/2$ in the plot. This slope is indicated by the trend line. The data is consistent with this value of the exponent.

However, the value of the nonuniversal constant $D$ that we extract from fitting this data is $D = 2.01$ for Haar and $D = 3.24$ for $\Delta t = 0.3$, which is far from that predicted below, for both ensembles. Experimenting with simpler toy models suggests that this may just be because of finite $r_c - r$ effects, i.e., not being close enough to $r_c$. The numerical method we use is afflicted by severe finite size effects (see Refs. [105–107] and Appendix C 1), associated with correctly sampling the right-hand tail of the distribution in Fig. 9, which mean we cannot approach too close to the critical point. Details of the numerical method are in Appendix C 1.

## 2. Nonlinear toy model

The nonlinear recursion relation in Eqs. (59) and (60) is not very approachable, even if expanded only to quadratic order. To make progress, we conjecture that the universal properties can be understood in a simpler model that
<figure><img src="image_17.png" /><figcaption>FIG. 14. Critical behavior for the Haar and $\Delta t = 0.3$ trees as obtained from $Z_{600}^{\text{typ}}$ and $Z_{700}^{\text{typ}}$, for Haar and $\Delta t = 0.3$, respectively, for a pool size $N = 3 \times 10^4$. The data suggest $Z_{k \to \infty}^{\text{typ}} \sim e^{-D(r_c - r)^b}$ with the best fits for $b$ being -0.5 and -0.53 for the Haar and $\Delta t = 0.3$ trees, respectively; the corresponding values of $D$ are 2.01 and 3.24, respectively.</figcaption></figure>

retains a few basic features. We study a recursion relation satisfying two requirements. First, it contains both linear terms and nonlinear terms of order $Z^2$, which tend to suppress $Z$ (naively, the terms of higher order than $Z^2$ should be negligible when we are parametrically close to the transition and $Z \ll 1$). Second, its linearized form is in the glass regime, as for the circuit. (Though in fact

<figure><img src="image_18.png" /><figcaption>FIG. 15. Critical behavior of the classical nonlinear tree and the continuum equation in the glass phase. Similar to the quantum trees, the data for the classical tree (blue dots) suggests a scaling approximately $e^{-c(a_c-a)^b}$ for $Z_{k \to \infty}^{\text{typ}}$. The best fits are $b = -0.52$ and $c = 3.86$. For the parameters for the tree used, $p_0 = 0.2$, $p_1 = p_2 = 0.4$, $\epsilon = 1.411$, and $\gamma = 0.5$, the theoretical result shown in gray corresponds to $c = 3.247...$ and $b = 0.5$. Data obtained by numerically solving the continuum equation (red dots) is in excellent agreement with the theoretical prediction close to the critical point, although some deviations are observed further away from the critical point. Inset: enlargement of the data for the classical tree.</figcaption></figure>

we study both this case and the paramagnetic case for completeness.)

We first write down a toy model for a tree with a discrete generation number $k$, but as in Ref. [58] it will be convenient to take a continuum limit in $k$. We assume that this continuum limit preserves the universal properties, as is the case for the linear problem.

For the toy model, define the random variable $Z_{k+1}$ at level $k+1$ in terms of a sum of $\ell$ random variables $Z_k^{(\ell)}$, ..., $Z_k^{(\ell)}$ at level $k$. Here $\ell$ is the branching number of a node, and is taken to be random with a distribution $p_\ell$ for $\ell \ge 0$. The precise range allowed for $\ell$ is not important, so for simplicity we allow $\ell = 0, 1, 2$. We also include nonlinearity of strength $\gamma$, and a multiplicative random variable written as $e^V$, with Gaussian $V$:

$$Z_{k+1} = \left[ e^V \sum_{i=1}^{\ell} Z_k^{(i)} \right] \exp \left\{ -\gamma \left[ e^V \sum_{i=1}^{\ell} Z_k^{(i)} \right] \right\}. \quad (84)$$

This can be viewed as the composition of a linear transformation analogous to Eq. (61),

$$Z_{k+1} = e^V \sum_{i=1}^{\ell} Z_k^{(i)} \quad (85)$$

(but slightly simpler because we avoid having correlated random variables) and a nonlinear one,

$$Z_{k+1} \rightarrow Z_{k+1} e^{-\gamma Z_{k+1}}. \quad (86)$$

The exponential form is arbitrary: for the continuum limit below it will anyway be sufficient to expand only to order $\gamma$, giving a quadratic recursion relation for $Z$. However, the above form guarantees that $Z_{k+1}$ is positive for any input values, which is important for our numerical explorations.

We conjecture that by solving this simple nonlinear system we also capture universal scaling for the problem of interest (Sec. E).

### 3. Continuum traveling wave equation

The equation for the generating function [cf. Eq. (70)] that follows from expanding Eq. (84) to order $\gamma$ is

$$G_{k+1}(x) = \exp(\gamma \partial_x e^x \partial_x) \sum_{\ell} p_{\ell} \langle G_k(x-V)^{\ell} \rangle_V, \quad (87)$$

where $p_0$ and $p_2$ are the probabilities of a termination and a branching, respectively. Now we take the continuum limit in the “time” $k$. When $\gamma = 0$, this gives the Fisher-KPP traveling wave equation [58]. We introduce a “time” step $\delta\tau$, which will be sent to zero and define $\tau = k\delta\tau$. The probabilities $p_0$ and $p_2$ are taken to be of order $\delta\tau$ (i.e., $p_1 = 1 - p_0 - p_2$ is close to 1) so that in the limit the tree
becomes a continuous time branching process. The parameter $\gamma$ is taken of order $\delta\tau$ (note that the prefactor does not matter: it can be absorbed into the normalization of $Z$) and the strength of the random potential is also taken to vanish with $\delta\tau$. It is convenient to parameterize its mean and second moment as

$$ \langle V \rangle = \frac{a}{2} \langle V^2 \rangle \quad \langle V^2 \rangle = 2b \times \delta\tau. \qquad (88) $$

Finally we absorb some constants into the generating function by defining [108]:

$$ G_k(x) = 1 - \frac{p_2 - p_0}{p_2} H_k(x). \qquad (89) $$

The asymptotics of $H$ may be taken to be

$$ H(-\infty) = 1, \quad H(\infty) = 0. \qquad (90) $$

After absorbing a constant into the definition of $\tau$, and shifting $x$ by a constant, $H$ satisfies

$$ \partial_{\tau} H = \partial_{x} [D(x) \partial_{x} - a] H + \Delta H (1 - H), \qquad (91) $$

where the growth rate is $\Delta = 2(p_2 - p_0)/\langle V^2 \rangle$ (which is finite in the $\delta\tau \to 0$ limit), the drift coefficient is $a$ in Eq. (88), and there is a spatially varying diffusion coefficient

$$ D(x) = 1 + e^{x}. \qquad (92) $$

The exponential term in Eq. (92) is the effect of the nonlinearity $\gamma$ in the tree problem.

Note that nonlinearity in the tree is unrelated to nonlinearity in the Fisher-KPP field $H(x, \tau)$ (which instead reflects branching of the tree).

$H$ forms a traveling wave, whose speed $c$ sets the exponential growth rate of $Z$ [cf. Eqs. (68) and (72)]. On their own, the combination of ordinary diffusion and logistic growth (the $\Delta$ term) in Eq. (91) would give a traveling wave propagating to the right ($c > 0$), which corresponds to exponential growth of $Z$. Here, in one phase, this wave instead propagates backwards ($c < 0$): this is possible because of the drift term in Eq. (91). In the other phase, the wave attempts to propagate to the right but is stopped by the exponential growth of the diffusion constant at positive $x$, which prevents the buildup of $H$ at large $x$. This results in $c = 0$.

There is therefore a transition between a phase where $Z$ is exponentially small at large generation number and a phase where $Z$ remains order 1. This is the toy model's version of the entanglement transition.

In the absence of the $e^x$ term in $D(x)$, the velocity of a traveling wave with tail $H \sim e^{-\lambda x}$ is [98]

$$ v(\lambda) = \lambda + a + \Delta\lambda^{-1} \qquad (93) $$

[as we see by keeping only the order $H$ terms in Eq. (91)], with a minimum at $\lambda_* = \sqrt{\Delta}$. Therefore, the linearized tree is in the glass regime [58] (Sec. G), where the traveling wave travels at speed $v_{\min} = v(\lambda_*)$, so long as $\Delta < 1$. This is the case we are interested in for the current circuit models, where $\lambda_* = 1/2$ at the entanglement transition.

The wave speed is then $c = 2\sqrt{\Delta} + a$, so in this toy model the analog of the entanglement transition is at $a_c = -2\sqrt{\Delta}$. Let us therefore write

$$ a = -2\sqrt{\Delta} + \sigma. \qquad (94) $$

We are interested in small positive $\sigma$, just inside the entangled phase. In principle we would like to solve for the stationary solution at late times,

$$ \partial_x [D(x) \partial_x + 2\sqrt{\Delta} - \sigma] H + \Delta H (1 - H) = 0, \qquad (95) $$

which we expect to exist when $\sigma > 0$. In the absence of a full solution, we consider the equation piecewise [109].

Let the position of the front, whose scaling with $\sigma$ we wish to determine, be denoted $x_{\text{front}}(\sigma)$. We assume (and confirm below) that $x_{\text{front}}(\sigma)$ is large and negative at small $\sigma$.

First, at large positive $x$, the leading term in the equation is simply $\partial_x e^x \partial_x H = 0$, so the only solutions satisfying $H \to 0$ at large $x$ have $H \sim e^{-x}$.

Second, consider $-|x_{\text{front}}(\sigma)| \ll x \ll 0$. In this regime we neglect both the variation of the diffusion coefficient and the $O(H^2)$ term. From Eq. (93), we can find a stationary solution for positive $\sigma$ only by making $\lambda$ complex [109]. Keeping only the leading $\sigma$ dependence,

$$ H \sim e^{-\sqrt{\Delta}(x-x_0)} \sin(\phi + \Delta^{1/4} \sqrt{\sigma x}). \qquad (96) $$

We would like to use the as-yet-undetermined constants $x_0$ and $\phi$ in order to allow this solution to match onto the solutions at large positive and negative $x$. Note that the slope of this solution on a logarithmic plot is

$$ \partial_x \ln H = -\sqrt{\Delta} + \frac{\sqrt{\sigma} \Delta^{1/4}}{\tan (\phi + \Delta^{1/4} \sqrt{\sigma x})}. \qquad (97) $$

For generic $x$, this slope is close to $\sqrt{\Delta}$, because of the small factor $\sqrt{\sigma}$ in the second term. However, close to the zeroes of the tangent this is not true. This allows us to match on the right-hand side of the range [110], where the slope is steeper, so long as we take $\phi = \pi$ to leading order in $\sigma$.
Similar considerations on the left show the argument of the tangent must approach 0 as the vicinity of the front is approached. Therefore, to leading order in $\sigma$, the position of the front is

$$x_{\text{front}}(\sigma) = -\frac{\pi}{\Delta^{1/4}\sqrt{\sigma}}. \qquad (98)$$

The constant $x_0$ in Eq. (96) then has the same leading term, to ensure that $H$ is of order 1 in the front region.

Since $x_{\text{front}}(\sigma)$ also sets the average value of $\ln Z$,

$$Z^{\text{typ}} \sim \exp\left(-\frac{\pi}{\Delta^{1/4}\sqrt{\sigma}}\right). \qquad (99)$$

By considering the tail of the distribution, we see that in the regime we are discussing, where $\Delta < 1$, the mean scales as

$$\langle Z \rangle \sim (Z^{\text{typ}})^{\sqrt{\Delta}} \sim \exp\left(-\frac{\pi \Delta^{1/4}}{\sqrt{\sigma}}\right). \qquad (100)$$

Notice that Eq. (99) is

$$Z^{\text{typ}} \sim \exp\left(-\frac{\pi}{|\text{Im } \lambda_{\sigma}|}\right) \qquad (101)$$

[also $\langle Z \rangle \sim (Z^{\text{typ}})^{\lambda_0}$], where $\lambda_\sigma$ solves $v_\sigma(\lambda) = 0$. That is, it depends only on the function $v(\lambda)$ for the linear problem! Indeed the strength of the nonlinearity $\gamma$ in Eq. (84) cannot appear, since it can be absorbed into a rescaling of $Z$ (which does not affect $\ln Z$ at leading order).

This suggests that we can apply the result to the quantum tree of Sec. E, using Eq. (75) for $v$. This gives

$$Z^{\text{typ}} \approx \exp\left(-\frac{C}{\sqrt{p_c - p}}\right), \qquad (102)$$

with

$$C = \frac{\pi(1 - p_c)}{\sqrt{2}} \sqrt{\langle \sum_i A_i^{1/2} (\ln A_i)^2 \rangle}. \qquad (103)$$

For the Haar-random case $C$ is given exactly by Eqs. (64) and (65). In terms of $r$ [Eq. (43)],

$$Z^{\text{typ}} \approx \exp\left(-\frac{1.482 \cdots}{\sqrt{r_c - r}}\right). \qquad (104)$$

Numerical results obtained by simulating the recursion relation in Eq. (84), and by solving the continuum traveling wave equation are shown in Fig. 15. These results indeed confirm the scaling of $Z^{\text{typ}}$ close to the critical point.

<figure><img src="image_19.png" /><figcaption>FIG. 16. The behavior of $Z_k^{\text{typ}}$ with $k$ at the critical point for two initial conditions, $Z_0$, and two pool sizes, $N$. The data is consistent with a $k^{1/3}$ scaling of $\ln Z_k^{\text{typ}}$. Results are shown for the Haar ensemble and errorbars are smaller than the data points.</figcaption></figure>

### 4. Tree entanglement at critical point

So far we discuss scaling in the two phases. Exactly at the transition, we might expect that $Z_k$ tends to zero with $k$, but more slowly than in the distantangled phase.

Figure 16 shows data for this for the Haar ensemble. The data is compatible with, though it does not clearly establish, the scaling $\ln Z_k^{\text{typ}} \sim k^{1/3}$, which is suggested by the following argument for the continuum model.

We expect that for the time-dependent equation $x_{\text{front}}$ drifts sub-ballistically to the left. Let us conjecture that at a given time $t$, and in the range $x_{\text{front}} \ll x \ll 0$, the instantaneous solution of the nonlinear equation approximates sufficiently closely the traveling-wave solution $G^{(\lambda)}$ of the linear equation with the same instantaneous speed, $v = \dot{x}_{\text{front}}$. At the critical point ($\sigma = 0$), $v(\lambda)$ has a double zero at $\lambda = \sqrt{\Delta}$, so this means that $\lambda(t) \simeq \sqrt{\Delta} \pm i\Delta^{1/4}\sqrt{|\dot{x}_{\text{front}}|}$. This gives a solution like Eq. (96), but with $\sqrt{|\dot{x}_{\text{front}}|}$ in place of $\sqrt{\sigma}$. Equation (98) then becomes

$$x_{\text{front}} \sim -\frac{\pi}{\Delta^{1/4} |\dot{x}_{\text{front}}|^{1/2}}, \qquad (105)$$

which gives $x_{\text{front}} \sim (3\pi^2 t/\sqrt{\Delta})^{1/3}$. These values for the exponent and the prefactor are in good agreement with a numerical solution of Eq. (91) at $a = -2\sqrt{\Delta}$ (we check the case $\Delta = 1/4$).

If $\sigma$ is small but positive there must be a crossover at a large time $t_{\text{sat}}$ from $x_{\text{front}} \sim -t^{1/3}$ to $x_{\text{front}} \sim -1/\sqrt{\sigma}$. This suggests $t_{\text{sat}} \sim \sigma^{-3/2}$, which also agrees well with numerical solutions.

### I. Quantum trees: other universality classes?

Above we noted that a priori there are two possibilities according to whether the entanglement transition takes place within the glass or the paramagnetic regime
of the linearized recursion relation: $\lambda_* < 1$ and $\lambda_* > 1$, respectively.

However, our approach required the statistical invariance of the node tensor $t_{b_1,b_2,b_3,...,b_\ell}^a$ under U(2) rotations on a leg. (We are free to allow for an arbitrary branching number $\ell$.) This invariance is necessary so that we can write a recursion relation for singular values only: otherwise we need a combined recursion relation for singular values and singular vectors. For any such tree, the argument of Sec. IV F and Appendix 2 shows that $\lambda_* = 1/2$ at the transition. That is, the recursion relation is of the form [111]

$$Z_{k+1} = \sum_{i=1}^{\ell} A_i Z_k^{(i)} + \mathcal{O}(Z^2), \quad (106)$$

with $\langle A_i \rangle = 1$ and $\langle A_i^{1/2} \ln A_i \rangle = 0$, which is sufficient to ensure $\lambda_* = 1/2$ at the critical point (Sec. IV F).

In this class of trees the weak correlations between the top and base in the disentangled phase are dominated by only a subgraph of the tensor network that contains a few paths from top to bottom. For this broad class of trees we expect the universal scaling described above. Therefore, within the class of trees that our formalism applies to there is no freedom to vary $\lambda_*$.

However, it is interesting to ask what happens in trees where the unitary invariance property is broken. Breaking this invariance introduces correlations between singular values and singular vectors. A plausible guess, at least if these correlations are not too strong, is that in this setting the same toy model nevertheless captures the universal scaling. If this is the case (which we do not determine here) then the next question is whether in these more general models it is possible to vary the critical value of $\lambda_*$ away from 1/2.

With this somewhat speculative motivation (and for completeness), below we extend the analysis of critical scaling in the toy model to the regime $\lambda_* > 1$.

Our analysis is also restricted to trees with bond dimension 2. A recursion relation (for the subleading squared singular values squared) may be formulated for trees with larger bond dimension, but has a more complicated structure, even at lowest order. It would be interesting to study this further.

## 1. Scaling of nonlinear recursion: $\Delta > 1$

We return to the toy model of Sec. 2 in the continuum limit, now with $\Delta > 1$. The near-critical regime inside the entangled phase is now

$$a = -(\Delta + 1) + \sigma, \quad (107)$$

with $0 < \sigma \ll 1$ representing the control parameter that drives the entanglement transition.

<figure><img src="image_20.png" /><figcaption>FIG. 17. Main panel: the exponent $\kappa$ that defines the critical divergence of $Z^{\text{typ}}$ [see Eq. (108)] as a function of the parameter $\Delta$ in the power-law regime $\Delta > 1$. Black points show results from a numeric solution of Eq. (91) whereas the gray dashed line shows $\kappa = \max\{1/(\Delta - 1), 1\}$. Details of the numeric solution and definition of error bars are given in Appendix 3. Inset: the critical divergence of $Z^{\text{typ}}$ from the discrete toy tree for two sets of parameters in the paramagnetic phase, PM - 1 and PM - 2, see text for details. For PM - 1, the expected exponent is -1 and the best fit exponent is -0.96 whereas for PM - 2, the expected and the fitted exponents are -1.97 and -1.53, respectively.</figcaption></figure>

The difference from the case studied above [cf. Eq. (96)] is that the solutions $\lambda$ of $v_\sigma(\lambda) = 0$ are no longer complex: instead there is a real solution at $\lambda = 1 + \mathcal{O}(\sigma)$, and a larger real solution at $\lambda_+ = \Delta + \mathcal{O}(\sigma)$. That is, if we neglect both the nonlinearity in $H$ and the $x$ dependence of the diffusion constant, the stationary solution is a sum of two exponentials, in contrast to Eq. (96).

In Appendix 3 we study this regime via the equation for $R = \partial_x \ln H$, which interpolates between 0 for $x \ll x_{\text{front}}$ and -1 for $x \gg 0$. We conclude that

$$Z^{\text{typ}} \sim \sigma^{\kappa}, \quad \kappa = \max\left\{\frac{1}{\lambda_{+}-1}, 1\right\}, \quad (108)$$

where the denominator appearing in $\kappa$ is the difference of the two solutions to $v(\lambda) = 0$ at the critical point $\sigma = 0$. In the present model, $\kappa = \max\{1/(\Delta - 1), 1\}$, but we conjecture that the form in Eq. (108), which requires only knowledge of the speed function $v(\lambda)$ of the linearized problem, applies to a wider set of models.

Our argument in Appendix 3 is not rigorous, so we compare the formula $\kappa = \max\{1/(\Delta - 1), 1\}$ with a numerical solution of the continuum equation. Results are shown in Fig. 17 and are in fairly good agreement with the prediction.

Numerical solution of the continuum equation suggests that the above exponent $\kappa$ also determines the decay of $Z$ right at $r_c$,

$$Z \sim t^{-\kappa}. \quad (109)$$
<figure><img src="image_21.png" /><figcaption>FIG. 18. Schematic: tree tensor-network wave function for a chain of spins, with a subset A of spins and the corresponding minimal cut indicated. We propose a modified minimal cut formula for the entanglement.</figcaption></figure>

We also study the discrete toy tree model in Eq. (84) numerically in the regime with $\lambda_* > 1$. We find polynomial scaling of $Z^{typ}$ near the critical point as expected. The numerical estimates of the exponents differ somewhat from the predicted ones, which we attribute to finite size limitations. See Fig. 17 (inset) for examples. The parameters corresponding to PM - 1 are $p_1 = 0.15$, $p_2 = 0.85$, $\epsilon = 0.95$, and $\gamma = 0.5$, as such $\Delta = 3.97391$ and we expect $Z_{k \to \infty}^{typ} \sim \sigma^{-1}$. Indeed the best-fit exponent from our simulation is $\kappa = -0.96$. On the other hand, the parameters corresponding to PM - 2 are $p_2 = 1$, $\epsilon = 1.5$, and $\gamma = 0.5$ such that $\Delta = 1.5066$ as such the expected exponent is $\kappa = -1.97$. We however find a best-fit exponent of $\kappa = -1.53$ and attribute the discrepancy to finite size of the pool and distance from the critical point.

### J. Trees, entanglement, and min cut

So far we characterize the entanglement between the top of the tree and the base. We now apply this to more general entanglement quantities in the tree.

Figure 18 is a schematic of a wave function for a chain of spins that is given by a tree tensor network (note that there is no longer a free bond at the top). Here we consider the entanglement $S(R)$ of a set $A$ of $R \gg 1$ contiguous spins in a much larger chain. This problem has also been tackled recently in Ref. [68] using a different method: see Sec. II D.

As is well known, in such a geometry the minimal cut sketch suggests the scaling $S \sim \ln R$ [67–69], which is the number of bonds cut for “typical” choices of the placement of the region $A$ (the tree strongly breaks translational invariance). Figure 18 shows an example of a minimal cut in a small tree. The logarithmic scaling is presumably correct in the entangled phase, but what happens close to the transition? For simplicity we consider the second Rényi entropy.

Note that the minimal cut in Fig. 18 lops off a disjunct set of smaller subtrees, marked in red and thick. We assume that the region is placed so that this is the case. In this setting, a natural conjecture for the tree is that the

universal scaling forms for the entanglement close to the transition and in the disentangled phase are given by a “modified minimal cut” formula: we first find the geometrical minimal cut, but then weight the contribution to the entanglement of a bond at height $k$ ($k$ generations above the base) by an amount that depends on $Z_k$. Since we are interested in the region close to the critical point and large $k$, we assume $Z_k \ll 1$.

What should this weight be? The simplest case is where the minimal cut breaks only one bond, i.e., where region $A$ corresponds to a single connected subtree. In such cases the minimal cut breaks the full tree into two subtrees, and each one is characterized by a $Z$ value. (In general one of them has an irregular structure, with different numbers of generations for different branches, but we can still use the recursion relation to compute its $Z$ value.) A $2 \times 2$ matrix calculation shows that the second Rényi entropy $S_2$ is proportional to the product of these $Z$ values, $S_2 \propto ZZ'$, to leading order [112]. Using the fact that $Z_k^{typ}$ is asymptotically nonincreasing in $k$, the $Z$ values of the subtrees are both of typical size $Z_k^{typ}$ for $k$ equal to the height of the cut bond.

This suggests the conjecture

$$S_2(R) \sim \sum_{k=1}^{c \ln R} Z_k Z_k'. \quad (110)$$

In this schematic formula, $c \ln R$ is the maximum height reached by the minimal cut, and $Z_k$, $Z_k'$ are random variables. All order-1 constants have been neglected, since we aim only to capture the asymptotic scaling with $R$ and with the distance from the critical point. We can confirm Eq. (110) explicitly in an artificial limit in which the scale of the $Z$s tends to zero, with $R$ arbitrary but fixed: this is described in Appendix 4. However, in the physical problem we wish to take $R$ to infinity, so this does not prove the conjecture.

Consider the first class of trees (including those with the statistical invariance property of the node tensors, for example those appearing in the Haar circuit). Let $r$ be an arbitrary parameter that drives the tree’s entanglement transition. The results in the previous sections and Eq. (110) yield

$$\overline{S_2(R)} \sim \exp\left(-\frac{\text{const}}{\sqrt{r_c - r}}\right) \ln R \quad (r \lesssim r_c), \quad (111)$$

$$\overline{S_2(R)} \sim \mathcal{O}(1) \quad (r = r_c). \quad (112)$$

Surprisingly, the entanglement is order 1 at large $R$ at the critical point, because of the rapid decay of $Z$ with $k$ (Sec. 4). This is also true in the disentangled phase.

In the previous section we speculate about the existence of trees with an effective value of $\Delta > 1$. If such
trees exist, then the same reasoning as above gives $S_2(R) \sim (r_c - r)^{2\kappa}$ for $r \lesssim r_c$, with a variable exponent $\kappa = \max([\Delta - 1]^{-1}, 1)$. The entanglement right at $r_c$ is again $\mathcal{O}(1)$.

### K. Connecting back to the quantum circuit

Our original motivation for studying the tree is the conjecture that, for the forced-measurement circuit models described in detail in Sec. 3, the critical point $r_c$ of the appropriate tree ensemble is also the critical point for the circuit.

Here we give an argument that bounds the operator entanglement in the circuit in terms of the entanglement in the tree. This argument is very heuristic: a task for the future is to make the connection between the circuit and the tree more precise.

The basic idea is to imagine breaking a bond in the interior of the FMPT circuit, and to ask how much effect this can have on properties of the nonunitary time-evolution operator $V$. Let $b$ be a bond inside the circuit at time coordinate approximately $t/2$. Then $V_b$ is a modification of $V$ in which bond $b$ is broken.

Starting from $b$ we imagine marking the two trees $T$ and $T'$ attached to either end of it, using the convention in Sec. 4, where bonds with projectors on them are removed. We stop after $k$ generations, choosing the largest possible $k$ such that these are indeed two disjoint trees (no loops). Therefore, $k$ should be of order $\ln N$. We assume that the number of spins $N$ is very large, so that the typical size of $Z$ and $Z'$ for these trees (the minimal singular value squared) is given by the asymptotic large $k$ result. Close to $r_c$, this typical value is small.

Together $T$ and $T'$, connected by $b$, form a tensor network $\tilde{T}$. This can be seen as a state in a tensor product Hilbert space $\mathcal{H} \otimes \mathcal{H}'$ associated with the bonds on the boundary of $T$ and $T'$, respectively. We may form the corresponding singular value decomposition of $\tilde{T}$: the smaller of its two singular values is of order $\sqrt{ZZ'}$, in terms of the $Z$ values or $T$ and $T'$ (assumed small, since we are in the critical regime).

Breaking the bond is defined to mean dropping this minimal singular value. Formally, this induces an error that is of order $\sqrt{ZZ'}$ in $Z$ and $Z'$. After averaging over the local unitaries, the error in any physical quantity is—again, formally—of order $ZZ'$ [113]. This suggests that the average change in $S_2$ when we break a single bond is at most of order $\langle Z_\infty \rangle^2$. This is of course far from being a proof, because in principle the small term $ZZ'$ in the formal expansion could be systematically compensated by a large prefactor.

Assuming this bound, we may straightforwardly bound the plateau value $s(r)N$ of the operator entanglement close to $r_c$ (Sec. II, Sec. I). Since breaking all the $N$ bonds in a timeslice reduces the entanglement to zero, we must have

$(c_1$ and $c_2$ are constants):

$$s(r) \le c_1 \exp \left( - \frac{c_2}{\sqrt{r_c - r}} \right) \quad (113)$$

for small $r_c - r$. Assuming also our conjecture that $r_c$ is the same for the tree and the circuit, this indicates that $s(r)$ vanishes extremely rapidly as the critical point is approached. In turn, $s(r)$ is related to the scaling of the exponential timescale in the entangled phase (Sec. I), just as it is in the classical problem (Sec. III).

We note that the bound Eq. (113) on the scaling need not be tight. This can be understood by considering the analogous argument for the classical minimal cut problem.

Above, our bound used $\langle Z \rangle$. The analogous quantity in the classical problem is the probability that a given tree is infinite. This is essentially the order parameter in the classical problem, scaling like $f_\infty \sim (r_c^{\text{cl}} - r)$. We can bound the cost of the classical minimal cut $S_0$ as follows. Consider all the bonds of the percolation configuration that traverse some timeslice, say at time $t/2$. Each bond has a probability approximately $f_\infty^2$ that the two trees attached to either end of it are both infinite. These are the only bonds we need to cut (the others lie in disconnected clusters or dangling ends). This shows that $s_{\text{cl}}(r)$ goes to zero at least as fast as $(r_c^{\text{cl}} - r)^2$ close to the transition.

This bound is consistent with, but weaker than, what we argue is the true scaling in the classical problem, $s_{\text{cl}}(r) \sim (r_c^{\text{cl}} - r)^{5/2}$ (Sec. III D).

### V. SIMULATIONS OF QUANTUM CIRCUITS

Having made a connection between trees and all-to-all circuits, we now turn to the numerical simulation of the latter. Exponentially large in system size Hilbert-space dimensions restrict us to systems with $N \le 20$ spin $1/2$. We simulate both measurement circuits and forced-measurement circuits, keeping in mind that the results obtained from the tree apply only to the latter. Unless specified, the results shown here are for the Haar ensemble Eq. (37) [we also comment briefly on the $\Delta t = 0.3$ ensemble, Eq. (38)].

All-to-all circuits have no spatial structure. Consequently, the entanglement transition does not entail a volume-to-area law transition in the entanglement associated with a spatial bipartition of a state. Instead, we consider two observables, which quantify the amount of quantum information transmitted from the initial to the final time: (i) the time evolution of the operator entanglement entropies (opEE) of the nonunitary evolution operator $V$, and (ii) the overlap of two initially orthogonal states that are both evolved using $V$.

We show that the entanglement transition separates an entangled phase at $r < r_c$, wherein an extensive amount of quantum information is retained for an exponentially long
time, from a disentangled phase at $r > r_c$ wherein memory of the initial state is rapidly lost. This is in agreement with analytical results for the quantum problem in Sec. I, and is qualitatively similar to what we find in the classical toy model in Sec. III.

First, we give evidence for a plateau in the operator entanglement for $r$ below a critical value. We then turn to observable (ii) above: since this does not require exact diagonalization of $V$, it allows larger $N$ to be accessed. We use this observable to define a timescale $\tau(r, N)$, and show that this timescale scales exponentially with $N$ inside the entangled phase. Details of numerical calculations are relegated to Appendix D.

## A. Operator entanglement

The amount of information carried from the bottom of the circuit to the top can be quantified via the OEE of $V$. In the case of measurements, where we must choose an initial state in order to define the Born-rule probabilities, we take this state to be a product state (with spins aligned in the positive $x$ direction, $|\rightarrow \rightarrow \dots \rightarrow|$).

The OEE is obtained from the singular value decomposition

$$V = \sum_{j=1}^{D_H} \mu_j |j\rangle_t \langle j|_0, \quad (114)$$

where $D_H = 2^N$ is the Hilbert-space dimension, and $\{|j\rangle_0\}$ and $\{|j\rangle_t\}$ are bases corresponding to the initial and final time. (We leave the $t$ dependence of $V$ implicit.) The OEE is

$$S_n = \frac{1}{1-n} \ln \sum_{j=1}^{D_H} \lambda_j^{2n}, \quad (115)$$

where $\lambda_j \equiv \mu_j / \sqrt{\sum_j \mu_j^2}$. For a unitary $V$, $S_n$ takes on its maximal value of $N \ln 2$. Any reduction compared to this value reflects loss of information between initial and final time due to worldlines of the spins broken by measurements. $S_n$ is bounded from above by the minimal cut separating the initial and final times. Close to the FMPT transition, we also have the conjectural bound Eq. (113) on the scaling form for $s_2(r) = S_2/N$ in the plateau region.

Results for $S_1$ for the Haar ensemble with forced measurements are shown in Fig. 19. In the top panel, we plot the entanglement density $S_1/N$ vs $t$ for $r = 0.1$ and various systems sizes. After an initial linear decrease with $t$ associated with the first measurements, there is a time regime where $S_1/N$ increases with $N$. This suggests the emergence of a plateau in $S_1/N$ at large $N$ (Sec. II).

<figure><img src="image_22.png" /><figcaption>FIG. 19. The OEE, $S_1$, of the nonunitary time-evolution operator [see Eq. (115)] for the Haar ensemble with forced measurements. The top panel is for $r = 0.1$, deep in the entangled phase where there is a plateau in $S_1/N$ in the $N \to \infty$ limit. The extrapolated $N \to \infty$ value is shown as the black dashed line; the extrapolation in $N$ is shown for a few exemplary time points in the inset. The bottom panels correspond to $r = 0.3$ and $r = 0.75$, the latter being the putative $r_c$ obtained from the Haar tree. Note that the value of $S_1/N$ is already quite small in the entangled phase at $r = 0.3$. All data is averaged over 5000 realizations.</figcaption></figure>

Recall that in the entangled phase we expect a nonzero value for

$$s_n(r) = \lim_{t \to \infty} \lim_{N \to \infty} \frac{S_n(N, t, r)}{N}, \quad (116)$$

and that when $N$ is finite but large, $S_1/N$ remains close to $s_1(r)$ over a range of times that grows exponentially with $N$.

To give evidence for the nonzero value of $s_1(r)$ at $r = 0.1$, we extrapolate the data to $N = \infty$ for each value of the time [114]. This $N \to \infty$ extrapolation is shown as a dashed line in the figure. It is consistent with a plateau, extending to $t = \infty$, with $s_1(0.1) > 0$. (We defer an analysis of timescales to the following subsection.) A similar plateau was observed in Ref. [12] in Clifford circuits.

It is clear that the plateau value $s(r)$ decreases very rapidly with increasing $r$. In the lower left panel we plot $S_1/N$ for the same set of $N$ and $r = 0.3$. This $r$ value is still far from the conjectured location of the critical point obtained from the tree ($r_c \simeq 0.749$). An increase of $S_1/N$ with $N$ is still observed, but it is clear that $s_1$ (assuming it is nonzero) is small. It is tempting to associate this with the exponential scaling in Eq. (113), which suggests that $s(r)$
<figure><img src="image_23.png" /><figcaption>FIG. 20. As in Fig. 19(top), the OEE density, $S_1/N$, of the nonunitary time-evolution operator for the Haar ensemble but with measurements for $r = 0.1$. The data is consistent with there being a plateau in $S_1/N$ in the $N \to \infty$ and $t \to \infty$ limit. All data is averaged over 5000 realizations.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>FIG. 21. Heatmap: the squared overlap $\overline{O(t)}$ between the states $|\psi^{(1)}(t)\rangle$ and $|\psi^{(2)}(t)\rangle$ [see Eq. (117)], as a function of time $t$ and measurement rate $r$ for $N = 20$. The different lines show the contours for $\overline{O(t)} = 1/2$ for different $N$. The data corresponds to the Haar circuit with forced measurements. The red mark at $r = 0.75$ denotes the critical point from the Haar tree.</figcaption></figure>

goes to zero very fast as the critical point is approached, so that a plot of $s(r)$ against $r$ would be very flat for $r \lesssim r_c$.

On the other hand, at $r = 0.75$ (lower right panel), $S_1/N$ decays exponentially to zero, with a very weak $N$ dependence and no indication of saturation at large $t$. In fact, the trend with increasing $N$ is in the opposite direction to the cases $r = 0.1$ or $0.3$.

Thus, the OEE for these system sizes is consistent with an entanglement transition, occurring below the classical critical point, and with the expected plateau for $S/N$ in the entangled phase. However, the rapid decay of the plateau value $s(r)$, and the weak $N$ dependence, make it hard to pin down the position of the transition. We check (but do not show) that the data for $S_2$ is qualitatively similar to that for $S_1$.

We find the same qualitative features for the Haar circuit with true measurements. Figure 20 shows the case $r = 0.1$. In fact at this relatively small value of $r$, the data for forced measurements and measurements is almost indistinguishable.

## B. State overlap and timescales

We now turn to the overlap of two initially orthogonal states undergoing time evolution with the nonunitary operator $V$:

$$O(t) = |\langle\psi^{(2)}(t)|\psi^{(1)}(t)\rangle|^2. \qquad (117)$$

We also define the “distance” $D(t)$ as

$$D(t) = 1 - O(t). \qquad (118)$$

The states are initiated as product states in the $\sigma^x$ basis, $|\psi^{(1)}(0)\rangle = |\to\to\cdots\to\rangle$ and $|\psi^{(2)}(0)\rangle = |\leftarrow\leftarrow\cdots\leftarrow\rangle$

and are evolved using the nonunitary operator

$$|\psi^{(j)}(t)\rangle = \frac{V|\psi^{(j)}(0)\rangle}{\sqrt{\langle\psi^{(j)}(0)|V^\dagger V|\psi^{(j)}(0)\rangle}}. \qquad (119)$$

In the case of forced measurements the spins are always projected along the positive $\sigma^z$ direction, and there is a symmetry between $|\psi^{(1)}(t)\rangle$ and $|\psi^{(2)}(t)\rangle$. In the case of measurements we use $|\psi^{(1)}(t)\rangle$ to determine the Born-rule probabilities, so this symmetry is absent.

$O(t)$ is another way to quantify the amount of information retained from the initial state. In the limit $r = 0$, where $V$ is unitary, the two states remain orthogonal for all time, $O(t) = 0$. In the opposite limit $r = 1$, where no unitaries are applied, $O(t)$ will be exactly one as soon as all spins are measured. For any fixed $r > 0$, and for a fixed value of $N$, the states will inevitably converge to $1$ as $t \to \infty$, because they are being subjected to the same projections. However, we expect the timescale for this to grow exponentially with $N$ in the entangled phase.

For a broadbrush view, we first show $O(t)$ as a heatmap in the space of $r$ and $t$ for the Haar forced-measurement circuit with $N = 20$ spins: Fig. 21. $O(t)$ grows towards unity with an $r$-dependent timescale. One way to define a timescale is using the contour $\overline{O(t)} = 1/2$. This is shown not only for $N = 20$, but also for smaller values of $N$, in the figure. The conjectured $r$ value of the phase transition is marked by the red dot. The timescale grows rapidly as $r$ is decreased. It also shows a clear $N$ dependence in the entangling phase, which becomes much weaker on approaching the transition point.

Figure 22 shows the time dependence of the overlap in more detail for $r = 0.5$ and $r = 0.75$ [115]. In both cases the time required to achieve a given value of $O$ increases
with $N$, but in the latter case this is mostly a shift of the curve, whereas in the former case there is the clear sign of an increasing time constant for the exponential approach of $O$ to 1.

We use the exponential approach of the overlap to unity to define a timescale $\tau(r, N)$. Since at late times $D = 1 - O$ is exponentially small, and may have a broad distribution, we choose to look at its typical value. We define this by $\ln D^{\text{typ}}(t) = \ln \overline{D(t)}$, where instances in which $D(t)$ is exactly zero are excluded from the average (similar to the treatment of the singular value $Z$ in the quantum tree, Sec. D). At late times this shows an exponential decay,

$$ \ln D^{\text{typ}} \sim -\frac{t}{\tau(r, N)}. \qquad (120) $$

Data for $\ln D^{\text{typ}}(t)$ vs $t$ are shown in Fig. 23, for the same values of $N$ and $r$ as in Fig. 22. We see clear exponential decay. In fact, Fig. 23 vividly shows the qualitative difference between the cases of $r = 0.5$ and $0.75$: while $\tau$ grows with $N$ for $r = 0.5$, it appears essentially $N$ independent for $r = 0.75$. Data for the circuit with measurements (not shown) is qualitatively similar.

We now analyze $\tau(r, N)$ in the entangled phase. This is the asymptotic slope of plots like Fig. 23. We extract this from a plot of $\tau_{\text{eff}}(t) = -(d \ln D^{\text{typ}}/dt)^{-1}$, the time-dependent slope: at late times, $\tau_{\text{eff}}(t)$ should stabilize at the value $\tau$. Representative data for $\tau_{\text{eff}}(t)$ and the plateaux therein are shown in Appendix D, see Figs. 39 and 40.

It turns out that finite-time effects become significant at larger values of $r$, but for $r$ not too large we are able to obtain an estimate of $\tau$. The data (shown in Appendix D) is consistent at small $r$ with exponential-in-$N$ growth of the timescale:

$$ \ln \tau(r, N) \sim a(r)N. \qquad (121) $$

<figure><img src="image_25.png" /><figcaption>FIG. 22. The squared overlap between the states $\overline{O(t)}$ as a function of time $t$ for two exemplary values of $r$ for different $N$. The data corresponds to the Haar circuit with forced measurements. Note the difference in the range of times shown in the two panels. It is also possible to see the emergence of a plateau at $\overline{O(t)} = 0$ at early times.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIG. 23. The typical distance $D(t) = 1 - O(t)$ between the states $|\psi^{(1)}(t)\rangle$ and $|\psi^{(2)}(t)\rangle$ as a function of time $t$ for the same parameters as in Fig. 22, showing exponential convergence of the two states at late time. Note the difference in the scales shown in the two panels.</figcaption></figure>

The coefficient $a(r)$ is plotted against $r$ in Fig. 24. This figure also shows data for the case of true measurements.

We expect $a(r)$ to vanish at the critical point with $a(r) \sim s(r)$ (see Sec. I). Unfortunately, hamstrung by severe finite-size effects, we are not able to estimate the critical point accurately [116]. The data is certainly consistent with a critical point for the FMPT, which is below the conjectured value $\simeq 0.749$. However, we speculate that this is instead a symptom of $a(r)$ vanishing very rapidly as $r_c$ is approached, as is suggested by the essential singularity in Eq. (113). The dashed line in the figure shows this exponential form with $c_1 = 39.4$ and $c_2 = 3.8$. These values have no theoretical significance: this line is simply to indicate the possibility of $a(r)$ remaining very small even for $r$ considerably below $r_c$.

The data for $S_1$ at small values of $r$ is very close for measurements and forced measurements, as noted above.

<figure><img src="image_27.png" /><figcaption>FIG. 24. The coefficient $a(r)$ in the exponential dependence of the timescale $\tau \sim e^{a(r)N}$ on $N$, as a function of $r$ for the Haar circuit with measurements and that with forced measurements. The dashed line is a guide to eye for a function $c_1 e^{-c_2/\sqrt{r_c^{-r}}}$ with $r_c = 0.75$, $c_1 = 39.4$ and $c_2 = 3.8$. See Fig. 41 for the fits used to extract $a(r)$.</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>FIG. 25. Evidence that the forced-measurement (FM) protocol leads to higher entanglement than the measurement (M) protocol at the same value of $r$. The data is for $r = 0.3$. The left panel shows $S_1/N$ whereas the right panel shows the average overlap $\bar{O}(t)$.</figcaption></figure>

We do see differences between the two cases at intermediate $r$, with the forced-measurement circuit having slightly larger entanglement at a given $r$. This is shown for $S_1/N$ in Fig. 25(left). The comparison between the overlap data for the two cases at the same value of $r$ [Fig. 25(right)], is also consistent with the above, with $\bar{O}(t)$ growing more slowly in the forced-measurement case. This hints that $r_c$ for the measurement case may be lower than that for the forced-measurement case, but our data does not allow us to determine this.

While the data above is shown for the Haar circuit, we perform the same set of numerical calculations for the $\Delta t = 0.3$ circuit as well. The results are qualitatively similar. Consistent with the results from the quantum tree, $r_c$ appears to be smaller for the $\Delta t = 0.3$ circuit compared to the Haar.

## VI. FIELD THEORIES FOR MEASUREMENT AND ENTANGLEMENT TRANSITIONS

A key question about the MPT, not previously resolved, is whether there is a simple Landau-Ginsburg-Wilson-like field theory that captures its universal properties. This question is also unresolved for entanglement transitions in random tensor networks (RTNs), and for the closely related FMPT. In this section we propose candidates for these field theories. (In this section the space-time dimensionality $D = d + 1$ is allowed to be arbitrary.) We obtain two Lagrangians, one for the MPT, and one for both FMPT and RTN. Surprisingly, these two Lagrangians are quite different in their structure, having for example different values for the upper critical dimension.

Microscopically, random circuits and random tensor networks can be mapped to lattice statistical mechanics models [13,14,34–36,42,44,45]. These are effective spin models where the “spin” is a group element in the permutation group $S_N$ for $N$ objects (we review this below; here $N$ is a replica number and not the number of qubits as

in previous sections). However, using these lattice models to guess appropriate continuum field theories is nontrivial for various reasons, one of them being a replica limit that is necessary to handle randomness. “Replica” lattice models were described for a random tensor network in Ref. [36], for Haar circuits in Ref. [34], and for circuits with measurement in Refs. [13,14].

Previous work pointed out that in certain limits (either by artificially deforming the weights in the effective spin model [36], or by taking a $q \to \infty$ limit in the measurement problem [13,14]) one could access a fine-tuned point where the effective spin model had a simple continuum theory, namely that of percolation. While this was a useful step, this fine-tuned point has an infinite number of relevant perturbations [36] so unfortunately this does not provide a definite Lagrangian for the physical phase transitions of interest. Another approach is to study Ising models that are obtained by simply omitting the replica limit, roughly in the spirit of an annealed average in conventional disordered systems [24,27,35]. These are useful toy models for various phenomena in the entangled phase [24,27] (we give an explanation for why this is, building on Ref. [43]) but they cannot capture the correct critical properties. Therefore, we attempt here to formulate explicit continuum replica field theories.

We emphasize that these theories are speculative conjectures, based on writing down the simplest Lagrangians compatible with the basic symmetries of the problem. It is certainly possible that in fact something more complicated happens in the continuum. Indeed, the exponential scaling we find in the tree seems to mean that it is not described by the high-dimensional limit of the field theory for RTNs proposed below (see Sec. H). How to resolve this tension is a question for the future.

We first review the replica approach, the inevitable global symmetry of the field theories we are looking for, and the emergence of permutations in the simplest Haar-random models (Secs. IV A and IV B are largely review). We then discuss the coarse graining of these degrees of freedom (Sec. C). Next we note that these degrees of freedom have a more general meaning in terms of Feynman trajectories in the circuit [43,117]. This picture motivates an alternative derivation of a lattice field theory, which in turn suggests a simpler continuum formulation (Sec. VI D). Our discussion also suggests an alternative way of thinking about the effective statistical mechanics of random tensor networks, in a way that is closer to traditional replica formulations of random magnets.

Then we discuss the issue of “replica group theory” for the MPT on one hand, and the RTN and FMPT on the other: that is, constraints on the field theories associated with the replica symmetry [118]. We propose the simplest candidate Lagrangians in each case (Secs. E–G). We discuss some of the basic consequences of the simpler of these Lagrangians, that for the MPT (Sec. H). Our discussion of
these field theories is relatively schematic: further details will be given in Ref. [119].

Section I, which is independent of the field theories proposed here, addresses scaling within the two phases, not necessarily near the critical point. Finally Sec. J describes variations of the measurement problem that are in distinct universality classes, for example, models with free-fermion structure or with additional symmetries.

### A. Multilayer circuits and replica symmetry

The crucial symmetries of the problem arise when dynamical quantities are written in terms of a multilayer circuit, illustrated schematically in Fig. 26. (We use the language of a circuit, with *d* spatial dimensions and one time dimension, but analogous considerations apply to a *D* = *d* + 1 dimensional RTN.) This multilayer circuit is a discrete analog of a path integral with multiple forward and backward paths, and it arises when we write powers of the reduced density matrix, say for the final state, in terms of the circuit. Let us briefly review this.

The layers are *N* identical copies of the original circuit $V(t)$ and *N* copies of its complex conjugate $V^*(t)$. We call these “forward” and “backward” layers, respectively. Formally, the multilayer circuit with a given *N* may be written

$$V^{(N)} = \underbrace{V \otimes \dots \otimes V}_{N} \otimes \underbrace{V^* \otimes \dots \otimes V^*}_{N}. \quad (122)$$

The physical quantity of interest will dictate the boundary conditions at the top and bottom: for example, contractions of indices between layers, or contraction of the bond indices at the bottom of a layer with an initial wave function. We review this in a simple setting in Sec. IV A. An important feature is the replica trick: *N* must be left free at intermediate stages of any calculation and then sent to

a limiting value at the end [13,14,34,36]. (A special case where replicas can be omitted is mentioned in Sec. IV B below.) Replicas allow us to handle denominators that arise because of the normalization of states [36] [nonunitarity means these normalization factors are nontrivial, see Eq. (3)] and/or to deal with logarithms in the definition of the entanglement entropies. See Refs. [13,14,36] for detailed discussions of this for the RTN and MPT.

The global symmetry of the effective models arises ultimately from a simple invariance of $V^{(N)}$ under various operations. $V^{(N)}$ is clearly invariant under (i) permutations of the forward layers among themselves; (ii) permutations of the backward layers among themselves; and (iii) complex conjugation accompanied by exchange of all the forward layers with all the backward layers [43]. Together these make up the symmetry group:

$$G_N = (S_N \times S_N) \times \mathbb{Z}_2. \quad (123)$$

Here the $\mathbb{Z}_2$ is generated by (iii) above. $G_N$ is a symmetry of the bulk structure of the tensor network; it will in general be broken by boundary conditions, e.g., by a choice of index contractions at the boundary of $V^{(N)}$.

A formal way to see the importance of this symmetry is via explicit mappings of random circuits or random tensor networks onto effective lattice spin models. We review this next. We give an alternative picture below in Sec. VID, by introducing an Edwards-Anderson-like field in a multilayer tensor network (this alternative picture may be more intuitive for those familiar with random magnets).

In simple models, averaging over the random tensors or unitaries leads to effective lattice magnets in which the “spins” $\sigma$ (not to be confused with the physical spins that the circuit acts on) are valued in the permutation group [13,14,34–36,42,44,45]:

$$\sigma \in S_N. \quad (124)$$

<figure><img src="image_29.png" /><figcaption>FIG. 26. Schematic: multilayer circuit (or tensor network) with *N* copies of the circuit *V* and *N* copies of its complex conjugate $V^*$. Each layer of the circuit has free bond indices at the bottom and top (initial and final time), which are not shown. The actions of the left and right permutation symmetries and the $\mathbb{Z}_2$ exchange that make up the internal symmetry group $G_N = (S_N \times S_N) \times \mathbb{Z}_2$ of the replica field theories are illustrated.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>FIG. 27. Schematic: mapping a circuit with unitaries and possibly measurements to an effective spin model. Each physical unitary gives rise to a separate spin degree of freedom $\sigma \in S_N$ (yellow circles). These spins have interactions on downward pointing triangles (shaded). We refer to $\sigma$ and its continuum versions as the “pairing field.”</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>FIG. 28. Schematic: boundary conditions in the model for effective spins σ obtained for a 1D unitary circuit with or without measurements. Lines and triangles indicate interactions. The values of the boundary spins are fixed, and determined by the choice of contractions at the initial and final time in the multilayer circuit.</figcaption></figure>

We do not need details of the lattice construction, but Fig. 27 shows an example for a 1 + 1D circuit geometry. For each unitary in the original circuit, we obtain a spin degree of freedom σ in the effective statistical mechanical model. We may write the partition function for these spins schematically as

$$Z_N = \sum_{\{\sigma\}} W(\{\sigma\}). \qquad (125)$$

The boundary conditions on the σ depend on the observable (Sec. IV B). The Boltzmann weight W( { σ }) is a product of local weights on each of the shaded triangles in Fig. 27: the form of the weight J ( σ$_{a}$, σ$_{b}$, σ$_{c}$ ) for the three spins σ$_{a}$, σ$_{b}$, σ$_{c}$ on a given triangle, specifying how they interact, may be found in Refs. [13,14] for a circuit with measurements and in Refs. [34,42] for the purely unitary case. Constructions for the random tensor network with random Gaussian tensors were discussed earlier in Ref. [35] and extended to take into account the replica trick in Ref. [36]. In all these cases the interaction terms are, loosely speaking, ferromagnetic, in that the Boltzmann weight is maximized when the σ configuration is uniform.

Physically, the spin σ should be thought of as a way to label a choice of pairing of the forward layers with the backward layers. Let the permutation σ ∈ S$_{N}$ map a given element i ∈ { 1 , . . . , N } to σ ( i ). Then σ stands for the pairing in which forward layer i is paired with backward layer σ ( i ) and so on. For example the identity permutation, σ = I , denotes the pairing of 1 with 1, of 2 with 2, and so on, i.e., in the pattern:

$$\sigma = \mathbb{I}: \quad \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node[circle, fill, inner sep=1.2pt] (1) at (0,0) {};
\node[circle, fill, inner sep=1.2pt] (2) at (1,0) {};
\node[circle, fill, inner sep=1.2pt] (3) at (2,0) {};
\draw[->] (1) -- (2) node[left] {$1 \xrightarrow{\phantom{\sigma}} 2$};
\draw[->] (2) -- (3) node[right] {$2 \xrightarrow{\phantom{\sigma}} 3$};
\draw[->] (3) -- (1) node[right] {$3 \xrightarrow{\phantom{\sigma}} 1$};
\end{tikzpicture} \qquad (126)$$

We take N = 3 for this example, and we reorder the layers in comparison with Fig. 24 so the pairing can be drawn

without crossings. For the transposition, σ = (12), layer 1 is paired with 2 and layer 2 with 1:

$$\sigma = (12) : \quad \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node[circle, fill, inner sep=1.2pt] (1) at (0,0) {};
\node[circle, fill, inner sep=1.2pt] (2) at (1,0) {};
\node[circle, fill, inner sep=1.2pt] (3) at (2,0) {};
\draw[->] (1) -- (2) node[left] {$1 \xrightarrow{\phantom{\sigma}} 2$};
\draw[->] (2) -- (3) node[right] {$2 \xrightarrow{\phantom{\sigma}} 3$};
\draw[->] (3) -- (1) node[right] {$3 \xrightarrow{\phantom{\sigma}} 1$};
\end{tikzpicture} \qquad (127)$$

Since σ specifies a pairing of layers, we sometimes refer to it (and the continuum versions in the subsequent sections) as the “pairing field.” The physical interpretation of these pairings of layers is discussed in Sec. V D below. Heuristically, pairing Feynman histories in the discrete time evolution allows phase cancellation to be avoided [34,43], in the spirit of the diagonal approximation in periodic orbit theory [117].

That is, we may think of the multilayer circuit as a discrete path integral for N forward and N backward copies of the system. A Feynman trajectory is specified by a sequence of spin states in each of the copies. In a given layer, the corresponding product of matrix elements of local gates is the discrete analog of the exponentiated action for a continuum Feynman trajectory: e $^{iS}$or e − iS depending on whether it is a forward or a backward layer. After averaging (or, in some cases, even without averaging [43]) this multilayer path integral may be dominated by configurations in which forward and backward layers form “pairs” with similar spin configurations, contributing opposite phases to the total weight. Such a pairing allows the effect of phase cancellation to be reduced. (See also Sec. V D.) The pattern of pairing will in general differ at different locations in space time, corresponding to spacetime dependence of the pairing field σ . If the boundary conditions—say at the final time—involve pairwise index contractions of layers, as arise in the expressions for Rényi entropies (Sec. IV B), this will act as a boundary “magnetic field,” which selects out a particular value for the pairing field σ at the boundary (see for example Fig. 28).

$G_N$ acts on σ via both left and right multiplications, for the two $S_N$ factors, respectively, and via inversion for the

<figure><img src="image_32.png" /><figcaption>FIG. 29. Left: the simple random tensor network described around Eq. (147), with bond dimension 2, which reduces to a square-lattice Ising model with complex interaction constants (right). The Ising spin values on the right represent values of bond indices on the left.</figcaption></figure>
$Z_2$ generator, so that we have the symmetry transformations:

$$ \sigma \rightarrow g_L \sigma g_R^{-1}, \quad \sigma \rightarrow \sigma^{-1}, \qquad (128) $$

for permutations $g_L$ and $g_R$, together with combinations of the above.

The effective spin interactions in Eq. (125) are local for simple choices of the random tensors or gates, but in general depend nontrivially on $N$, and may even be negative [120]. However, there is a relatively simple picture of the entangling phase as a phase where $\sigma$ is ferromagnetically ordered, so that $G_N$ is spontaneously broken, and of the disentangling phase as a disordered phase. Entanglement entropies may be expressed as free-energy costs for nonuniform boundary conditions [34–36] (see Sec. IV B). We appeal only to these facts and the symmetry structure above. Note that the simplest nontrivial case is $N = 2$: then there are only two possible pairings, $\mathbb{I}$ and (12). Denoting these + and – leads to an effective Ising model [35,42]. In this case $G_N$ reduces to a simple $Z_2$ symmetry relating the two states.

Finally, we must specify the replica limits of interest. Loosely speaking, the required value of $N$ [14,36] can be seen by counting powers of $V$. It is

$$ N \rightarrow 1 \quad \text{for the MPT,} \qquad (129) $$

$$ N \rightarrow 0 \quad \text{for the RTN and FMPT.} \qquad (130) $$

$N \rightarrow 0$ is what we typically have for systems with quenched randomness (Sec. VI D). The additional power of $V$ and $V^*$ for the MPT comes from Born's rule factor (recall that we label $V = V_m$ by the sequence $m$ of measurement outcomes obtained in a given realization of the dynamics):

$$ P(\mathbf{m}) = \langle \psi | V_{\mathbf{m}}^{\dagger} V_{\bar{\mathbf{m}}} | \psi \rangle, \qquad (131) $$

which must be included in every average for the MPT. We review this more carefully in Sec. IV B.

## B. Boundary conditions in replica formalism

In order to review the replica formalism [13,14,36], let us express the operator entanglement $S_2$ of the nonunitary time-evolution operator $V$ in a measurement or forced-measurement circuit. The latter case is precisely analogous to a random tensor network, except that for the case of time evolution there is a natural division of the external legs of the tensor network into those associated with the initial time and those associated with the final time. We focus in this subsection only on reviewing how the boundary conditions in the effective partition function arise formally (see

Sec. VI D for more on how the “pairing field” arises in the bulk).

We define the operator entanglement in Sec. II. Recall that, if we view $V$ formally as a tensor network wave function for $2N$ spins, then $\rho_t$ is the unnormalized reduced density matrix associated with the final-time legs. Let us start with the case of the FMPT, where expectation values (denoted by $\mathbb{E}[\dots]$ or $[\dots]$), are simple averages over the unitaries and projections in $V$. The expectation value of the second Rényi entropy is

$$ \bar{S}_2 = -\mathbb{E} \ln \frac{\text{tr} \rho_t^2}{(\text{tr} \rho_t)^2} = -\mathbb{E} \ln \frac{\bigcirc}{\bigcirc} \qquad (132) $$

On the right we indicate the pattern of index contraction graphically. The vertical lines represent a stack of copies of $V$ and $V^*$, like that in Fig. 24, but viewed from the side. For convenience, we order the four layers in the stack as follows: $V^*$, $V$, $V^*$, $V$ (instead of grouping all of the $V^*$s together as we do in Fig. 24). The arcs at the top and bottom indicate the pattern of index contractions between layers. Index contractions are done separately for each of the physical sites.

Next let us define “partition functions” that are averages of the multilayer circuit with particular choices of boundary conditions. We use the notation $Z_N(\sigma|\tau)$ for the average of the circuit with $N$ layers of $V$ and $N$ layers of $V^*$, and with index contractions in the pairing pattern $\sigma$ at the top and $\tau$ at the bottom. For example,

$$ Z_2(\mathbb{I}|\mathbb{I}) = \mathbb{E} \bigcirc \bigcirc \qquad Z_2(\mathbb{I}|(12)) = \mathbb{E} \bigcirc \bigcirc. \qquad (133) $$

$Z_N(\sigma|\tau)$ maps to a partition function for the pairing field with an effective “magnetic field” favoring pairing state $\sigma$ at the final time (top) and $\tau$ at the initial time (bottom).

Equation (132) is not immediately written in terms of such partition functions, because of the logarithm and the fraction, but this can be dealt with using the replica trick [36]. Equation (132) is trivially equivalent to

$$ \bar{S}_2 = -\frac{1}{m} \mathbb{E} \ln \frac{\bigcirc^m}{\bigcirc^{2m}} \qquad (134) $$

for any $m > 0$, since the factors of $m$ cancel. But one may check (by expanding in $m$ in the numerator and denominator below) that in the limit $m \rightarrow 0$ the expectation value may be taken for the numerator and denominator
separately:

$$ \bar{S}_2 = - \lim_{m \to 0} \frac{1}{m} \ln \frac{\mathbb{E}[\mathcal{U}^m]}{\mathbb{E}[\mathcal{U}]^{2m}} \quad (135) $$

As usual, we treat $m$ as a positive integer at intermediate stages of the calculation. The above then becomes

$$ \bar{S}_2 = - \lim_{m \to 0} \frac{1}{m} \ln \frac{\mathcal{Z}_{2m}(\mathbb{I} | \tau_{2,m})}{\mathcal{Z}_{2m}(\mathbb{I} | \mathbb{I})} \quad (136) $$

Here $\tau_{2,m}$ denotes a permutation in $S_{2m}$ that is a product of $m$ commuting two cycles [34]

$$ \tau_{2,m} = (12)(34) \dots (2m - 1, 2m). \quad (137) $$

Equation (136) may now be interpreted as the free-energy cost of imposing distinct boundary conditions for the pairing field $\sigma$ (represented by the continuum field $X$ in the sections below) at the initial and final times. If the free-energy cost for given boundary conditions $\sigma$ and $\tau$ is [121] $\mathcal{F}_N(\sigma^{-1}\tau)$ then

$$ \bar{S}_2 = \lim_{m \to 0} \frac{1}{m} \mathcal{F}_{2m}(\tau_{2,m}). \quad (138) $$

This generalizes directly to higher Rényi entropies. (The von Neumann entropy can either be obtained using an additional limit $n \to 1$, or by a slightly different construction with a single replica limit [36].) Note that the total number $N$ of replicas (denoted $2m$ above) tends to zero as stated above for the FMPT and the RTN.

The simplest situation, discussed in the next subsection, is where the pairing field is well ordered across the entire sample. Then the free-energy cost $\mathcal{F}$ is essentially the free-energy cost of inserting a single domain wall in this order [34–36]. See for example Fig. 30 in Sec. I.

In fact, in this situation (the strongly ordered regime) results from the unitary case suggest that in the replica limit can be dispensed with: we can map the entanglement to the free-energy cost of a single domain wall in an effective classical disordered system [34]. The most direct way to understand this is to avoid the replica trick entirely [43]. It is possible to make a formal mapping of the multilayer circuit in Eq. (132) (with $N = 2$) to an “Ising model” without any averaging. In general this model has complicated long-range interactions, so that it is not useful for discussing the critical point. But in the strongly ordered regime we expect (assuming the considerations for the unitary case in Ref. [43] carry over) that the interactions are effectively local after sufficient coarse graining. $S_2$ in a given realization can then be understood as a domain wall cost in a

<figure><img src="image_33.png" /><figcaption>FIG. 30. At large times, a system in $d+1$ space-time dimensions is quasi-one-dimensional. The long timescale $\tau$ in the entangled phase, with $\ln \tau \sim sN$, is due to the free-energy cost, which is approximately $sN$ of a domain wall in the effective spin model (Sec. I). At early times, the replica trick can be avoided, giving a domain wall in a disordered Ising model. More generally we must use the replica spin model.</figcaption></figure>

disordered Ising model. This is a route for justifying the use of an Ising model to discuss, for example, subleading corrections to the volume law in the entangled phase [24,27]. When the critical point is approached we must however return to the replica description above.

The application to the MPT is similar to the case of the FMPT. Graphically, the Born probability $P(\mathbf{m}) = \langle \psi | V_{\mathbf{m}}^{\dagger} V_{\mathbf{m}} | \psi \rangle$ for a sequence of measurement outcomes may be denoted by

$$ P(\mathbf{m}) = \mathcal{U}, \quad (139) $$

where the dots represent contraction with $\psi$ or $\psi^*$ as appropriate. Let us absorb a trivial constant into “$\mathbb{E}$” so that it denotes the average over the structure of the circuit together with the unweighted sum over $\mathbf{m}$:

$$ \bar{S}_2 = -\mathbb{E} \ln \frac{\mathcal{U}}{\mathcal{U}^*}. \quad (140) $$

We may simplify the formulas slightly by averaging over the initial state, which yields

$$ \bar{S}_2 = -\mathbb{E} \ln \frac{\mathcal{U}}{\mathcal{U}^*}. \quad (141) $$

The replica trick then allows us to write

$$ \bar{S}_2 = - \lim_{m \to 0} \frac{1}{m} \ln \frac{\mathcal{Z}_{2m+1}(\mathbb{I} | \tau_{2,m})}{\mathcal{Z}_{2m+1}(\mathbb{I} | \mathbb{I})} \quad (142) $$

Formally this is similar to Eq. (136), but the total number of replicas $N = 2m + 1$ is taken to 1 rather than 0 [13,14].
### C. Permutations and coarse graining

We now focus on the critical properties. Let us first make a brief detour to consider coarse graining a lattice model of permutations, such as that shown schematically in Eq. (125), in an abstract sense, in order to understand one of the basic challenges. (This section is not an essential prerequisite for the following developments—the reader who wants to get to the concrete results may wish to skip it.)

We work throughout with a system in some finite number of dimensions $D = d + 1$ (the space-time dimension in the case of a circuit). Naively we might expect the limit of large $d$ to match the all-to-all circuit (as in Sec. III E) but this is unclear (Sec. H).

Let us first imagine attempting a block-spin RG procedure in a naive way, by simply “averaging” the spins $i$ within each $D$-dimensional local block:

$$ \mu_{\text{block}} \propto \sum_{i \in \text{block}} \sigma_i. \qquad (143) $$

What does this expression mean? At this point, each $\sigma_i$ on the rhs is a formal group element in $S_N$. Their linear combination, $\mu_{\text{block}}$, is no longer in $S_N$, since addition is not a group operation (only multiplication). Instead it is an element of the group algebra of $S_N$ [122]. A general element of the group algebra is a linear combination of the elements $g$ of the group with numerical coefficients $M_g$,

$$ \mu_{\text{block}} = \sum_{g \in S_N} M_g g, \qquad (144) $$

where in the present case $M_g \in \mathbb{R}$. In other words, we can think of the coefficients $M_g$ as forming a vector $M$ of length $N!$, which is the order of $S_N$. The coarse-grained spin above is equivalent to this vector.

However, $\mu_{\text{block}}$, or equivalently the vector $M$, is not a natural coarse-grained field in general. The reason for this is that $M$ does not form a single representation of the global symmetry $G_N$. Instead, the $N$-dimensional vector space splits into many distinct representations, in fact a number of representations that grows exponentially as $N$ grows. Standard results for the group algebra imply that the representations of $G_N$ that appear when we decompose $\mu_{\text{block}}$ are in one-to-one correspondence with the irreducible representations of $S_N$ [122]. To extract a particular representation of $G_N$, we simply replace the formal group elements in Eq. (143) with their matrix representatives in the corresponding representation of $S_N$.

This means that our initial attempt to form a block spin has led us not to a single coarse-grained field, but to an indeterminate number (because $N$ must be left free) of different coarse-grained fields, each in a different representation of the global symmetry group $G_N$.

In principle, we could try to write down a Lagrangian including all of these fields. However, since the number of these fields, and therefore the number of couplings, depends on $N$, this does not seem promising. Instead, it is natural to hope that only one or a small number of the fields become massless at the critical point, and the other fields do not need to be included in a continuum Lagrangian. This is the assumption we make, motivated by the more explicit picture in the following section.

This picture of splitting $\mu_{\text{block}}$ into separate fields gives an alternative view on the discussion of the percolation fixed point in Ref. [36]. The authors imagined starting with a lattice model with a much enlarged symmetry, $S_N!$ (not $S_N$ or $S_N \times S_N$ or $G_N$). This much larger symmetry group is allowed to arbitrarily permute all the $N!$ values $\sigma \in S_N$ that the spin can take. Such a lattice model is simply a Potts model with $Q = N!$ states, for which the continuum theory is well known (becoming percolation when $Q \to 1$). The authors then considered deforming model in the direction of the physical model of interest [cf. Eq. (125)], which does not have $S_N!$ symmetry. They found that the lowest-order perturbation that could be added was quadratic in the Potts field, and so relevant. However, there was considerable freedom in the index structure of this perturbation, which could be formed from any class function of $S_N$.

From the present point of view, this perturbation is a sum of mass terms, with one independent mass for each of the infinite number of fields that appear when we decompose $\mu_{\text{block}}$ above into representations of $S_N$ for arbitrary $N$.

### D. Motivating a simple Landau theory

A familiar way to represent a permutation in $S_N$ is as an $N \times N$ matrix $X_{a,b}$ of ones and zeros, with a single 1 in each row and in each column,

$$ \sum_a X_{ab} = 1, \quad \sum_b X_{ab} = 1. \qquad (145) $$

Under the global symmetries in Eq. (128), this matrix transforms as

$$ X \to LXR^{-1}, \quad X \to X^T, \qquad (146) $$

where $L$ and $R$ are permutation matrices representing $g_L$ and $g_R$.

We might hope that we can build a Landau theory from such a matrix. In terms of the discussion in the previous section, this will correspond to the simplest choice of representations of $G_N$ to include in the continuum theory (discussed below). In fact we can motivate such a Landau theory in a more direct way, without the need to go through the mappings discussed above involving permutations.

For this we appeal to the basic physical picture for why the pairings of layers arise in the multilayer circuit, which is to avoid phase cancellation. To make this explicit, let us
consider a particularly simple example of a tensor network $V$ (which we can interpret formally as a nonunitary time evolution for qubits) with the geometry in Fig. 29(left). Label the bond index values by $S = \pm 1$ (these are the spins' $\sigma_z$ values if $V$ is interpreted as a time evolution). Take the local gates $w$, with bond indices $S_1, S_2, S_3, S_4$, to have the simple form

$$w_{S_1,S_2,S_3,S_4} = \exp \left( \sum_{1 \le i \le 4} h_i S_i + \sum_{1 \le i < j \le 4} J_{ij} S_i S_j \right), \quad (147)$$

where each $h$ is an independent, identically distributed complex Gaussian variable with mean zero, and equal variance $\Delta_h^2/4$ for its real and imaginary parts, and similarly for the $J$'s, with variance $\Delta_J^2/2$. (These couplings are taken complex since tensors in a generic tensor network are complex.)

The tensor contraction defining $V$ involves a sum over all the indices carried by the internal bonds in Fig. 29, i.e., over all Feynman trajectories, if we think of the vertical direction as time. This tensor contraction is an Ising partition function for the indices $S_i$ on the bonds $i$. With the choices above, this Ising model lives on a rotated square lattice. We may write its partition function as

$$Z = \sum_{\{S_i\}} \exp(iS[\{S\}]), \quad (148)$$

where the exponentiated lattice “action” $e^{iS}$ is just a product of terms of the form Eq. (147), so that $iS[\{S\}]$ is an Ising Hamiltonian with random complex magnetic fields and random complex nearest-neighbor couplings. This is schematic as we have left the boundary conditions unspecified. (Fixed boundary conditions on the spins give a matrix element of $V$, for example; in practise we are interested in taking several layers of $Z$, which are coupled at their boundaries.)

Quantities of interest involve the replicated partition function (cf. Fig. 24). Up to boundary conditions, this is given by averaging $Z^N \times Z^{*N}$ over all of the random $h$ and $J$ parameters, as in the standard application of the replica trick to the Ising model with random bonds or random fields [82]. Introducing $N$ replicas of the Ising spin for the forward layers, denoted $S^a$ for $a = 1, \dots, N$, and $N$ replicas for the backward layers denoted $\bar{S}^b$, the replicated partition function $Z_N$ has the form

$$Z_N = \sum_{\{S_i^a, \bar{S}_i^b\}} \exp(-S_{\text{eff}}) \quad (149)$$

(we do not include an $i$ in the definition) with

$$S_{\text{eff}} = -\Delta_J^2 \sum_{\langle ij \rangle} \sum_{ab} X_{ab}(i) X_{ab}(j) - \Delta_h^2 \sum_i \sum_{ab} X_{ab}(i), \quad (150)$$

where we define the “pairing field”

$$X_{ab}(i) = S_i^a \bar{S}_i^b. \quad (151)$$

This is similar to an Edwards-Anderson order parameter in an Ising spin glass. However, the usual Edwards-Anderson order parameter would be of the form $S^a S^b$ (as there would be no distinction between forward and backward layers) and the replica permutation symmetry would act on both $a$ and $b$ together. In the present case we have separate permutation symmetries for the row and column indices of $X$.

We defer an explicit discussion of coarse graining for this and other microscopic models to a separate publication [119]. Here we note only that the form of Eq. (150), with ferromagnetic interactions between the pairing field $X$ for different sites, motivates writing a continuum Lagrangian for an $N \times N$ matrix, as discussed above.

In the present microscopic formulation, $X$ is not a permutation matrix, but the action of symmetry is the same [Eq. (146)]. This is what we use, together with the assumption that the pattern of symmetry breaking in the entangled phase is the simplest one corresponding to pairing, i.e., to a choice of permutation.

Without loss of generality, let this permutation be the identity permutation (other cases are related by symmetry). Then the pattern of symmetry breaking is captured by an expectation value of the form

$$X_{ab} = f \delta_{ab} + c, \quad (152)$$

where $f$ is the order parameter. Here $c$ is a constant, which is generically nonzero even in the disordered phase, since (unlike $f$) it does not break any symmetry. This order breaks $G_N$ down to $S_N \times \mathbb{Z}_2$, where the remaining permutation group is the subgroup of diagonal $S_N \times S_N$ transformations with $g_L = g_R$. (We briefly discuss more complex possibilities for symmetry breaking in Sec. J.)

The physical interpretation of $X$ is simple: if in some region the spin configuration in the forward layer $a$ is close to that in backward layer $b$, then the coarse-grained $X_{ab}$ in this region will be large. Heuristically, we expect repulsive interactions between $X_{ab}$ and $X_{ac}$ for $b \neq c$: if the configuration in $a$ is close to that in $b$, the phases from the $a$ layer are already (partially) cancelled, so there is less gained by also pairing with $c$.

Let us briefly mention a caveat to the above discussion. A “random tensor network” is by definition a statistical