# Entanglement and charge-sharpening transitions in U(1) symmetric monitored
quantum circuits

Utkarsh Agrawal  Department of Physics, University of Massachusetts, Amherst,
MA 01003, USA  Aidan Zabalo  Department of Physics and Astronomy, Center for
Materials Theory, Rutgers University, Piscataway, NJ 08854, USA  Kun Chen
Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue,
New York, NY 10010, USA  Justin H. Wilson  Department of Physics and
Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ
08854, USA  Andrew C. Potter  Department of Physics, University of Texas at
Austin, Austin, TX 78712, USA  Department of Physics and Astronomy, and
Quantum Matter Institute, University of British Columbia, Vancouver, BC,
Canada V6T 1Z1  J. H. Pixley  Department of Physics and Astronomy, Center for
Materials Theory, Rutgers University, Piscataway, NJ 08854, USA  Department of
Physics, Princeton University, Princeton, NJ 08544, USA  Center for
Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York,
NY 10010, USA  Sarang Gopalakrishnan  Department of Physics, The Pennsylvania
State University, University Park, PA 16802, USA  Romain Vasseur  Department
of Physics, University of Massachusetts, Amherst, MA 01003, USA

###### Abstract

Monitored quantum circuits can exhibit an entanglement transition as a
function of the rate of measurements, stemming from the competition between
scrambling unitary dynamics and disentangling projective measurements. We
study how entanglement dynamics in non-unitary quantum circuits can be
enriched in the presence of charge conservation, using a combination of exact
numerics and a mapping onto a statistical mechanics model of constrained hard-
core random walkers. We uncover a charge-sharpening transition that separates
different scrambling phases with volume-law scaling of entanglement,
distinguished by whether measurements can efficiently reveal the total charge
of the system. We find that while Rényi entropies grow sub-ballistically as
$\sqrt{t}$ in the absence of measurement, for even an infinitesimal rate of
measurements, all average Rényi entropies grow ballistically with time $\sim
t$. We study numerically the critical behavior of the charge-sharpening and
entanglement transitions in U(1) circuits, and show that they exhibit emergent
Lorentz invariance and can also be diagnosed using scalable local ancilla
probes. Our statistical mechanical mapping technique readily generalizes to
arbitrary Abelian groups, and offers a general framework for studying
dissipatively-stabilized symmetry-breaking and topological orders.

###### Contents

  1. I Introduction
  2. II Overview of Results
    1. II.1 Models
    2. II.2 Observables and averaging
    3. II.3 Results
      1. II.3.1 Entanglement transition
      2. II.3.2 Charge-sharpening transition
      3. II.3.3 Entanglement dynamics
  3. III Numerics on qubit chains
    1. III.1 Entanglement transition
    2. III.2 Charge sharpening transition
  4. IV Statistical mechanics model
    1. IV.1 Statistical model for systems without symmetry
    2. IV.2 Statistical Model with U(1) qubits – Summary
  5. V Numerical results from the statistical mechanics model
    1. V.1 Charge-sharpening transition
      1. V.1.1 Entanglement transition in the qubit sector
      2. V.1.2 Charge-sharpening transition
      3. V.1.3 Local ancilla probe
    2. V.2 Entanglement dynamics
  6. VI Discussion
  7. A Mapping to the statistical model with U(1) qubits
    1. A.1 Link weights
    2. A.2 Replicated model
    3. A.3 Replica limit
    4. A.4 $p=0$ limit
  8. B Statistical mechanics model for general Abelian symmetries
    1. B.1 Haar average
    2. B.2 Dual entangling phases classical and quantum orders
      1. B.2.1 $1+1d$: Symmetry Protected Topology (SPT)
      2. B.2.2 $1+1d$: Spontaneous Symmetry Breaking
      3. B.2.3 $2+1d$: Discrete Gauge Theories
  9. C Additional numerical results
    1. C.1 Entanglement dynamics in the statistical mechanics model
    2. C.2 Charge sharpening dynamics in the fuzzy phase and near the charge-sharpening transition
  10. D Finite size scaling

##  I Introduction

The dynamics of quantum information has become a central theme across multiple
branches of physics ranging from condensed matter and atomic physics to
quantum gravity Amico  _et al._ (2008); Horodecki  _et al._ (2009); Eisert
_et al._ (2010); Calabrese and Cardy (2009); Laflorencie (2016). Of particular
recent interest is the exploration of entanglement dynamics in open quantum
systems, motivated by the advent of noisy intermediate-scale quantum
simulators Preskill (2018). An open system is continuously “monitored” by its
environment, so its dynamics consists of two competing processes: unitary
evolution, which generates entanglement and generically leads to chaotic
dynamics Nahum  _et al._ (2017, 2018); von Keyserlingk  _et al._ (2018); Zhou
and Nahum (2019); Rakovszky  _et al._ (2018); Khemani  _et al._ (2018), and
non-unitary operations resulting from measurements and noisy couplings to the
environment, that tend to irreversibly destroy quantum information by
revealing it.

A minimal model that captures these competing processes consists of a quantum
circuit made up of random unitary gates interlaced with local projective
measurements. Remarkably, this minimal model undergoes a dynamical phase
transition as the rate of measurements is increased. This transition occurs in
individual quantum trajectories (i.e., the state of the system conditional on
a set of measurement outcomes), and separates two phases where typical
trajectories have very different entanglement properties Li  _et al._ (2018);
Skinner  _et al._ (2019). When measurements are frequent enough, they rapidly
extract quantum information from any initial state, and collapse it into a
weakly entangled pure state. Below a critical measurement rate, however,
initial product states grow highly entangled over time, while initial mixed
states remain mixed for extremely long times. In this “entangling” phase,
unitary dynamics scrambles quantum information into nonlocal degrees of
freedom that can partly evade local measurements. These nonlocal degrees of
freedom span a decoherence-free subspace in which the dynamics is effectively
unitary Choi  _et al._ (2020); Gullans and Huse (2020a); Li and Fisher (2020);
Fan  _et al._ (2020): this subspace can be regarded as the code space of a
quantum error correcting code. The effective size of the protected subspace
vanishes at the measurement-induced phase transition. The critical properties
of the measurement-induced transition are still under investigation, but key
qualitative features of the transition including the emergence of conformal
invariance are by now well-established Bao  _et al._ (2020); Jian  _et al._
(2020); Li  _et al._ (2020); Zabalo  _et al._ (2021). Meanwhile, different
variants of this transition have been investigated involving various ensembles
of gates, circuit geometries, etc., establishing that such transitions are
generic aspects of the quantum trajectories of open systems Li  _et al._
(2018); Skinner  _et al._ (2019); Li  _et al._ (2019); Chan  _et al._ (2019);
Li  _et al._ (2020); Cao  _et al._ (2019); Gullans and Huse (2020a);
Szyniszewski  _et al._ (2019); Choi  _et al._ (2020); Bao  _et al._ (2020);
Jian  _et al._ (2020); Gullans and Huse (2020b); Zabalo  _et al._ (2020);
Nahum and Skinner (2020); Ippoliti  _et al._ (2020); Lavasani  _et al._
(2020); Sang and Hsieh (2020); Tang and Zhu (2020); Lopez-Piqueres  _et al._
(2020); Nahum  _et al._ (2021); Turkeshi  _et al._ (2020); Fuji and Ashida
(2020); Lunt  _et al._ ; Lunt and Pal (2020); Fan  _et al._ (2020); Vijay
(2020); Li and Fisher (2020); Turkeshi  _et al._ (2021); Ippoliti and Khemani
(2021); Lu and Grover (2021); Jian  _et al._ (2020); Gopalakrishnan and
Gullans (2021); Turkeshi (2021); Bao  _et al._ (2021); Block  _et al._ (2021);
Bentsen  _et al._ (2021); Zabalo  _et al._ (2021); Noel  _et al._ (2021).

Given the central role of scrambling in the measurement-induced transition, it
is natural to ask how the transition (and the entangling phase) are affected
if one constrains the scrambling dynamics by imposing symmetries or
conservation laws. This question remains largely open, despite recent hints
that symmetries may allow for distinct volume-law entangled states of matter
Bao  _et al._ (2021). Even in the absence of measurements, conservation laws
severely constrain the scrambling of local operators Rakovszky  _et al._
(2018); Khemani  _et al._ (2018); Friedman  _et al._ (2019). Moreover,
conservation laws can parametrically slow down entanglement growth. For
example, the dynamics of the Rényi entropies

$S_{n}=\frac{1}{1-n}\ln{\rm tr}\rho_{A}^{n}$ (1)

(with $\rho_{A}$ the reduced density matrix of a subsystem $A$) in systems
with a conserved charge have been shown to grow sub-ballistically as
$S_{n>1}\sim\sqrt{t}$ Rakovszky  _et al._ (2019); Huang (2019); Zhou and
Ludwig (2020); Rakovszky  _et al._ (2020); Žnidarič (2020), while the von
Neumann entropy remains linear in time $S_{1}\sim t$. These results suggest
that slow hydrodynamic modes might fundamentally alter, not just the nature of
the measurement-induced phase transition, but also the nature of the
entangling phase.

In this work we study the many-body dynamics of monitored quantum circuits
with a conserved charge (or equivalently a U(1) symmetry). These systems
exhibit not only a measurement-induced entanglement transition but also a new
type of “charge-sharpening” transition that separates two distinct entangling
phases. These two phases are distinguished by whether it is easy or hard (in a
way we will make precise below) for measurements to reveal the charge of the
system. In the “charge-fuzzy” phase, there are large charge fluctuations even
_conditional_ on the measurement history: i.e., the history of measurement
outcomes does not suffice to fix the charge profile. In the “charge-sharp”
entangling phase, by contrast, the measurement outcomes fix the charge
distribution. The charge-sharp phase is nevertheless highly entangled because
of the scrambling of neutral degrees of freedom. We find that both the
entanglement transition and the charge-sharpening transition are Lorentz
invariant, with a dynamical exponent $z=1$. This ballistic dynamics may seem
surprising in light of the diffusive growth of Rényi entropies in the absence
of measurements. However, we give general arguments that, for any finite
measurement rate, $p>0$ all Rényi entropies grow linearly in time, $S_{n}\sim
t$.

Our conclusions are supported by exact numerics on Haar random U(1) monitored
circuits, and by a replica statistical mechanics model Bao  _et al._ (2020);
Jian  _et al._ (2020) obtained from the study of a U(1) symmetric qubit degree
of freedom coupled to a $d$-dimensional qudit. Remarkably, in the limit
$d\to\infty$, we are able to analyze the replica limit analytically, and the
contributions to entanglement from the qubit and qudit degree of freedom
decouple. This enables us to study the entanglement dynamics of the qubit
directly in this limit by simulating an effective statistical mechanics model
of a symmetric exclusion process constrained by the measurements and
entanglement cuts. We also discuss scalable probes of the charge-sharpening
transition using ancilla qubits, and find evidence for a new universality
class for the charge-sharpening transition in the limit of small local Hilbert
space dimension.

The plan of the rest of the paper is as follows. In Sec. II we specify the
models we have explored, and present some general considerations on their
steady state phases and dynamics. In Sec. III we present numerical results for
U(1)-symmetric qubit chains. In Sec. IV we present a tractable limit in which
the model can be mapped onto the statistical mechanics of constrained random
walkers. In Sec. V we present numerical results for the transfer matrix of
this statistical model. Finally in Sec. VI we summarize our results and
discuss their broader implications. In particular, we note that these methods
can be readily generalized to general Abelian symmetry groups, as detailed in
Appendix B, where we leverage duality relations to discuss implications for
systems with highly-entangled phases with symmetry-breaking and topological
orders.

##  II Overview of Results

In this section we will introduce a family of U(1)-symmetric circuits, and
present some general observations concerning their steady-state phase
structure and entanglement dynamics. The numerical evidence supporting these
observations will be presented below, in Secs. III and V.

###  II.1 Models

Following Khemani  _et al._ (2018), we consider a one-dimensional chain in
which each site hosts a two-level system (“qubit”) and a $d$-level system
(“qudit”), i.e., the on-site Hilbert space is
$\mathbb{C}^{2}\otimes\mathbb{C}^{d}$ for $d>1$ and $\mathbb{C}^{2}$ for
$d=1$. The dynamics will consist of local unitary gates and measurements,
which are chosen to conserve the U(1) charge

$\mathcal{Q}\equiv\sum_{i}\mathfrak{q}_{i}\otimes\mathbb{I}_{i},\,\,\,\mathrm{where}\,\,\,\mathfrak{q}_{i}=(\sigma^{z}_{i}+1)/2$
(2)

is acting on the $i$th site of the chain of length $L$ and $\mathbb{I}$ is the
identity matrix on the qudits. These chains evolve under (i) unitary two-qubit
gates, acting on nearest neighbor sites, which conserve the global charge
$\mathcal{Q}$, and (ii) single-site projective measurements in which the qubit
is measured in its $Z$ basis and the qudit is simultaneously measured in some
reference basis 111Since the unitaries acting on the qudits are random, the
randomizing measurement basis is superfluous.. At each time-step, a given site
is measured with probability $p$; for specificity, we assume that when this
happens both the qubit _and_ the qudit are measured, so the measurement acts
on that site as a rank-1 projector. The symmetry-preserving two-site unitary
gates are arranged in a brickwork geometry and take the form

$\displaystyle U_{i,i+1}=\begin{pmatrix}U^{0}_{d^{2}\times d^{2}}&0&0\\\
0&U^{1}_{2d^{2}\times 2d^{2}}&0\\\ 0&0&U^{2}_{d^{2}\times
d^{2}}\end{pmatrix},$ (3)

where $i$ labels a site, $U^{\mathfrak{q}}_{D\times D}$ is a unitary matrix of
size $D\times D$ acting on the charge
$\mathfrak{q}_{1}+\mathfrak{q}_{2}=\mathfrak{q}\in\\{0,1,2\\}$ sector (a local
charge is defined to take values $0$ and $1$), and $D$ is the dimension of the
Hilbert space of the charge sector. Each matrix is drawn independently from
the Haar random ensemble of unitary matrices of the appropriate size.

We present numerical results for this class of circuits in two limits. First,
we consider the limit $d=1$, where there is no qudit degree of freedom, and
one simply has a chain of qubits interacting via gates that conserve the
charge $\mathcal{Q}$. In this limit, we obtain numerical results by direct
time evolution. Second, we consider the complementary limit $d=\infty$, in
which we can map the problem to a statistical mechanics model and explicitly
write down a transfer matrix that generates the observables of interest. The
phase diagrams in the two complementary limits are similar.

The qubit-only ($d=1$) model is directly realizable in existing quantum
processors. The $d>1$ model is perhaps less natural experimentally, but could
be realized in circuit quantum electrodynamics setups Blais  _et al._ (2021)
in which superconducting transmon qubits are coupled to multilevel
superconducting cavities (qudits), or by blocking multiple qubits together
(e.g. $d=2$ could be realized as a two-leg ladder of qubits). Regardless of
experimental implementation, we expect the $d>1$ models to capture the generic
universal behavior of phases and transitions, while allowing greater
theoretical control in the large-$d$ limit.

###  II.2 Observables and averaging

For a given choice of unitary gates and measurement locations $\mathbf{X}$ (in
spacetime), the unitary-measurement dynamics of the density matrix can be
written as a quantum channel $\mathcal{N}_{t}$, in the Kraus form

$\rho(t)=\mathcal{N}_{t}(\rho)=\sum_{\\{{\cal
M}(\mathbf{X})\\}}K_{\mathbf{m}}\rho K_{\mathbf{m}}^{\dagger},$ (4)

where ${\cal M}(\mathbf{X})$ indexes the outcomes of all projective
measurements. Here $\mathbf{m}$ denotes a “quantum trajectory”, associated
with a fixed configuration of measurement locations $\mathbf{X}$, and
measurement outcomes ${\cal M}(\mathbf{X})$. We will write
$\mathbf{m}=\\{\mathbf{X},{\cal M}(\mathbf{X})\\}$. The Kraus operators
$K_{\mathbf{m}}$ consist of random unitary gates and projection operators onto
the measurement outcomes ${\cal M}(\mathbf{X})$, and we have $\sum_{\\{{\cal
M}(\mathbf{X})\\}}K_{\mathbf{m}}^{\dagger}K_{\mathbf{m}}=\mathbb{I}$. We will
be concerned with general properties of the single-trajectory state
$\rho_{\mathbf{m}}\equiv K_{\mathbf{m}}\rho K_{\mathbf{m}}^{\dagger}$. (As a
concrete example, consider the purity
$\Pi_{\mathbf{m}}=\mathrm{Tr}(\rho_{\mathbf{m}}^{2})/(\mathrm{Tr}\rho_{\mathbf{m}})^{2}$.)
We then average over quantum trajectories, weighting each set of measurement
outcomes by its probability of occurrence, i.e., the Born probability
$p_{\mathbf{m}}=\mathrm{Tr}(\rho_{\mathbf{m}})$. Finally, we average the
answers across the ensemble of channels.

A few comments are in order here.

  1. 1.

For an initially pure state $|\psi\rangle$ the Born probability takes the
familiar form $p_{\mathbf{m}}=\|K_{\mathbf{m}}|\psi\rangle\|^{2}$, i.e., it is
just the norm of the projected state. For a series of measurements
interspersed with unitary gates, one can pick each measurement outcome based
on the Born probability or _equivalently_ apply a set of projectors at random
and evaluate the probability of the entire measurement history by computing
the norm of the state at the end of the trajectory.

  2. 2.

There are four different types of average that we will consider here: (i) the
quantum expectation value of an observable $A$ in a single trajectory
$\rho_{\mathbf{m}}$, namely
$\mathrm{Tr}(A\rho_{\mathbf{m}})/\mathrm{Tr}\rho_{\mathbf{m}}$, which we will
write as $\langle A\rangle_{\mathbf{m}}$; (ii) the (Born-weighted) average of
a single-trajectory function, such as purity or entanglement entropy, over the
trajectories that make up a particular channel; (iii) the average over unitary
gates, chosen with Haar measure; and (iv) the average over spacetime points
where the measurements occur (measurement locations). In much of this work, we
will present results for which averages (ii)-(iv) have been done. We will use
the notation $[\cdot]$ for this full average. At some points it will be useful
to separate these averages. In these cases we will use the explicit notations
$\sum_{\\{{\cal M}(\mathbf{X})\\}}(\cdot)$, $\mathbb{E}_{U}(\cdot)$, and
$\mathbb{E}_{\mathbf{X}}(\cdot)$ for averages over measurement outcomes,
gates, and measurement locations respectively. We will also use a shorthand
notation $\mathbb{E}_{\mathbf{m}}\equiv\sum_{\mathbf{m}}p_{\mathbf{m}}(\dots)$
to denote summation over all possible measurement locations $\mathbf{X}$
including appropriate probability factors of $p$ and $1-p$, and over all
measurement outcomes ${\cal M}(\mathbf{X})$, including the associated Born
probability factor $p_{\mathbf{m}}$.

  3. 3.

It is crucial that the quantities of interest to us are _nonlinear_ functions
of $\rho_{\mathbf{m}}$ Skinner  _et al._ (2019), such as $\Pi_{\mathbf{m}}$.
To see the significance of this, let us compare the quantity
$\Pi_{\mathbf{m}}$ to that of some simple expectation value. In
$\rho_{\mathbf{m}}$, the expectation value of a local operator $A$ would be
$\mathrm{Tr}(A\rho_{\mathbf{m}})/\mathrm{Tr}\rho_{\mathbf{m}}$. Averaging this
over trajectories (measurement outcomes) with the Born probabilities would
simply give infinite temperature behavior 222The ensemble-averaged $\rho(t)$
resulting from maximally-random, local, open-systems dynamics is
indistinguishable from an infinite temperature state over distances $\sim t$.:
$\sum_{\\{{\cal M}(\mathbf{X})\\}}(p_{\mathbf{m}}\langle
A\rangle_{\mathbf{m}})=\sum_{\\{{\cal
M}(\mathbf{X})\\}}\mathrm{Tr}(A\rho_{\mathbf{m}})=\mathrm{Tr}(A\rho(t)){\approx\mathrm{Tr}(A)/\mathrm{Tr}(1)}$.
By contrast, $\sum_{\\{{\cal
M}(\mathbf{X})\\}}\Pi_{\mathbf{m}}p_{\mathbf{m}}=\sum_{\\{{\cal
M}(\mathbf{X})\\}}\mathrm{Tr}(\rho_{\mathbf{m}}^{2})/\mathrm{Tr}\rho_{\mathbf{m}}$,
which cannot simply be written in terms of $\rho(t)$. The averaged density
matrix $\rho(t)$ is blind to measurement transitions; only nonlinear functions
of single-trajectory wavefunctions detect it.

###  II.3 Results

In the following, we unveil a charge sharpening transition that takes place
before the entanglement transition in two distinct models of monitored U$(1)$
symmetric random quantum circuits. Our main results are summarized in Fig. 1
and discussed in more detail below.

####  II.3.1 Entanglement transition

A general feature of unitary-projective circuits is the presence of an
_entanglement transition_ , separating a phase where initially unentangled
states develop volume-law entanglement from one where their entanglement
remains area-law at all times. We briefly review the general properties of
this transition and discuss how they are modified by the presence of a
conservation law.

This transition occurs at some critical measurement rate $p_{c}$. In the
volume-law phase, the half-system entanglement entropy grows linearly in time
and saturates on timescales $t\sim L$. At times $t>L$, the entanglement
entropy (averaged over circuits and trajectories) reaches a steady state value
$[S(L/2)]=cL$, where $c$ is a constant that decreases continuously to zero at
$p_{c}$. At $p=p_{c}$, we have that $[S_{n}(L/2)]=\alpha_{n}\log L$, where
$\alpha_{n}$ (which generally depends on the Rényi index $n$) is part of the
universal critical data Skinner  _et al._ (2019); Zabalo  _et al._ (2020).

An equivalent way to understand the entanglement transition is as a
“purification transition” for an initially mixed state Gullans and Huse
(2020a). For $p<p_{c}$, an initially mixed state for a system of size $L$
evolves to a pure state on a timescale $t_{\pi}\sim\exp(L)$, whereas for
$p>p_{c}$ purification happens on a timescale that grows sub-linearly in $L$.
If one takes the limits $L\rightarrow\infty$, $t/L=\mathrm{constant}$, the
purity of an initially mixed state for $p<p_{c}$ is essentially constant in
time: i.e., the steady state is defined to be at early times compared with the
slow purification dynamics for $p<p_{c}$.

The purity of an initially mixed state on timescales $t\sim L$ can be used to
define an effective order parameter for the volume-law phase, as follows
Gullans and Huse (2020b); Zabalo  _et al._ (2020). Consider evolving a pure
state along some trajectory until times $t\sim L$, and then entangling some
local degree of freedom with an ancilla qubit. The reduced density matrix of
the system is now a rank-2 mixed state. For $p<p_{c}$, this mixed state
remains mixed for an exponentially long time. Therefore, by evolving for
another $t\sim L$ and then measuring the entanglement entropy of the ancilla
(which is equivalent to measuring the purity of the system density matrix in
this setup), one can extract a local order parameter for the volume-law phase
Gullans and Huse (2020b). By studying the correlations of this local order
parameter—e.g., by coupling in two ancillas at distinct spacetime points and
tracking their mutual information—it was established (in the absence of the
U$(1)$ symmetry) that the critical theory has an emergent Lorentz invariance
with dynamical scaling exponent $z=1$.

One of our results is to locate and characterize this entanglement transition
in the presence of the U$(1)$ conservation law. In the $d\to\infty$ limit, we
find that $p_{c}=1/2$, exactly as in the absence of a conservation law.
Moreover, the entanglement transition corresponds to a percolation transition.
For $d=1$, we find that $p_{c}=0.105(3)$ moves to substantially lower
measurement rate than in the generic circuit without a conservation law
($p_{c}^{\mathrm{Haar}}\approx 0.17$ Zabalo  _et al._ (2020)). The correlation
length exponent $\nu=1.32(6)$ is close to the percolation value $4/3$, but the
coefficients $\alpha_{n}$ differ from the percolation value as well as the
value for Haar-random circuits without a $U(1)$ symmetry. Finally, we find
compelling numerical evidence that the dynamical scaling $z=1$ holds at this
critical point. That this holds regardless of the diffusive ($z=2$) dynamics
of the U$(1)$ conserved charge is perhaps puzzling at first sight. We return
below to the resolution of this puzzle.

Figure 1:  Phase diagram and crossover time scales in U(1) symmetric monitored
quantum circuits. Our numerical results indicate that there are two distinct
phases in the entangling (volume-law) regime $p<p_{c}$, separated by a charge-
sharpening critical point as $p=p_{\\#}$. In the charge-fuzzy phase
($p<p_{\\#}$), we identify three relevant time scales in the dynamics: For a
large enough system, first (1) average Rényi entropies crossover from
diffusive $[S_{n>1}]\sim\sqrt{t}$ to ballistic $\sim t$ scaling over a time
scale $\sim p^{-3/2}$, then (2) charge sharpens after the crossover time scale
$t_{\\#}\sim L$, and finally (3) the system purifies over a much long time
scale $t_{\pi}\sim{\rm e}^{L}$.

####  II.3.2 Charge-sharpening transition

In addition to changing the critical properties of the entanglement transition
at $p_{c}$, the conservation law gives rise to a _distinct_ “charge-
sharpening” transition at a measurement rate $p_{\\#}$ inside the volume-law
phase. The charge-sharpening transition separates a “charge-sharp” phase for
$p>p_{\\#}$, in which the measurements along a typical trajectory can rapidly
collapse an initial superposition (or mixture) of different charge sectors,
and a “charge-fuzzy” phase where this collapse is parametrically slower
occuring on a time scale $t_{\\#}\sim L$. Specifically, we can distinguish
charge- sharp or fuzzy behavior by the variance of the conserved charge
$\cal{Q}$ in Eq. (2) over a single trajectory, averaged across trajectories
and samples, i.e.,

$\displaystyle[\delta{\cal Q}^{2}]=[\langle{\cal
Q}^{2}\rangle_{{\mathbf{m}}}-\langle{\cal Q}\rangle_{{\mathbf{m}}}^{2}],$ (5)

where the quantity in parentheses is the _quantum_ number variance in a given
trajectory. In the sharp phase, $[\delta{\cal Q}^{2}]=0$ while in the fuzzy
phase it remains non-zero at times of order $t\sim L$.

The dynamics of charge sharpening at small $p$ can be qualitatively understood
in terms of a simple classical model, in which one ignores the spatio-temporal
correlations between measurements. One can then ask how many independent
density measurements $\mathcal{N}_{M}$ are required to distinguish systems
with $N$ particles on $L$ sites from those with $N-1$ particles on $L$ sites,
where $n\equiv N/L={\cal O}(1)$. Assuming Gaussian density fluctuations (as in
the $p=0$ thermal state) we expect the $N$-particle and $(N-1)$-particle
states to become distinguishable when $\mathcal{N}_{M}\sim L^{2}$. Since
$\mathcal{N}_{M}=pLt$ in the circuits we consider, sharpening happens on a
crossover timescale $t_{\\#}\sim L/p$. For timescales $t\geq t_{\\#}$, we
expect that $[\delta\mathcal{Q}^{2}]\sim\exp(-t/t_{\\#})$. This follows, e.g.,
from using the central limit theorem to estimate the probability that an $N$
particle state will give an average density of $n\pm 1/L$ after $pLt$
measurements. This simple model of the volume law phase predicts that a
crossover to charge sharpening should take place on a timescale $t_{\\#}\sim
L/p$, consistent with our numerical findings (see Sec. III and V), and
parametrically faster than purification. Importantly, at any finite $t/L$,
$[\delta\mathcal{Q}^{2}]$ remains non-zero in the fuzzy phase (albeit
exponentially small for $t\gg t_{\\#}$).

By contrast, for $p>p_{\\#}$, charge-sharpening happens on a timescale that is
sublinear (logarithmic) in system size. In the limit
$L\to\infty,t/L=\mathrm{constant}$, each trajectory has a definite charge.
Thus there is a sharp phase transition at $p_{\\#}$, for which
$[\delta\mathcal{Q}^{2}]$ acts as an order parameter. In addition to
sharpening global superpositions of charge, we find evidence from the
statistical mechanics model, that charge sharpening also occurs on a local
level, i.e. the charge fluctuations, $\delta\mathcal{Q}_{A}^{2}$ in a finite
but large (compared to correlation length) region $A$ exhibit area-law
scaling, $\delta\mathcal{Q}_{A}^{2}\sim|\partial A|$ in the charge-sharp
phase, rather than extensive volume-law scaling
$\delta\mathcal{Q}_{A}^{2}\sim|A|$ in the charge-fuzzy phase. In this sense,
the charge variance of can also be used as an order parameter for the charge-
sharpening transition. Our numerical results also indicate that charges are
completely frozen by the measurements in some regions of space-time exhibiting
locally minimal spacing of measurements (see Sec. V).

As with the entanglement transition, one can probe the charge-sharpening
transition by coupling an ancilla to the circuit. One entangles the ancilla
with the system such that each ancilla state is coupled to a system state with
a different value of $\mathcal{Q}$. The system-ancilla entanglement vanishes
when $\mathcal{Q}$ sharpens under the circuit dynamics.

####  II.3.3 Entanglement dynamics

We now turn to the dynamics of entanglement at times of order unity. Recall
that, absent measurements, the Rényi entropies $S_{n}\sim\sqrt{t}$ for all
$n>1$ in random circuits with U$(1)$-symmetric gates. This diffusive
entanglement dynamics appears to be a generic property of random circuits, so
one might expect it to hold throughout the volume-law phase. If it held at the
critical point, it would prevent the critical theory from being a conformal
field theory (CFT). We now discuss why Rényi entropies in fact scale
_ballistically_ for any non-zero measurement probability, $p>0$, allowing both
the sharpening and entanglement transitions to obey relativistic $z=1$ dynamic
scaling.

First, we review the argument for diffusive scaling in the absence of
measurements Rakovszky  _et al._ (2019); Huang (2020); Zhou and Ludwig (2020).
This phenomenon arises from rare fluctuations that leave a region empty (or
maximally filled), as follows. Consider, for concreteness, the dynamics of the
initial product state for the qubit
$|\psi\rangle=\otimes_{i=1}^{L}|+_{x}\rangle_{i}$ where
$|+_{x}\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$. Suppose we are
interested in the entanglement across a cut at $L/2$ at some later time $t$.
We can divide the system into three regions: a central region of radius
$\ell=\sqrt{Dt}$ centered at the entanglement cut, and regions to the left and
right. Define

$|\psi_{\text{dead}}\rangle=\bigotimes_{i=1}^{L/2-\ell}|+_{x}\rangle_{i}\bigotimes_{i=L/2-\ell+1}^{L/2+\ell}|0\rangle_{i}\bigotimes_{i=L/2+\ell+1}^{L}|+_{x}\rangle_{i}.$
(6)

Initially, $|\langle\psi_{\text{dead}}|\psi\rangle|=2^{-\ell}$. After evolving
for time $t$,
$|(\langle\psi_{\text{dead}}|U^{\dagger}_{t})(U_{t}|\psi\rangle)|=2^{-\ell}$
by unitarity. However, $U_{t}|\psi_{\text{dead}}\rangle$ is a product state
with respect to the cut at $L/2$: by construction, $t$ is not long enough for
particles to have diffused to the entanglement cut, and unless there is a
$|10\rangle$ or $|01\rangle$ configuration at the cut the gates acting across
the cut cannot generate entanglement. The largest Schmidt coefficient of
$U_{t}|\psi\rangle$ is its maximal overlap with _any_ product state, so we can
lower-bound the largest Schmidt coefficient of $U_{t}|\psi\rangle$ as
$2^{-\ell}=2^{-\sqrt{Dt}}$, and therefore $S_{\infty}\leq\ell\ln
2\sim\sqrt{Dt}$. All Rényi entropies with $n>1$ are dominated by this largest
Schmidt coefficient and grow as $\sqrt{t}$. The Von Neumann entropy $S_{1}$ is
dominated instead by _typical_ Schmidt coefficients: the number of these grows
exponentially in $t$, but they are also exponentially small in $t$ and are
therefore subleading for $n>1$.

We now address how this argument changes when $p>0$. In a typical trajectory,
on a timescale $t$, there are $p\ell t\sim pt^{3/2}$ measurements in the
putative dead region near the entanglement cut, and about half the
measurements observe a qubit to be in the charge state $|1\rangle$. There are
rare circuits with few measurements near the cut, as well as rare histories in
a typical circuit where all the measurements yield the same outcome
$|0\rangle$. However, both are _at least_ exponentially suppressed in $\ell$
and cannot dominate the trajectory-averaged entanglement (since _any_
trajectory contributes at most $\sim t$ entanglement, and in any case these
atypical trajectories have unusually slow entanglement growth). Therefore, to
compute the trajectory-averaged Rényi entropies it suffices to consider
trajectories with typical measurement locations and typical outcomes. In
typical trajectories, one observes a $|1\rangle$ charge after ${\cal O}(1)$
measurements in the region near the cut, so the putative dead region survives
only until a time $t\sim p^{-2/3}$. At longer times, the overlap of the
wavefunction with dead regions is zero. We conclude that the trajectory-
averaged Rényi entropies $[S_{n}]$ grow linearly in time whenever $p>0$
333Though other quantities such as $\log[e^{-S_{n}}]$ are dominated by rare
dead-region contributions and do exhibit $\sqrt{t}$ growth due to rare dilute
measurement locations. The parametrically strong discrepancy between the
average purity and the average Rényi entropies is also seen numerically in our
statistical model approach (Sec. V).. We also note that, the existence of
diffusive hydrodynamic modes, which are a purely classical phenomena, does not
affect the $z=1$ dynamical scaling at the volume-to-area law entanglement
transition at $p_{c}$. Our numerical estimates of the dynamic exponent in Sec.
III are consistent with this result that $z=1$ scaling applies and diffusive
hydrodynamics decouples also at the charge-sharpening transition at $p_{\\#}$.

##  III Numerics on qubit chains

In this section we present numerical results on a model of random
U(1)-conserving gates acting on a chain of qubits (i.e., the $d=1$ limit of
the general model in Sec. II.1). Specifically, in the basis of the adjacent
qubits
$\\{\ket{\downarrow\downarrow},\ket{\uparrow\downarrow},\ket{\downarrow\uparrow},\ket{\uparrow\uparrow}\\}$
the two-qubit gates at site $i$ take the block diagonal form

$U_{i,i+1}=\begin{pmatrix}e^{i\phi_{0}}&&\\\ &U^{1}_{2\times 2}&\\\
&&e^{i\phi_{1}}\end{pmatrix}$ (7)

where $\phi_{1}$ and $\phi_{0}$ are chosen at random from the interval
$\left[0,2\pi\right)$ and $U^{1}_{2\times 2}$ is a Haar-random $2\times 2$
unitary matrix that can be parameterized by 4 angles

$U^{1}_{2\times
2}(\alpha,\phi,\psi,\chi)=e^{i\alpha}\begin{pmatrix}e^{i\psi}\cos\phi&e^{i\chi}\sin\phi\\\
-e^{-i\chi}\sin\phi&e^{-i\psi}\cos\phi\end{pmatrix}$ (8)

where $0\leq\phi_{0,1}\leq\frac{\pi}{2}$ and $\alpha$, $\psi$, $\chi$, and
$\phi$ are chosen so that $U^{1}_{2\times 2}$ is uniformly sampled from U$(2)$
444To obtain a uniform distribution over $U(2)$ we must pick
$\alpha,\psi,\chi\in\left[0,2\pi\right)$ and $\xi\in\left[0,1\right]$
uniformly and compute $\phi=\arcsin\sqrt{\xi}$.. In between layers of gates,
projective measurements are performed: with a probability $p$, the qubit is
projected onto ${\lvert\uparrow\rangle\mbox{ or }\lvert\downarrow\rangle}$
given by the Born rule. Utilizing the conservation law, we work in definite
number-sectors to reduce the memory load of the exact numerics.

The conservation law leads to different charge sectors defined by eigenspaces
of $\mathcal{Q}$ in Eq. (2). We will typically focus on charge sectors near
the central subspace ($\mathcal{Q}=L/2$).

###  III.1 Entanglement transition

We begin by locating the entanglement transition point $p_{c}$ in this model.
In order to probe the location of the critical point and the correlation
length critical exponent, we study two quantities that have been identified as
good measures of the transition: the tripartite mutual information Zabalo  _et
al._ (2020) and an order parameter defined through the use of an ancilla
Gullans and Huse (2020b) that is coupled to one charge-$\mathcal{Q}$ sector.
In the following it is essential that we use an accurate estimate of $p_{c}$
to be able to numerically disentangle it from the charge sharpening transition
at $p_{\\#}$.

First, the tripartite mutual information for the Rényi index $n$ is defined as

$\displaystyle\mathcal{I}_{3,n}(A,B,C)\equiv$
$\displaystyle~{}~{}S_{n}(A)+S_{n}(B)+S_{n}(C)-$ $\displaystyle-S_{n}(A\cup
B)-S_{n}(A\cup C)-S_{n}(B\cup C)+$ $\displaystyle+S_{n}(A\cup B\cup C),$ (9)

where we have chosen regions $A,B,\mbox{and }C$ to be adjacent regions of size
$L/4$ and $S_{n}(A)$ is the Rényi entropy defined in Eq. (1). For
$\mathcal{I}_{3,n}$ at late times ($t=4L$) we apply the finite size scaling
hypothesis $\mathcal{I}_{3,n}\sim f(L^{1/\nu}(p-p_{c})),$ to locate the
critical point, where $f(x)$ is a scaling function and $\nu$ is the
correlation length exponent. The data for $n=1$ is shown in Fig. 2(a) where we
find the data collapses with the minimum $\chi^{2}$ for the choice of
$p_{c}=0.105(3)$ and $\nu=1.32(6)$. A similar analysis can be performed for
the other Rényi entropies and the resulting values of $p_{c}$ and $\nu$ are
similar for all $n\geq 1$ investigated, we find
$p_{c}=0.103(4),0.12(2),0.12(1)$ and $\nu=1.37(8),1.47(3),1.5(2)$ for
$n=2,5,\infty$, respectively.

Figure 2:  Entanglement transition in qubit chains. (a) Data and collapse of
the tripartite mutual information, $\mathcal{I}_{3,n=1}$, used to determine
the critical point of the entanglement transition, $p_{c}=0.105(3)$ and the
correlation length exponent, $\nu=1.32(6)$. (b) Data and collapse of the
entanglement transition order parameter, $\left[S_{1,E}\right]$, used as an
alternative method to determine the critical point of the entanglement
transition, $p_{c}=0.110(3)$, and the correlation length exponent
$\nu=1.42(16)$. (c) At the critical point, the bipartite entanglement entropy
shows logarithmic scaling with the system size. The coefficient of the
logarithm has strong Rényi index dependence that can be described by a
functional form $\alpha(n)=0.65(1)\left(1+\frac{1}{n}\right)+0.04(1)$. This
closely resembles the standard result for the groundstate of a CFT, but has an
offset slightly larger than zero.  Figure 3: Charge sharpening transition in
qubit chains. (a) Data and collapse of $N_{0}$, the fraction of trajectories
with $\delta\mathcal{Q}^{2}<\epsilon$ used to determine the critical point of
the charge sharpening transition, $p_{\\#}=0.094(3)$ and correlation length
exponent, $\nu_{\\#}=2.0(3)$. The value of $\epsilon=10^{-2}$ and is chosen
such that it maximizes the quality of collapse at $t/L=4$. (b) Data and
collapse of the entanglement transition order parameter,
$\left[S_{1,Q}\right]$, used as an alternative method to determine the
critical point of the entanglement transition, $p_{\\#}=0.088(3)$, and the
correlation length exponent $\nu_{\\#}=2.15(15)$. (c) Data and collapse of the
fraction of trajectories where the ancilla qubit is purified $N_{\text{pure}}$
at the charge-sharpening transition. The transition point $p_{\\#}=0.087(4)$
and the correlation length exponent $\nu_{\\#}=2.1(2)$ are consistent with the
ancilla probe.  Figure 4: Dynamical exponent. Plot of the rescaled time-
dependence of the ancilla-circuit entanglement entropy for the entanglement
transition at $p=p_{c}\approx 0.11$ (blue curve) and the charge-sharpening
transition at $p=p_{\\#}\approx 0.088$ (red curve). The finite size collapse
indicates a dynamical exponent $z=1$ for both transitions.  Figure 5: Two
transitions. Plot of the $68\%$ confidence interval of the critical point $p$
and the correlation length exponent $\nu$ for both the entanglement and charge
transitions. The mutual information $I_{3,1}$ (solid blue circle) and the
ancilla probe (dashed blue circle) are for the entanglement transition, while
the fraction of trajectories $N_{0}$ (solid red circle) and the ancilla probe
(dashed red circle) are for the charge-sharpening phase transition. The two
transitions appear to be different with statistical significance, although we
cannot exclude systematic finite-size effects that would change this
conclusion in the thermodynamic limit.

At the critical point, the bipartite entanglement entropy shows a logarithmic
dependence on the system size and the coefficient of the logarithmic
divergence shows strong Rényi index dependence (Fig. 2(c)). This behavior can
be described by

$S_{n}(p_{c},L)\sim\alpha(n)\ln
L,\,\,\alpha(n)=0.65(1)\left(1+\frac{1}{n}\right)+0.04(1).$ (10)

Apart from the small offset, this Rényi index dependence matches the result
one expects for the ground state of a CFT Calabrese and Cardy (2009),
$\alpha_{\mathrm{CFT}}(n)=\frac{c}{6}\left(1+\frac{1}{n}\right)$. The
coefficients in Eq. (10) clearly differ from those at the measurement induced
transition without a conservation law Zabalo  _et al._ (2020).

As an alternative way of locating the entanglement transition, we also study
the “order parameter” Gullans and Huse (2020b). In order to have the ancilla
couple to the system within a particular global charge sector, we consider the
qubits at two adjacent sites $i$ and $i+1$ to be in the entangled state with
an ancilla:
$|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}\ket{1}+\ket{\downarrow\uparrow}\ket{0})$
where the ancilla has orthogonal basis states $\ket{1}$ or $\ket{0}$. We then
evolve the system in time $t=2L$ without measurements in order to create a
state
$\ket{\Psi}=\frac{1}{\sqrt{2}}(\ket{\psi_{1}}\ket{1}+\ket{\psi_{0}}\ket{0})$
where $\ket{\psi_{0,1}}$ are orthogonal and in the same charge sector. We then
run the circuit with measurements for an additional time $2L$ and compute the
von Neumann entropy of the ancilla, which we denote as $S_{1,E}$ (as it probes
the entanglement transition).

The results for the von Neumann entanglement entropy of the ancilla $S_{1,E}$
are shown in Fig. 2(b), and are consistent with $\mathcal{I}_{3,n}$ data: From
the scaling ansatz $S_{1,E}\sim f_{E}((p-p_{c})L^{1/\nu})$, where $f_{E}(x)$
is a universal scaling function, we obtain $p_{c}=0.110(3)$ and $\nu=1.4(2)$
in good agreement with $\mathcal{I}_{3}$.

Summarizing these results, the entanglement transition in U(1)-symmetric
circuits has a critical exponent $\nu$ that is consistent with the value for
Haar-random circuits with no symmetries, although the nonuniversal $p_{c}$ has
drifted down from the Haar value $p^{\mathrm{Haar}}_{c}\approx 0.17$ (as one
might expect since each gate cannot generate as much entanglement). At
$p=p_{c}$ we extract the dynamical exponent of the entanglement transition
using the scaling ansatz $[S_{1,E}]\sim g_{E}(t/L^{z})$, which shows a good
quality as seen in Fig. 4 for $z=1$ and $g_{E}(x)$ some universal scaling
function. Again, this result is consistent with the non-conserving case. While
we have focused on $\mathcal{Q}=L/2$ we have checked that for the largest
system sizes considered $p_{c}$ is only very weakly affected for
$\mathcal{Q}=L/2-1$ (not shown).

###  III.2 Charge sharpening transition

We now turn to estimating $p_{\\#}$ in two ways: the charge variance of a
state and the entropy of an ancilla entangled with two different number
sectors.

First, we compute the variance of the total charge (Eqs. (2) and (5)). For a
trajectory that lies in a well defined charge sector, $\delta{\cal Q}^{2}=0$,
otherwise $\delta{\cal Q}^{2}\neq 0$. Therefore, we start with an initial
state that is spread out over all of the different $\mathcal{Q}$-sectors,
$|\psi_{0}\rangle=\bigotimes_{i=1}^{L}\frac{1}{\sqrt{2}}\left(\ket{\uparrow}_{i}+\ket{\downarrow}_{i}\right)$,
and run the conserving hybrid dynamics to late times to determine if the
system has sharpened into a single charge sector for some measurement
probability $0\leq p_{\\#}\leq p_{c}$, where $p_{c}\approx 0.11$ is the
critical point of the entanglement transition. In this situation, the critical
point of the charge sector transition can be determined by the studying the
probability, $P(\delta{\cal Q}^{2}=0)$. (Recall that $\delta\mathcal{Q}^{2}$
is a quantum uncertainty that is a property of each trajectory; the
probability distribution $P(\delta\mathcal{Q}^{2})$ is over trajectories and
circuits, where each trajectory is weighted by its Born probability.) For
large systems, $P(\delta{\cal Q}^{2}=0)\rightarrow 0$ when the system is
distributed over multiple sectors while $P(\delta{\cal Q}^{2}=0)\rightarrow 1$
when the system has been constrained to a single sector. In Fig. 3(a), the
fraction $N_{0}$ of trajectories having a variance $x=\delta{\cal Q}^{2}\leq
s$ (with $s=10^{-2}$) is shown for various system sizes and measurement
probabilities. The critical point can be identified by the crossing near
$p=0.1$. Performing a finite size scaling analysis, we find the data for
different system sizes collapses onto a universal curve for the critical point
$p_{\\#}=0.094(3)$ and correlation length exponent $\nu_{\\#}=2.0(3)$.

Lastly, we consider an ancilla coupled to two different charge sectors, in
particular we take
$\ket{\Psi}=\ket{\psi_{\mathcal{Q}}}\ket{0}+\ket{\psi_{\mathcal{Q}-1}}\ket{1}$
where $\ket{\psi_{\mathcal{Q}}}$ represents states within the charge sector
$\mathcal{Q}$ (while $\ket{1}$ and $\ket{0}$ are states of the ancilla as
before). Since there is no unitary that mixes these sectors, we can say
definitively that the reduced density matrix has the form

$\rho_{\mathrm{anc}}=\begin{pmatrix}|\braket{\psi_{\mathcal{Q}}(t)}{\psi_{\mathcal{Q}}(t)}|^{2}&0\\\
0&|\braket{\psi_{\mathcal{Q}-1}(t)}{\psi_{\mathcal{Q}-1}(t)}|^{2}\end{pmatrix}.$
(11)

This formulation is convenient for the numerical algorithm we have developed
that conserves charge since if the ancilla were just considered an extra
qubit, $\ket{\Psi}$ would be in the conserving sector $M$ for $L+1$ qubits.
Doing this, we compute the von Neumann entanglement entropy of the ancilla
qubit that we denote as $S_{1,\mathcal{Q}}$, that is shown in Fig. 3(b). Based
on the crossing in the data and the ansatz $S_{1,\mathcal{Q}}\sim
g_{\mathcal{Q}}((p-p_{\\#})L^{1/\nu_{\\#}})$ we obtain $p_{\\#}=0.088(3)$ and
$\nu_{\\#}=2.2(2)$, which matches the $p_{\\#}$ and $\nu_{\\#}$ found by
$P(\delta{\cal Q}^{2}=0)$. In addition, we extract the charge sharpening
transition from the probability that the ancilla has fully disentangled from
the circuit by computing the fraction of trajectories $N_{\mathrm{pure}}$ that
have fully purified the ancilla [Fig. 3(c)]. From the crossing of
$N_{\mathrm{pure}}$ we find a third consistent estimate of $p_{\\#}$ and
$\nu_{\\#}$. Thus, we have identified the charge sharpening transition across
all sectors of $\mathcal{Q}$ with the transition in $S_{1,\mathcal{Q}}$ across
$\mathcal{Q}=L/2,L/2-1$. The dynamical exponent of the charge sharpening
transition is also $z=1$, as $[S_{1,\mathcal{Q}}]\sim g_{\mathcal{Q}}(t/L)$ at
criticality (Fig. 4).

The two critical points we have identified in this model at $p_{\\#}$ and
$p_{c}$ are at least $\sim 3.5$ error bars from each other, providing evidence
that a charge sharpening transition occurs _before_ full purification. This is
further exemplified by the estimated confidence intervals for the two
transitions for the various probes we have considered as shown in Fig. 5.
Moreover, the correlation length exponent for charge-sharpening quantities is
distinct from that of the entanglement purification transition, further
suggesting that these represent distinct critical points with different
universality classes.

We note, however, that the close proximity of the putative two transitions
make them challenging to cleanly separate numerically in small scale systems,
and acknowledge that this data could in principle be accounted for by large,
systematic finite-size errors in the critical exponents that affected the
charge and entanglement properties differently555In the absence of a
conservation law, it has been shown that the probes we have used for the
entanglement transition ($\mathcal{I}_{3,n}$ and $[S_{1,E}]$) have weak finite
size drifts in Clifford circuits Zabalo  _et al._ (2020) by examining small
and large system sizes.. In the following sections, we will see that for the
model with large-$d$ qudits, the location of the two transitions become
clearly distinct.

##  IV Statistical mechanics model

In this section, we show that in the $d\to\infty$ limit, the calculation of
entanglement in monitored U(1) circuits can be mapped exactly onto a classical
statistical model defined on a square lattice. In this limit, the
contributions to the entanglement entropy from the qubit with conserving
dynamics and the qudit decouple. The resulting qubit contribution can then be
obtained from a constrained symmetric exclusion process.

Our main goal is to compute averaged Rényi entropies $\left[S_{n}\right]$. The
Rényi entropies of a spatial sub-region, $A$, for a fixed quantum trajectory
are given by

$\displaystyle S_{n}(A,\mathbf{m})$
$\displaystyle=\frac{-1}{n-1}\ln\text{Tr}\left(\rho_{{\mathbf{m}}}^{\otimes
n}T_{n,A}\right),$ (12)

where
$\rho_{\mathbf{m}}=\ket{\psi_{\mathbf{m}}(t)}\bra{\psi_{\mathbf{m}}(t)}$,
$\ket{\psi_{\mathbf{m}}(t)}=K_{\mathbf{m}}\ket{\psi_{0}}$ is the state
(“trajectory”) of the system after evolution by time $t$ for a measurement
history $\mathbf{m}$, and $T_{n,A}$ is a “SWAP” operator permuting the $n$
copies of the input state in the entanglement region $A$:

$\displaystyle
T_{n,A}=\prod_{i}|s_{\sigma_{i}(1)}s_{\sigma_{i}(2)}...s_{\sigma_{i}(n)}\rangle\langle
s_{1}s_{2}...s_{n}|$
$\displaystyle\sigma_{i}=\begin{cases}\text{identity}=e,&i\notin A\\\ (12\dots
n),&i\in A\end{cases},$ (13)

where the index $i$ runs over all physical sites, $\ket{s_{i}}$ are members of
the onsite Hilbert space, $\sigma_{i}$ is an element of the permutation group
$\mathcal{S}_{n}$, and $(12\dots n)$ denotes a cyclic permutation of the $n$
copies of $\rho$. The key technical difficulty in this problem is to perform
the average over gates, measurement locations and outcomes, and to normalize
the state after the projective measurements since entanglement is
intrinsically non-linear in the density matrix. To bypass this problem, we
follow Refs. Jian  _et al._ (2020); Bao  _et al._ (2020) (see also Hayden  _et
al._ (2016); Vasseur  _et al._ (2019) in the context of random tensor
networks) and introduce $k$ replica copies of the system. The average Rényi
entropy $S_{n}$ is then written as:

$\displaystyle\left[S_{n}\right]$ $\displaystyle=\lim_{k\rightarrow
0}\frac{-1}{k(n-1)}\sum_{\mathbf{m}}\left(Z_{A}(\mathbf{m})-Z_{\emptyset}(\mathbf{m})\right),$
(14)

where

$\displaystyle Z_{A}(\mathbf{m})$
$\displaystyle=\mathbb{E}_{U}\left[\text{Tr}\left(\left(K_{\mathbf{m}}\ket{\psi_{0}}\bra{\psi_{0}}K_{\mathbf{m}}^{\dagger}\right)^{\otimes
nk+1}T_{n,A}^{\otimes{k}}\right)\right]$ $\displaystyle
Z_{\emptyset}(\mathbf{m})$
$\displaystyle=\mathbb{E}_{U}\left[\text{Tr}\left(\left(K_{\mathbf{m}}\ket{\psi_{0}}\bra{\psi_{0}}K_{\mathbf{m}}^{\dagger}\right)^{\otimes
nk+1}\right)\right],$ (15)

with $T_{n,A}$ defined in (13), and $\ket{\psi_{0}}$ is the quantum state of
the system at $t=0$. As the notation suggests, $Z_{A,\phi}$ will correspond to
the partition function of an effective statistical model, where $Z_{A}$ and
$Z_{\emptyset}$ only differ with respect to the boundary condition at the top
boundary region A (see Appendix A Fig. 11). We will denote the total number of
replicas as $Q=nk+1$ in the subsequent discussion. The additional replica is
due to the Born probability factor, which ensures quantum trajectories are
weighted appropriately Jian  _et al._ (2020). Also note that since the
original non-linear quantity has been converted to a linear quantity defined
on $Q$ copies, we are free to do various averages in any order we want.

The rest of this section is organized as follows. We start by giving a very
brief overview of the statistical model for random monitored circuits without
any symmetries, following Refs. Jian  _et al._ (2020); Bao  _et al._ (2020),
before moving on to summarize the result for the U(1) symmetric system. We
include a detailed and technical derivation of the above model in Appendix A.
This technical section can be skipped without breaking any continuity.

###  IV.1 Statistical model for systems without symmetry

We briefly review the mapping for random monitored circuits without symmetries
to a statistical model Jian  _et al._ (2020); Bao  _et al._ (2020). We focus
on the details required for our subsequent discussion, in particular on the
large dimension limit $d\rightarrow\infty$. To calculate Eq. (15) we need to
average over $Q$ copies of the circuit over Haar gates and measurement
outcomes (but _not_ over measurement locations). Since the random Haar gates
are drawn independently, we can individually average $Q$ copies of each gate.
The combinatorial results of the averaging can be captured as a partition
function that can be computed as follows: each unitary gate in the circuit is
replaced by a vertex associated with a pair of permutation “spins”
$\sigma_{a},\bar{\sigma}_{a}$, each belonging to the permutation group
$\mathcal{S}_{Q}$. In the $d\rightarrow\infty$ limit, these spins become
locked together in a single $\mathcal{S}_{Q}$ degree of freedom, $\sigma_{a}$.
Vertices from adjacent gates, i.e. those which share an input/output qubit,
are connected by links. The weight associated with a vertex in the partition
function is given by $V_{a}=1/d^{2Q}$. The weight of the links connecting
vertices with elements $\sigma_{a,b}$ is given by

$\displaystyle
W_{ab}[\sigma^{-1}_{b}\sigma_{a}]=\begin{cases}d^{|\sigma^{-1}_{b}\sigma_{a}|}&\text{if
link (ab) is not measured}\\\ d&\text{if link (ab) is measured}\end{cases},$
(16)

where $|\sigma^{-1}_{b}\sigma_{a}|$ is equal to the number of cycles in the
cycle decomposition of
$\sigma^{-1}_{a}\sigma_{b}=C_{1}...C_{|\sigma^{-1}_{b}\sigma_{a}|}$. Note that
the above weights are symmetric under left and right multiplication by
elements of $\mathcal{S}_{Q}$ so that the full symmetry of the statistical
mechanics model includes $\mathcal{S}_{Q}\times\mathcal{S}_{Q}$.

We see that in this $d\to\infty$ limit, spins (permutations) connected by
unmeasured links are forced to be the same, whereas spins on measured links
are effectively decoupled, i.e. a measurements “break” the links connecting
spins, diluting the lattice. This naturally yields a picture of the
purification transition in terms of classical percolation of clusters of
aligned permutation “spins” Skinner  _et al._ (2019); Jian  _et al._ (2020);
Bao  _et al._ (2020), though of course this simple percolation picture is
special to $d\rightarrow\infty$: $1/d$ fluctuations are a relevant
perturbation to the percolation critical point such that finite $d$
transitions are described by a distinct universality class from percolation
Jian  _et al._ (2020); Zabalo  _et al._ (2021).

As we saw in Eq. (13), the calculation of $S_{n}$ requires taking the
difference between two partition functions of the model described above but
with different boundary condition (see Fig. 11); in the replica limit, this
difference in partition functions becomes equivalent to a difference in free
energies $F_{A,\emptyset}=-\log Z_{A,\emptyset}$ (since the partition
functions approach unity in the replica limit). The boundary condition for the
calculation of $Z_{A}$ forces a different boundary condition in region $A$,
and thus introduces a domain wall (DW) near the top boundary. In the limit
$d\to\infty$, the DW is forced to follow a minimal cut, defined as a path
cutting a minimum number of unmeasured links (assumed to be unique for
simplicity). This can be seen as follows: due to the boundary condition in
$Z_{\emptyset}$, all vertex elements in $Z_{\emptyset}$ are equal666Any
difference in the vertex elements will lead to the creation of DW which are
suppressed as ${\cal O}(1/d)$ and whose contribution goes to zero as
$d\rightarrow\infty$., and $Z_{\emptyset}=d^{(1-Q)N_{m}}$, where $N_{m}$ is
the number of measured sites. $Z_{A}$ would be same as $Z_{\emptyset}$ except
for the fact that due to the DW some links contribute different weights to
$Z_{A}$. More precisely, we have $Z_{A}=d^{(k+1-Q)\ell_{\rm
DW}}Z_{\emptyset}$, where $\ell_{\rm DW}=\ell_{\rm DW}(\mathbf{X})$ is equal
to the number of unmeasured links that the DW crosses. Since $k+1-Q=(1-n)k<0$
for $n>1$, the DW will follow the path that minimizes $\ell_{\rm DW}$ 777Note
that the DW permutation element $(1...n)^{\otimes k}$ has $k+1$ cycles and
each cycle can follow an independent path (to the leading order) Zhou and
Nahum (2019). However, this subtlety will not change the final result about
the minimal cut, but only leads to fluctuations contributing sub-leading
logarithmic corrections.. Using the expression of $Z_{A,\emptyset}$ in Eq.
(14) and taking the replica limit $k\to 0$, we find $S_{n}=\ell_{\rm
DW}(\mathbf{X})\ln d$, which is valid for each configuration of measurement
locations. Averaging over measurement locations, we have

$\left[S_{n}\right]=\left(\mathbb{E}_{\mathbf{X}}\ell_{\rm
DW}(\mathbf{X})\right)\ln d,$ (17)

where $\mathbf{X}$ denotes a configuration of measurement locations
(measurement outcomes and Haar gates have been averaged over to get the
statistical model). In the language of the statistical model, $\mathbf{X}$
denotes a percolation configuration. $\ell_{\rm DW}$ is the length of the
minimal cut from one end of the sub system $A$ to the other end.

In the $d\to\infty$ limit, equation (17) is valid for any measurement
probability, $p$ 888Note that this description of the $d\to\infty$ differs
from that of Ref. Jian  _et al._ (2020). There the measurement locations
$\sum_{\mathbf{X}}$ were averaged over directly in the partition function, in
an annealed way, while we chose here to keep the measurement locations as
quenched disorder. Our approach predicts a minimal cut picture consistent with
Ref. Skinner  _et al._ (2019). We leave a discussion of the validity of the
replica trick in this limit to future work. . For $p=0$, there are no measured
links and hence $\ell_{\rm DW}=|A|$, where $|A|$ is the length of subsystem
$A$. In fact, $\ell_{\rm DW}$ undergoes a percolation transition at
$p_{c}=1/2$, where $\ell_{\rm DW}$ is extensive in $|A|$ for $p<1/2$, and
becomes ${\cal O}(1)$ for $p>1/2$ Skinner  _et al._ (2019).

Figure 6: Statistical mechanics model. (a) The average of
$\mathcal{U}=U^{\otimes Q}\otimes U^{*\otimes Q}$ over Haar gates is non-zero
if and only if the conjugate (bra) replicas are permutations of the non-
conjugate (ket) replicas. Hence we can conveniently write each leg in the
circuit as a set of $Q$ copies of non-conjugate states combined with a
permutation group element (see Eq. (23)). In the large $d$ limit, the
permutation group elements for in-going and out-going legs become locked
together in a single permutation, and the corresponding permutation group
element $\sigma$ can be associated with a vertex (one per gate), while the
charge states live on links. The U(1) charges ${\alpha,\beta}$ are constrained
by charge conservation. (b)The charge dynamics in each replica are given by an
effective 6-vertex model with weights $v$, corresponding to a symmetric
exclusion process constrained by the measurements and entanglement cut. (c)
Example of charge configuration.

###  IV.2 Statistical Model with U(1) qubits – Summary

Here we provide a concise summary of the statistical model in the case of U(1)
circuits, deferring the technical details to Appendix A.

Introducing a U(1) qubit on top of each qudit modifies the above model by
introducing an additional degree of freedom (per replica) $\alpha_{ij}$
defined on links, which can take value $0$ or $1$ and correspond to the charge
of the U(1) qubits. The weight of each vertex is modified according to the
input and output U(1) charges as follows,

$\displaystyle(1,1)$ $\displaystyle\rightarrow(1,1)$ $\displaystyle(0,0)$
$\displaystyle\rightarrow(0,0)$ $\displaystyle(1,0)$
$\displaystyle\rightarrow\frac{1}{2}\left((1,0)+(0,1)\right)$
$\displaystyle(0,1)$
$\displaystyle\rightarrow\frac{1}{2}\left((1,0)+(0,1)\right),$ (18)

where the left-hand side denotes the two input charges and the right-hand side
denote output charges. The constants before the output spins are the
contribution to the vertex weights. For all other configurations of charges,
the weight is equal to 0 thereby enforcing charge conservation. These rules
can also be seen as a special case of a 6-vertex model where the states $0,1$
denote two species of links and the weight of the vertex depends on the
configuration of the links around the vertex; see Fig. 6. Alternatively, those
weights can be interpreted as describing hard-core random walkers (symmetric
exclusion process), where each state “$1$” corresponds to a walker (solid
link), with the number of walkers being conserved as a function of time
(vertical direction in the statistical mechanics model).

We cannot directly average over the measurement outcomes of the U(1) qubits
due to the non-local nature of the vertex weights. Hence, we only write a
statistical model for a given set of measurement locations $\mathbf{X}$ and
outcomes ${\cal M}(\mathbf{X})$ for the U(1) qubits; we collectively denote
this set by $\mathbf{m}$, as above. The charges in the statistical model at
broken links of the percolation sample are pinned by the measurement outcome
of the qubit on that link. In other words, for a given configuration
$\mathbf{m}$, all measured links (broken links in the percolation cluster)
carry a fixed value of the local charge $0$ or $1$ determined by the
measurement outcome of the qubit, which is fixed in $\mathbf{m}$.

The statistical model is then given by

$\displaystyle
Z(\mathbf{m})=\sum_{\\{\alpha\\}}\prod_{i\in\text{vertices}}V_{i}(\\{\alpha\\}),$
(19)

where the sum over $\\{\alpha\\}$ denotes the sum over the set of charges
$\alpha$ on all links, $V_{i}$ is the 6-vertex model weight corresponding to
the rules (18), $\mathbf{m}$ represents a percolation configuration combined
with a set of values of pinned charges on broken links, corresponding to the
measurement outcomes of the qubits on those links. This statistical model has
a straightforward physical interpretation: it counts histories of the charge
degrees of freedom compatible with a given set of measurement locations and
measurement outcomes.

To calculate $S_{n}(\mathbf{m})$, we first need to find the minimal cut in the
percolation configuration. Recall that $\ell_{\rm DW}$ is the number of
unbroken links (not measured) along the cut. There are $2^{\ell_{\rm DW}}$
different charge configurations along the cut; we denote this set of different
configurations by ${\\{\beta_{\rm DW}\\}}$. From the partition function (19),
one can straightforwardly compute the probability of finding configuration
$\beta_{\mathrm{DW}}$ along the minimal cut. We denote this
$p_{\beta_{\mathrm{DW}}}$. Taking the replica limit exactly (see Appendix A),
we find that the Rényi entropy is given by

$\displaystyle S_{n}(\mathbf{m})=\frac{-1}{n-1}\ln\left(\sum_{\\{\beta_{\rm
DW}\\}}p_{\beta_{\rm DW}}^{n}\right)+\ell_{\rm DW}\ln d.$ (20)

The entropy $S_{n}$ averaged over all trajectories is then given by,

$\displaystyle[S_{n}]=\sum_{\mathbf{m}}Z(\mathbf{m})S_{n}(\mathbf{m}),$ (21)

where $Z(\mathbf{m})$ in Eq. (19) can be interpreted as some effective Born
probability for observing the trajectory $\mathbf{m}$, where unitary gates
have been averaged over. In particular, note that
$\sum_{\mathbf{m}}Z(\mathbf{m})=\mathbb{E}_{\mathbf{X}}\sum_{\\{{\cal
M}(\mathbf{X})\\}}Z(\mathbf{m})=1$.

Note that the second term in Eq. (20) is the entropy of a pure qudit system.
We thus interpret the first term as coming from the qubit sector and treat it
as the qubits’ contribution to the entanglement entropy. This first term also
has an appealing physical interpretation as the classical Rényi entropy of
qubit configurations along the minimal cut. This is a special feature of the
$d\to\infty$ limit. From now on, we will use $S^{T}_{n}$ to denote total
entropy of the qubits and qudits in (20) $S_{n}^{d}$ for the contribution to
the entropy from the qudit sector alone, and $S_{n}=S^{T}_{n}-S^{d}_{n}$ which
is equal to the first term in (20).

While this expression can, in principle, be computed using Monte Carlo
sampling with _no sign problem_ , for the one dimensional systems considered
here, we find it more convenient to use a disordered transfer matrix to evolve
the initial state up to some time $t$. Specifically, we fix $\mathbf{m}$ by
randomly generating a percolation configuration, and use the vertex rules
described in (18) to evolve the system in time. At each broken link (measured
qubit) encountered in the evolution, we choose the outcome of the measurement
(and hence the fixed value of the charge degree of freedom on that link) with
probability equal to the Born probability. This is equivalent to a Monte Carlo
sampling for the probability distribution given by $Z(\mathbf{m})$ in Eq.
(19). Many samples are generated and for each sample we calculate the
probability distribution $\\{p_{\beta_{\rm DW}}\\}$. Any physical quantity is
then calculated as
$\left[O\right]=\frac{\sum\displaylimits_{i=1}^{N_{s}}O(\mathbf{m})}{N_{s}}$,
where $N_{s}$ is the number of samples generated.

We remark that, in addition to the direct simulation of the transfer matrix
techniques we employ in this work, it could also be interesting to investigate
further the scaling of the transition using tensor network techniques applied
to the transfer matrix of the constrained 6-vertex model Schollwöck (2011).

##  V Numerical results from the statistical mechanics model

In this section, we present numerical results for the U(1) statistical
mechanics of constrained symmetric exclusion process described in the previous
section, valid in the $d\to\infty$ limit. Unless otherwise stated, we focus on
the contribution of the qubit to entanglement, and ignore the qudit
contribution $\ell_{\rm DW}\ln d$ which is entirely given by classical
percolation physics. We first present late time ($t\sim L$) entanglement data,
and present evidence for the existence of the charge-sharpening transition
occurring for $p_{\\#}=0.315\pm 0.01<p_{c}=\frac{1}{2}$. We also analyze the
time dependence of the Rényi entropies, and show that they all grow linearly
in time for any $p>0$, in sharp contrast with the $p=0$ behavior.

###  V.1 Charge-sharpening transition

Figure 7: Entanglement transition in the statistical model qubit contribution.
(a) Plot of the qubit entanglement entropy $S_{1}$ vs $p$. The inset shows the
finite size collapse using the ansatz discussed in the main text, with
$\nu\sim 1.3$ and $p_{\\#}\approx 0.29$. (b) Plot of $S_{1}$ vs $L$ near the
critical point. We find that at the critical point $S_{1}$ grows
logarithmically with $L$. (c) Tripartite mutual information
$\mathcal{I}_{3,1}$ vs $p$, showing a crossing around $p_{\\#}\approx 0.29$.
The inset shows the finite size collapse using the same correlation length
exponent as in (a).

In the statistical model, the total entanglement entropy of the subsystem $A$,
$S^{T}_{n}$, depends on the minimal cut which undergoes a percolation
transition at $p_{c}=\frac{1}{2}$; for $p<p_{c}$, the length of the minimal
cut scales with $L_{A}$ while for $p>p_{c}$ the measurement locations
percolate and $\ell_{\rm DW}$ becomes ${\cal O}(1)$. Clearly the total
entanglement entropy follows the area law for $p>p_{c}$, and is extensive (and
dominated by the qudit contribution) for $p<p_{c}$. As discussed below (21),
$S^{T}_{n}$ is given by the sum of two contributions from the qudit and qubit
sectors, respectively. In what follows, we will focus on the qubit
contribution $S_{n}=S^{T}_{n}-S^{d}_{n}$, and argue that this quantity
undergoes an entanglement transition from volume law to area law for
$p=p_{\\#}$. We will show that this entanglement transition from the qubit
sector coincides with a charge-sharpening transition, which can also be
diagnosed in a scalable way using a local ancilla probe, as in Sec. III.

####  V.1.1 Entanglement transition in the qubit sector

In this section we look at the Rényi entropies $S_{n}$ at long times, $t>4L$
as a function of $p$. We consider the qubit initial state
$\ket{\psi_{0}}=\left(\frac{\ket{0}+\ket{1}}{\sqrt{2}}\right)^{\otimes L}$. To
study the behavior of $S_{n}$ for $p>0$, we numerically run the statistical
model (20) and calculate the half system $S_{n}$ by averaging $S_{n}$ over
various time steps in the interval of $4$ for $t>4L$. We present results for
the $S_{1}$ in Fig. 7.

In analogy with the non-symmetric measurement transition Skinner  _et al._
(2019); Li  _et al._ (2019), we use the following scaling ansatz for $S_{n}$

$\displaystyle\left[S_{n}\right]-S^{c}_{n}=f(L/\xi),$ (22)

where $\xi\sim(p-p_{\\#})^{-\nu_{\\#}}$, and
$S^{c}_{n}=\left[S_{n}(p_{\\#})\right]\sim\alpha_{n}\ln L$. Using both the
entanglement entropy scaling and tripartite mutual information as in Sec. III,
we find that the qubit contribution shows an entanglement transition from
volume-law to area law at a critical value $p_{\\#}$ less than $p_{c}=1/2$.
Finite size collapses are compatible with $p_{\\#}=0.3\pm 0.02$ and
$\nu_{\\#}=1.3\pm 0.2$. We emphasize that this entanglement transition occurs
inside the entangling phase of the total system (the qudit contribution obeys
a volume-law scaling in this regime), and occurs only as a subleading
contribution to the total entanglement entropy.

From the point of view of the statistical mechanics model, this transition is
especially surprising, as it indicates that the entropy (20) of the charge
degrees of freedom along the minimal cut does not scale with its length for
$p>p_{\\#}$. Instead, our numerical results indicate that measurements are
enough to constrain most charges along the cut, so the charges are almost
completely “frozen” by the measurements near the minimal cut.

Figure 8: Charge-sharpening transition in the statistical mechanics model. (a)
Plot of $\left[\delta\mathcal{Q}^{2}\right]/L$ vs $p$ plotted for $t/L=2$. We
find that with increasing $L$ the value of
$\left[\delta\mathcal{Q}^{2}\right]$ approaches zero for $p>p_{\\#}\approx
0.3$. This is consistent with the entanglement transition in the qubit sector
where we observed area-law scaling for $\langle S_{n}\rangle$ for $p>p_{\\#}$.
Inset: charge variance in half of the system, behaving in the same way. (b)
Charge variance $\left[\delta\mathcal{Q}^{2}\right]/L$ vs time $t$. For
$p\lesssim p_{\\#}$, $\left[\delta\mathcal{Q}^{2}\right]/L$ decays
exponentially with a decay rate decreasing with $L$ . For $p>p_{\\#}$, the
decay rate is same for all $L$ suggesting that $\langle\sigma^{2}\rangle/L$
goes to zero faster with increasing system size (faster in units of $t/L$).
(c) Histogram of the charge variance in the charge fuzzy phase. (d) Plot of
$N_{0}$ vs $p$ with finite size collapse in the inset. $N_{0}$ was calculated
at $t/L=2$. We find excellent collapse for $p_{\\#}=0.315$ and $\nu=1.3$. (e)
Time evolution of $N_{0}$. We clearly see a reversal in trend with system size
$L$ around $p_{\\#}\approx 0.31$. At the transition $p=p_{\\#}$ we find that
$N_{0}\sim h(t/L)$, with $h(x)$ some scaling function, consistent with a
dynamical exponent $z=1$. (f) Histogram of the charge variance in the charge
sharp phase.

####  V.1.2 Charge-sharpening transition

Following Sec. III we probe charge-sharpening by following the dynamics of the
single-trajectory charge variance $\delta{\cal Q}^{2}_{{\mathbf{m}}}$ starting
from an initial state that is a superposition over charge sectors. We first
discuss the average of this quantity over all trajectories. We compute this
quantity using the statistical model and plot $[\delta{\cal Q}^{2}]/L$ as a
function $p$ in Fig. 8.a. We see that for $p\gtrsim p_{\\#}\approx 0.3$, which
is the threshold of the area law phase in the qubit sector, $[\delta{\cal
Q}^{2}]/L$ goes to zero exponentially as a function of time in a way that is
independent of $L$. This implies that the time scale $t_{\\#}$ for charge
sharpening for $p>p_{\\#}$ (defined as the time it takes for the charge
variance to reach a given small value $\epsilon$) scales logarithmically with
system size. In contrast, for $p<p_{\\#}$, this charge sharpening time scales
as $t_{\\#}\sim L$ (see Appendix C.2). Fixing $t=2L$, $[\delta{\cal Q}^{2}]/L$
behaves as an order parameter for the charge sharpening transition, coinciding
with the entanglement transition in the qubit sector described in the previous
section. We observe the same behavior in the bipartite charge variance.

To extract $p_{\\#}$ it is useful to analyze a quantity that has a
discontinuity at the transition. To this end, we consider $N_{0}$, the
fraction of trajectories with $\delta{\cal Q}^{2}_{{\mathbf{m}}}<\epsilon$ for
a given threshold $\epsilon$, as in Sec. III. We check that the results does
not depend on $\epsilon$ for small enough values. We plot this quantity in
Fig. 8.b and find a crossing around $p_{\\#}=0.31$. Note that for all $L$, we
chose $\epsilon$ to be small enough so that $N_{0}$ counts only configurations
where the charge is essentially perfectly sharp to numerical accuracy. Defined
in this way, $N_{0}$ approaches 0 in the fuzzy phase, while it goes to 1 in
the sharp phase. If we increase the threshold $\epsilon$, instead, we find
that $N_{0}$ behaves more like an order parameter, being fixed to $N_{0}=1$ in
the sharp phase and continuously decreasing in the fuzzy phase.

Using a scaling ansatz $N_{0}=f\left((p-p_{\\#})L^{1/\nu_{\\#}}\right)$, we
find the best collapse for $\nu=1.3\pm 0.15$, consistent with the entanglement
data of the qubit. We also look at the evolution of $N_{0}$ with $t/L$ in Fig.
8. We find that $N_{0}$ goes to $1$ for all $p$ at long times but the rate of
increase of $N_{0}$ decreases with $L$ for $p<p_{\\#}$ and increase with $L$
for $p>p_{\\#}$, while remaining constant for $p=p_{\\#}$. We check that the
exponent $\nu$ and the critical probability $p_{\\#}$ do not vary much with
the time chosen for calculating $N_{0}$ as long as it is not too large. The
crossing value of $N_{0}$ tends to increase with increasing $t/L$: We focus
here on the regime where the thermodynamic limit is taken first so $t/L$ is
“small” (in practice, $t/L=2$ is small enough to obtain stable results). We
thus conclude that the volume- to area-law transition of $[S_{n}]$ in the
qubit sector can be interpreted as a charge sharpening transition wherein
starting from a mixed superposition of all charge sectors, the measurements
collapse the wave function to one charge sector for $p>p_{\\#}=0.315\pm 0.01$.

Figure 9: Ancilla probe in the statistical mechanics model.  Top: entanglement
entropy of the ancilla qubit, which behaves as an order parameter for the
charge-sharpening transition. Bottom: Finite size scaling of the number of
trajectories where the ancilla qubit is purified $N_{\text{pure}}$, probing
the charge-sharpening transition.

####  V.1.3 Local ancilla probe

As in Sec. III for the qubit-only ($d=1$) model, we now present a scalable
probe of the charge-sharpening transition by entangling a reference ancilla
qubit to different charge sectors
$\ket{\Psi_{0}}=\ket{\psi_{\mathcal{Q}}}\ket{0}+\ket{\psi_{\mathcal{Q}-1}}\ket{1}$.
Our numerical protocol is identical to that of Sec. III (Fig. 9). Those
results are obtained by taking the minimal cut to be always at the link
connecting the ancilla to the system: this is correct in the thermodynamic
limit below the percolation threshold $p<p_{c}=1/2$, and removes spurious
finite size effects due to percolation physics. Our results for this quantity
are qualitatively different from the $d=1$ model of Sec. III, which showed a
possible crossing in that quantity, while we observe here a behavior
consistent with that of an order parameter for the charge-sharpening
transition. This difference might be due to different “magnetization”
exponents $\beta$, with the $d=1$ model being closer to percolation. Analogous
to the $N_{0}$ quantity defined above, we introduce $N_{\text{pure}}$ which is
equal to number of trajectories where the reference qubit is purified. We
observe a crossing $N_{\text{pure}}$ near $p_{\\#}\approx 0.315$ and plot the
finite size collapse in Fig. 9.

To conclude this section we compare our results for the charge sharpening
transition in the two limits of $d=1$ and $d\rightarrow\infty$. In both cases
we have found Lorentz invariant critical points with $z=1$ to within numerical
accuracy. In the limit of $d\rightarrow\infty$ we have a correlation length
exponent $\nu_{\\#}=1.3\pm 0.15$, which is consistent with the percolation
universality class that is also found in the qudit sector at $p_{c}=1/2$.
Whereas in the limit of $d=1$ we have $\nu_{\\#}\approx 2.0$, which points to
a unique universality class that is distinct from both the limit of
$d\rightarrow\infty$ and the entanglement transition at $p_{c}$.

###  V.2 Entanglement dynamics

Figure 10: Entanglement dynamics in the statistical mechanics model. Ratio
$R_{s}=\frac{S_{1}}{S_{\infty}}$ vs $\sqrt{t}$ for $L=12$. We find that
$R_{s}$ saturates implying that $S_{\infty}$ and $S_{1}$ are growing at the
same rate $\sim t$. As expected, the saturation time is longer at low $p$.

Finally, we briefly turn to the dynamics of the Rényi entropies $S_{n}(t)$
using the statistical model. As before, $S_{n}$ is the contribution of the
U(1) qubits to the entanglement entropy: the total entropy $S_{n}^{T}$ always
grows linearly for $p<p_{c}=1/2$ due to the qudit sector. The results below
should be interpreted as sub-leading corrections to the growth of the total
entanglement entropy arising due to the slow dynamics of the U(1) qubits.

In the absence of measurements, the statistical mechanics model predicts that
all Rényi entropies scale diffusively $S_{n>1}\sim\sqrt{t}$, whereas
$S_{n=1}\sim t$, in agreement with earlier results Rakovszky  _et al._ (2019);
Huang (2020); Zhou and Ludwig (2020) (see Appendix C.1). As argued in Sec. II,
in the presence of measurements, the “dead regions” responsible for this
unusual behavior survive only until a time $t\sim p^{-2/3}$ in typical
trajectories. At long times, the overlap of the wavefunction with dead regions
is zero, and we expect the trajectory-averaged Rényi entropies $[S_{n}]$ to
grow linearly in time for all $p>0$.

To confirm this, we plot the ratio $R_{s}=[S_{1}]/[S_{\infty}]$ in Fig. 10.
The average von Neumann entropy $[S_{1}]$ is expected to grow linearly for all
$p$. The quantity $R_{s}$ is thus a measure of the growth of $[S_{\infty}]$:
if $[S_{\infty}]$ were to increase as $\sqrt{t}$, then we would expect $R_{s}$
to grow as $\sqrt{t}$ too. This is indeed what we observe at $p=0$. At higher
$p$ we find that $R_{s}$ saturates to a constant value implying that
$[S_{1}]\sim[S_{\infty}]\sim t$, in agreement with our general argument. Other
observables confirming this scaling are presented in Appendix C.1.

##  VI Discussion

In this work, we have studied measurement-induced phases and phase transitions
in monitored quantum circuits with charge conservation. We argued that
measurements can have a dramatic effect on entanglement growth. While all
Rényi entropies with index $n>1$ grow diffusively in the absence of
measurements, for any $p>0$, the effect of these rare regions are washed out
by measurements leading to ballistic scaling $S_{n}\sim t$ at long times.

Whereas, in the absence of symmetry, there can only be two possible steady-
states, entangling or purifying, charge conservation enriches this dynamical
phase diagram. We uncovered a new type of charge-sharpening transition that
separates distinct entangling phases. Even as the dynamics remain scrambling
and lead to a volume-law entangled state, the U(1) charge can either be
“fuzzy” or “sharp” depending on the rate of measurements. This charge-
sharpening transition occurs at a critical measurement rate $p_{\\#}$ that is
generically smaller than $p_{c}$, corresponding to the purification
transition. This new transition is also fundamentally different from the
purification entanglement transition, as for any $p>0$, the charge will
eventually become sharp with exponentially small corrections for $t\gg
t_{\\#}\sim L$ (up to logarithmic corrections) for a system of size $L$,
whereas the purification time diverges exponentially in the system size in the
entangling phases.

We presented evidence for the existence of this transition using both exact
numerical results in a symmetric qubit model ($d=1$), and from the numerical
analysis of an emergent statistical mechanics model describing the evolution
of charged qubits coupled to large qudits ($d\to\infty$). For the model in the
$d\rightarrow\infty$ limit, the correlation length exponent $\nu$ of the
charge-sharpening transition is consistent with that of percolation. In
contrast, in the qubit-only model we showed that the charge-sharpening
correlation length exponent is distinct from the that found for the
entanglement transition with $\nu_{\\#}\approx 2$. Understanding the critical
properties of this transition represents a clear challenge for future works.
Although our numerical results indicate that the charge sharpening transition
occurs at $p_{\\#}<p_{c}$, we cannot rule out more exotic scalings and slow
finite size drifts that would lead to $p_{\\#}=p_{c}$ in the thermodynamic
limit even in the statistical model with $d=\infty$ where the two transitions
appear to be well separated. Another conceivable scenario would be that the
charge-sharpening and entanglement transitions could merge into a single
transition below a critical qudit dimension, $d<d_{c}$ (where $d_{c}$ could
potentially be infinite). Establishing on firmer grounds the existence of a
distinct charge-sharpening transition would also be an important task for
future works.

The conservation law has not affected the universality class of the
entanglement transition in the limit of $d\rightarrow\infty$. Whereas, in the
limit of $d=1$ we have shown that the log-scaling of the Renyi entropy at
criticality $S_{n}\sim\alpha(n)\log L$, has an $\alpha(n)$ that is clearly
distinct from the transition with Haar random gates Zabalo  _et al._ (2020),
which implies the (boundary) universality class is distinct in the presence of
a conservation law. Interestingly, we have found that $\nu\approx 1.3$, which
is not sensitive enough to discern between percolation, stabilizer dynamics,
and the Haar universality class. It will be interesting in future work to
probe other critical exponents of the entanglement transition with a
conservation law to discern other uniques properties of this transition.

It would be interesting to extend our results to other symmetry groups or
kinetic constraints. Our results can be readily generalized to arbitrary
Abelian groups (see Appendix B). As we discuss in Appendix B, if one assumes
the existence of charge-sharp phases for other symmetry groups and spatial
dimensionalities, standard duality relations immediately imply the existence
of monitored random circuit classes that exhibit volume-law entangled phases
with symmetry-breaking, symmetry-protected topological, and intrinsic
topological orders in a typical trajectories. Such trajectory-ordered but
volume law entangled phases are clearly forbidden in any equilibrium or
closed-system dynamical setting, and are a new feature of non-unitary open
system dynamics.

Moreover, it is clear by now that new types of dynamical phases can be
obtained in the steady state of monitored quantum circuits, from the
combination of different competing (non-commuting) measurements and unitary
dynamics Nahum and Skinner (2020); Lavasani  _et al._ (2020); Ippoliti  _et
al._ (2020); Sang and Hsieh (2020). The full phase structure allowed by the
microscopic symmetry group and the dynamical symmetries of such monitored
quantum circuit appears to be particularly rich Bao  _et al._ (2021), and
remains largely unexplored. We expect non-Abelian symmetries to be especially
interesting, as they could lead to fundamental constraints on the entanglement
structure of the steady-state, as in the case of many-body localized systems
Potter and Vasseur (2016).

_Acknowledgments._ We thank Michael Gullans, David Huse, Chaoming Jian, Vedika
Khemani, and Andreas Ludwig for useful discussions and collaborations on
related projects. We acknowledge support from NSF DMR-1653271 (S.G.), NSF
DMR-1653007 (A.C.P.), the Air Force Office of Scientific Research under Grant
No. FA9550-21-1-0123 (R.V.), and the Alfred P. Sloan Foundation through Sloan
Research Fellowships (A.C.P., J.H.P., and R.V.). A. Z. and J.H.P. are
partially supported by Grant No. 2018058 from the United States-Israel
Binational Science Foundation (BSF), Jerusalem, Israel. A. Z. is partially
supported through a Fellowship from the Research Discovery Informatics
Institute. The Flatiron Institute is a division of the Simons Foundation.

##  Appendix A Mapping to the statistical model with U(1) qubits

In this appendix, we present a in detail discussion of the mapping to the
statistical model, and derive Eq. (20) in the limit $d\to\infty$. To evaluate
the quantities in Eq. (15), we need to calculate the average of
$\mathcal{K}\equiv K_{\mathbf{m}}^{\otimes Q}\otimes
K_{\mathbf{m}}^{\dagger\otimes Q}$, corresponding to $Q$ copies of the random
circuit. Each unitary gate in $\mathcal{K}$ is repeated $Q$ times and since
they are drawn independently we can individually average them over the random
unitary ensemble. Let us denote the tensor product of $Q$ copies of a gate by
$\mathcal{U}\equiv U^{\otimes Q}\otimes U^{\dagger\otimes Q}$. We view
$\mathcal{U}$ as a super-operator which acts on two sites with each leg
containing $Q$ ket states and $Q$ bra states; let
$\ket{g^{i}\alpha^{i}}\bra{g^{i*}\alpha^{i*}}$ be a basis where $g^{i}$ is a
basis of the qudit Hilbert space ${\mathbb{C}}^{d}$, and $\alpha^{i}$ is the
computational basis for the qubit. The index $i$ labels replicas, and runs
from $1$ to $Q$. Using standard Haar calculus and Weingarten formulas, we find
that the action of $\overline{\mathcal{U}}$ on the above basis is non trivial
after averaging if and only if $g^{i*}=g^{\sigma(i)}$ and
$\alpha^{i*}=\alpha^{\sigma(i)}$, where $\sigma\in\mathcal{S}_{Q}$ is a
permutation. Therefore, we introduce a shorthand notation for writing the
relevant members of the basis as

$\displaystyle(g^{i}\alpha^{i};\sigma)\equiv\ket{g^{i}\alpha^{i}}\bra{g^{\sigma(i)}\alpha^{\sigma(i)}}.$
(23)

More precisely, each unitary gate in the circuit is replaced by a vertex
associated with a pair (corresponding to in- and out-going legs) of
permutation “spins” $\sigma,\bar{\sigma}$, each belonging to the permutation
group $\mathcal{S}_{Q}$. In the $d\rightarrow\infty$ limit, these spins become
locked together in a single permutation degree of freedom, $\sigma_{a}$, that
we associate with that vertex. Vertices from adjacent gates, i.e. those which
share an input/output qubit and qudits, are connected by links in a way that
will be explained below. In the large $d$ limit, the weight associated with a
vertex in the partition function is given by $V_{a}=1/D^{Q}$, where $D$ is the
size of the block of the relevant symmetry sector. We have $D=d^{2}$ if all
incoming and outgoing charges are the same, and $D=2d^{2}$ otherwise, see eq.
(3).

The results for $\overline{\mathcal{U}}$ to leading orders are summarized in
Fig. 6; the sub-leading corrections are suppressed as ${\cal O}(1/d^{2})$
which we will ignore in rest of the paper. The factor of
$\frac{\delta_{\alpha_{1}^{i}\beta_{1}^{i}}\delta_{\alpha_{2}^{i}\beta_{2}^{i}}+\delta_{\alpha_{1}^{i}\beta_{2}^{i}}\delta_{\alpha_{2}^{i}\beta_{1}^{i}}}{2}$
in Fig. 6 enforces U(1) charge conservation, and follows from the size of the
different blocks in eq. (3). In fact, if we view charge $0$ as vacuum and
charge $1$ as a particle, then the dynamics of the U(1) degree of freedom can
be understood as a hard core random walk these particles, known as the
symmetric exclusion process. Alternatively, it can be seen as a special case
of the 6-vertex model (see Fig. 6.b). Though we have focused mainly on the
case of U(1) symmetry groups, our approach readily extends to other Abelian
groups. In Appendix B we provide a general derivation for arbitrary Abelian
symmetry groups.

###  A.1 Link weights

Combining Fig. 6 with the brick wall geometry of the circuit leads to a model
described on a square lattice as shown in Fig. 11. Each vertex has an element
from the permutation group $\mathcal{S}_{Q}$ and each link has $Q$ copies of
the elements of the basis of the local Hilbert space. The vertex weights
$V_{a}$ are given by the rule described in Fig. 6.b. The link weight $W_{ab}$
has two kind of contributions: 1) due to the presence of domain wall (DW) in
the permutation group elements $\sigma_{a,b}$ (DW constraint), 2) the state at
the link $\langle ab\rangle$ is being measured (measurement constraint). We
describe these constraints in detail in the following.

Figure 11: Replicated statistical mechanics model. (a) The replicated
statistical model is defined on a tilted square lattice, with permutation
degrees of freedom $\sigma_{a}\in\mathcal{S}_{Q}$ living on vertices, and
charge degrees of freedom $\\{\alpha^{i}\\}_{i=1,\dots,Q}$. The Boltzmann
weights have contributions from both vertices $V_{a}$, see Eq. (32), and links
$W_{ab}$, Eq. (31). (b) Fixed boundary conditions in $Z_{\emptyset}$ at the
top layer (ending the circuit at a given time $t$), with all permutations
fixed to $e$. (c) The partition function $Z_{A}$ differs from $Z_{\emptyset}$
by the boundary condition fixed to $\sigma_{0}$ in the entanglement interval
$A$. This creates a domain wall (DW) that follows a minimal cut in the limit
$d\to\infty$.

1) DW constraint. We first consider a link joining two vertices that we label
“1” and “2”. If we integrate out the qudit degrees of freedom, we find the
following weight for the links

$\displaystyle W_{12}(\sigma_{2}^{-1}\sigma_{1})$
$\displaystyle=\sum_{g_{1}^{i},g_{2}^{i}}\text{Tr}\left[(g_{1}^{i}\alpha_{1}^{i};\sigma_{1})(g_{2}^{i}\alpha_{2}^{i};\sigma_{2})^{\dagger}\right]$
$\displaystyle=d^{p}\delta_{C_{1}}...\delta_{C_{p}},$ (24)

where $C_{1}...C_{p}$ is the cycle structure of the permutation element
$\sigma_{2}^{-1}\sigma_{1}$, $p$ is the number of cycles in that permutation,
and $\delta_{C_{i}}$ is equal to $1$ if all charge states of the replicas
within cycle $C_{i}$ are the same, and otherwise equal to $0$. Note that we
cannot sum over the charges $\alpha^{i}$ as these depends on the states at
neighboring links (see Fig. 6).

Intuitively we can interpret the above result as follows. Since the link is
shared between vertices with different permutations $\sigma_{1}$ and
$\sigma_{2}$ then we must have the following constraint

$\displaystyle(g^{i}\alpha^{i};\sigma_{1})=(g^{i}\alpha^{i};\sigma_{2}),$ (25)

which is true if and only if
$\sigma_{2}^{-1}\sigma_{1}(\\{\alpha_{i}\\})=\\{\alpha_{i}\\}$, and
$\sigma_{2}^{-1}\sigma_{1}(\\{g_{i}\\})=\\{g_{i}\\}$. Let us take the simple
case where $\sigma_{2}^{-1}\sigma_{1}$ is equal to a transposition, say,
$(12)$. We can interpret the above equation as saying that $g_{1}=g_{2}$ and
$\alpha_{1}=\alpha_{2}$. The $g_{1}=g_{2}$ condition will reduce the number of
allowed basis qudit states from $d^{Q}$ to $d^{Q-1}$, and since all qudit
contributes equally, the weight of the link in this case will be reduced by a
factor $1/d$ due to the reduced configurational entropy in the qudit sector.
For the qubit degrees of freedom, we cannot sum over all spins due to the non-
local charge conservation constraint. The more general case of
$\sigma_{2}^{-1}\sigma_{1}\in\mathcal{S}_{Q}$ follows similarly, giving (24).
An important thing to note is that each transposition in
$\sigma_{2}^{-1}\sigma_{1}$ reduces the weight by $1/d$ and the weight is
strongest when $\sigma_{2}=\sigma_{1}$, that is when there is no DW –
corresponding to a ferromagnetic interaction. Thus at large $d$ limit, it is
expensive to have a DW and the system will remain in an ordered phase unless
DW are forced, for example, at the entanglement cut (see Fig. 11). This will
play an important role in the subsequent discussion.

2) Measurement constraint. If the link $\langle 12\rangle$ happens to be
measured, then the measurement outcomes are same in all replicas, that is, all
copies are acted on with the same projection operator. If the projection
operator is denoted as $P=P^{q}\otimes P^{d}$, then we have the weight

$\displaystyle
W_{12}(\sigma_{2}^{-1}\sigma_{1};P)=\text{Tr}\left[(g^{i}_{2}\alpha^{i}_{2};\sigma_{2})^{\dagger}P^{\otimes{Q}}(g^{i}_{1}\alpha^{i}_{1};\sigma_{1})P^{\otimes
Q}\right].$ (26)

Averaging over all measurement outcomes results in

$\displaystyle W_{12}$
$\displaystyle=\sum_{s=0,1}\sum_{x=1,...,d}W_{12}(\sigma_{2}^{-1}\sigma_{1};P^{q}_{s}\otimes
P^{d}_{x}),$ $\displaystyle=d\sum_{s=0,1}\prod_{i}\delta_{\alpha^{i},s},$ (27)

where $\delta_{\alpha^{i},s}$ ensures that the charge state of the $i^{\rm
th}$ replica is compatible with the measurement outcome $s=0,1$ of the qubit.
Whenever a measurement occurs, all $Q$ charges on the corresponding link are
constrained to be same. This gives the $\delta$ factor in (27). For the qudit
sector, this leads to a decrease in configurational entropy from $d^{Q}$ to
$d$. An important observation is that the link weights $W_{ab}$ do not depend
on the permutation $\sigma_{b}^{-1}\sigma_{a}$, a result of crucial importance
for the discussion below.

###  A.2 Replicated model

Combining all these results we can write a statistical model with the
partition function given by,

$\displaystyle Z$ $\displaystyle=\sum_{\mathbf{m}}Z[\mathbf{m}],$ (28)
$\displaystyle Z[\mathbf{m}]$
$\displaystyle=\sum_{\text{configurations}}\left(\prod_{(ab)\in\text{links}}\prod_{v\in\text{vertices}}W_{ab}(\sigma_{b}^{-1}\sigma_{a})V_{v}\right),$
(29)

where
$\sum_{\mathbf{m}}\equiv\sum_{\\{\mathbf{X}\\}}p^{N_{\mathbf{X}}}(1-p)^{LT-
N_{\mathbf{X}}}\sum_{\\{{\cal M}(\mathbf{X})\\}}$, where $\mathbf{X}$ denotes
a configuration of measurement locations, $N_{\mathbf{X}}$ is the number of
links being measured (number of bonds in the percolation configuration
$\mathbf{X}$), $L$ is the spatial length of the system, $T$ is the number of
time steps; and ${\cal M}(\mathbf{X})$ is the set of qubit measurement
outcomes at measurement locations $\mathbf{X}$. The sum over configurations is
given by

$\displaystyle\sum_{\text{configurations}}$
$\displaystyle\equiv\sum_{\\{\sigma_{v}\\}\in\mathcal{S}_{Q}}\sum_{\\{\alpha^{1}\\}=0,1}...\sum_{\\{\alpha^{Q}\\}=0,1}$
$\displaystyle\equiv\sum_{\\{\sigma_{v}\\}\in\mathcal{S}_{Q}}\sum_{\\{\alpha\\}},$
(30)

corresponding to permutation and charge degrees of freedom in each replica.
The link weights $W_{ab}$ are given by

$\displaystyle
W_{ab}(\sigma_{b}^{-1}\sigma_{a})=\begin{cases}d^{|\sigma^{-1}_{b}\sigma_{a}|-Q}\delta_{C_{1}}...\delta_{C_{|\sigma_{b}^{-1}\sigma_{a}|}}&\text{if
$(ab)$ not in $\mathbf{X}$}\\\ d^{1-Q}\delta_{\alpha^{i},s_{ab}}&\text{if
$(ab)$ in $\mathbf{X}$}\end{cases},$ (31)

with $s_{(ab)}\in{\cal M}(\mathbf{X})$ the measurement outcome of the qubit on
link $(ab)$. Finally, the vertex weight $V_{v}$ is given by

$\displaystyle
V_{v}=\prod_{i=1}^{Q}\frac{\delta_{\alpha_{1}^{i}\beta_{1}^{i}}\delta_{\alpha_{2}^{i}\beta_{2}^{i}}+\delta_{\alpha_{1}^{i}\beta_{2}^{i}}\delta_{\alpha_{2}^{i}\beta_{1}^{i}}}{2}=\prod_{i=1}^{Q}V_{v}^{i},$
(32)

where $\alpha_{1,2}$ and $\beta_{1,2}$ are incoming and outgoing charges (see
Fig. 6). We note that $V_{v}$ factorizes over the replicas, that is,
$V_{v}=\prod_{i=1}^{Q}V_{v}^{i}$; this will play an important role in
factorizing $Z[\mathbf{m}]$ in the discussion below.

Note that we have integrated out the qudit sector from the model. This was
possible due to each qudit on a given link being independent of the values at
other links. However, this is not possible for the U(1) sector on account of
non-local constraints due to the charge conservation. Importantly, the
statistical model $Z(\mathbf{m})$ should be thought of as a quenched
disordered model where the measurement locations and outcomes (for the qubit)
are quenched “impurities”; averaged quantities in the original problem have
become quenched average in the statistical model. From now on,
$\mathbf{m}=\\{\mathbf{X},{\cal M}(\mathbf{X})\\}$ will denote the system’s
quantum trajectory with measurement locations + U(1) measurement outcomes
fixed, corresponding to a fixed “disorder” realization of the statistical
model. This is unlike the previous-works on the non symmetric problem where
the randomness in the measurement locations were absorbed in the statistical
model in an annealed way.

$\alpha^{i}_{l}$ | Charge at link $l$ and copy $i$
---|---
$\\{\alpha^{i}\\}$ | Set of charges on all links for copy $i$
$\\{\alpha_{l}\\}$ | Set of charges on link $l$ for all copies
$\\{\alpha\\}$ | Set of charges on all links and copies
$\delta^{\\{l\\}}_{\\{\alpha\\}}$ | All copies of $\alpha$ on set of links $\\{l\\}$ are equal
$\delta^{\\{l\\}}_{\\{\alpha\\},\\{{\cal M}\\}}$ | All copies of $\\{\alpha\\}$ on links $\\{l\\}$ are equal to $\\{{\cal M}\\}$
Table 1: Table summarizing the meaning of various notations used in this
appendix.

###  A.3 Replica limit

We now proceed to take the replica limit, and will use various notations
summarized in Table 1.

We first focus on the partition function $Z_{\emptyset}(\mathbf{m})$, where
the links at the top boundary are restricted to be of the form
$(g^{i}\alpha^{i};e)$. The permutation identity element $e$ represents the
fact that we are tracing over all the system and is equal to the Born
probability of observing the particular trajectory $\mathbf{m}$. As mentioned
above, a DW in the statistical model is suppressed by $1/d^{Q-p}$, where $p$
is the number of cycles in the DW. Thus, the leading order contribution to
$Z_{\emptyset}$ comes when all vertex elements being equal to $e$. This
simplifies $Z_{\emptyset}(\mathbf{m})$ dramatically as we do not need to sum
over the permutation elements. We have

$\displaystyle Z_{\emptyset}$
$\displaystyle(\mathbf{m})=d^{(1-Q)N_{m}}\left(\sum_{\\{\alpha\\}=0,1}\delta^{\mathbf{m}}_{\\{\alpha\\},{\cal
M}(\mathbf{m})}\prod_{v}V_{v}\right),$

where $N_{m}$ is the number of measured links,
$\delta^{\mathbf{m}}_{\\{\alpha\\},{\cal M}(\mathbf{m})}$ is non-zero and
equal to $1$ if and only if the charges $\\{\alpha\\}$ on the measured sites
are equal to the measurement outcomes ${\cal M}(\mathbf{m})$ of the qubit, and
$V_{v}$ is given in (32). Intuitively, we should only sum over charge
configurations compatible with the measurement outcomes. The partition
function can be factorized over replicas to give,

$\displaystyle
Z_{\emptyset}(\mathbf{m})=d^{(1-Q)N_{m}}\left(Z_{\emptyset}^{(1)}(\mathbf{m})\right)^{Q},$
(33)

where $Z_{\emptyset}^{(1)}(\mathbf{m})$ is given by

$\displaystyle
Z_{\emptyset}^{(1)}(\mathbf{m})=\sum_{\\{\alpha\\}}\delta^{\mathbf{m}}_{\\{\alpha\\},{\cal
M}(\mathbf{m})}\prod_{v}V_{v}^{(1)}.$ (34)

The superscript $(1)$ denotes the fact that the quantity is for a single
replica.

We can similarly factorize $Z_{A}(\mathbf{m})$ with the caveat that we now
have a minimal cut for the permutation degrees of freedom running through the
system (see discussion in Section IV.1). A DW between $e$ and
$(1...n)^{\otimes k}$ reduces the link weight by $d^{k+1-Q}=d^{-(n-1)k}$ (31)
and the contribution of the cut to the partition function is thus given by
$d^{-(n-1)k\ell_{\rm DW}}$, where $\ell_{\rm DW}$ is the length of the minimal
cut. There are $k+1$ cycles in the DW; $k$ cycles of the type $(1...n)$ and
the last one being an identity on a single copy. Thus we can factorize
$Z_{A}(\mathbf{m})$ as

$\displaystyle Z_{A}(\mathbf{m})$ $\displaystyle=d^{(1-Q)N_{m}-(n-1)k\ell_{\rm
DW}}$
$\displaystyle\times\sum_{\\{\alpha^{i}\\}=0,1}\delta^{\mathbf{m}}_{\\{\alpha\\},{\cal
M}(\mathbf{m})}\prod_{C_{a}}\delta^{\rm
DW}_{\\{\alpha^{C_{a}}\\}}\prod_{v}V_{v},$ (35)

the where $\delta^{\rm DW}_{\\{\alpha^{C_{a}}\\}}$ is non-zero (and equal to
$1$) if and only if the charges within the cycle $C_{a}$ are the same on
(unmeasured) links on the minimal cut. We can further factorize the above
equation to get,

$\displaystyle Z_{A}(\mathbf{m})=d^{(1-Q)N_{m}-(n-1)k\ell_{\rm
DW}}\left(Z_{A}^{(n)}(\mathbf{m})\right)^{k}Z_{\emptyset}^{(1)}(\mathbf{m}),$
(36)

where

$\displaystyle Z_{A}^{(n)}(\mathbf{m})$
$\displaystyle=\sum_{\\{\alpha\\}}\delta^{\rm
DW}_{\\{\alpha\\}}\prod_{i=1}^{n}\left(\delta^{\mathbf{m}}_{\\{\alpha^{i}\\},{\cal
M}(\mathbf{m})}\prod_{v}V_{v}^{i}\right)$
$\displaystyle=\sum_{\beta_{1}...\beta_{\ell_{\rm
DW}}}\prod_{i=1}^{n}\left(\sum_{\\{\alpha^{i}\\}}\delta^{\rm
DW}_{\\{\alpha^{i}\\},\\{\beta\\}}\delta^{\mathbf{m}}_{\\{\alpha^{i}\\},{\cal
M}(\mathbf{m})}\prod_{v}V_{v}^{i}\right)$
$\displaystyle\equiv\sum_{\\{\beta\\}}Z_{A}\left[\mathbf{m};\\{\beta\\}\right]^{n}$
(37)

with $\delta^{\rm DW}_{\\{\alpha\\},\\{\beta\\}}$ non-zero (and equal to $1$)
if and only if all copies of charges on the unbroken (not measured) links
along the minimal cut are equal to $\\{\beta\\}$. The superscript $(n)$
denotes the fact that we have $n$ charge copies. Using the above results and
(14), we find

$\displaystyle S_{n}(\mathbf{m})$
$\displaystyle=\frac{-1}{n-1}\lim_{k\rightarrow
0}d^{(1-Q)N_{m}}Z_{\emptyset}^{(1)}(\mathbf{m})\frac{d^{-(n-1)k\ell_{\rm
DW}}\left(Z_{A}^{(n)}\right)^{k}-\left(Z_{\emptyset}^{(1)}\right)^{nk}}{k}.$
(38)

Remarkably, this factorized form allows us to take the replica limit exactly:

$\displaystyle S_{n}(\mathbf{m})$
$\displaystyle=\frac{-1}{n-1}Z_{\emptyset}^{(1)}(\mathbf{m})\ln\frac{Z_{A}^{(n)}}{\left(Z_{\emptyset}^{(1)}\right)^{n}}+\ln
d\ \ell_{\rm DW},$
$\displaystyle=\frac{-1}{n-1}Z_{\emptyset}^{(1)}(\mathbf{m})\ln\left(\sum_{\\{\beta\\}}\frac{Z_{A}[\mathbf{m};\\{\beta\\}]^{n}}{\left(Z_{\emptyset}^{(1)}\right)^{n}}\right)+\ln
d\ \ell_{\rm DW},$ (39)

where $\\{\beta\\}$ represents all possible configuration of the charge on the
unmeasured links along the minimal cut. We can further think of
$\frac{Z_{A}[\mathbf{m};\\{\beta\\}]}{\left(Z_{\emptyset}^{(1)}\right)}$ as
the probability for the charges along the unbroken links of the minimal cut to
be equal to $\\{\beta\\}$ in the statistical model described by the partition
function $Z_{\emptyset}^{(1)}$. Denoting this probability by $p_{\\{\beta\\}}$
we have our final result

$\displaystyle
S_{n}(\mathbf{m})=\frac{-1}{n-1}Z_{\emptyset}^{(1)}(\mathbf{m})\ln\left(\sum_{\\{\beta\\}}p_{\\{\beta\\}}^{n}\right)+\ln
d\ \ell_{\rm DW}.$ (40)

###  A.4 $p=0$ limit

To illustrate the meaning of the statistical model (40), we compute $S_{n}$
for $p=0$. Let us start from the following product state,

$\displaystyle\ket{\psi_{0}}=\left(a_{0}\ket{0}+a_{1}\ket{1}\right)^{\otimes
L}.$ (41)

In terms of the statistical model, this corresponds to the bottom links being
in charge states $1$ or $0$ with probability $a_{1}^{2}$ and $a_{0}^{2}$
respectively. The minimal cut will be spatial in nature as we are considering
late times and $\ell_{\rm DW}=L_{A}$ since the permutations are fully ordered.
Since the vertex weights (32) are SU$(2)$ symmetric, the link charge states
are invariant under time evolution. This immediately gives
$p_{\\{\beta\\}}=a_{0}^{2N_{0}}a_{1}^{2N_{1}}$, where $N_{0,1}$ are the number
of links with charge $0,1$ in $\\{\beta\\}$. Using Eq. (40), we find the
following expression for the Rényi entropies at late times:

$\displaystyle S_{n}$
$\displaystyle=\frac{-1}{n-1}\ln\left(|a_{0}|^{2n}+|a_{1}|^{2n}\right)^{L_{A}}+L_{A}\ln
d.$ (42)

This result is consistent with thermalization to a density matrix
$\rho_{A}=e^{-\mu\mathcal{Q}}/\text{Tr}e^{-\mu\mathcal{Q}}$, where the
chemical potential $\mu$ is set by charge conservation

$\displaystyle\langle\mathfrak{q}\rangle=a_{1}^{2}={\rm
Tr}\left[\mathfrak{q}\rho_{A}\right]=\frac{e^{-\mu}}{1+e^{-\mu}}.$ (43)

We check that the Rényi entropies are indeed given by
$S_{n}=\frac{-1}{n-1}\ln\text{Tr}\rho_{A}^{n}$, since

$\displaystyle S_{n}$ $\displaystyle=L_{A}\left(\ln
d+\frac{1}{n-1}\ln\left(\langle\mathfrak{q}\rangle^{n}+(1-\langle\mathfrak{q}\rangle)^{n}\right)\right),$
(44)

which coincides with (42).

##  Appendix B Statistical mechanics model for general Abelian symmetries

In this section, we generalize the statistical mechanics models to general
Abelian groups, and discuss consequences of charge-sharp phases under duality
transformations. Notably, our results suggest the existence of volume-law
entangled phases with symmetry protected and intrinsic topological order.

###  B.1 Haar average

Consider a general Abelian group $G$, with $\alpha\in\\{1\dots|R|\\}$ labeling
the different combinations of total charge for pairs of sites (e.g. for the
U(1) model $\alpha\in\\{-1,0,+1\\}$, for $\mathbb{Z}_{N}$ $\alpha\in\\{0\dots
N-1\\}$ etc…). We can decompose a symmetric two-site unitary into a direct sum
of reps: $U=\sum_{\alpha}U_{\alpha}P_{\alpha}$, where $P_{\alpha}$ is a
projector onto the $\alpha^{\text{th}}$-charge-sector subspace. The main
object in the statistical mechanics model is the unitary average of
$\mathbb{E}_{U}\left[U^{Q}\otimes U^{*Q}\right]$ where $Q$ is the number of
replicas, which decomposes into a direct sum of all $R^{2Q}$ charge-sector
combinations. Since Haar-averaging requires that each $U_{\alpha}$ is “paired”
with a complex-conjugated partner $U^{*}_{\alpha}$ of the same total charge,
only terms in which the $U^{*Q}$ charge-sectors form a permutation of the
$U^{Q}$ charge-sectors contribute. For each of these surviving combinations of
charge-sectors, denote by $n_{\alpha}$ the number of times that charge-sector
$\alpha$ appears in $U^{Q}$, and choose permutation elements $\sigma,\tau\in
S_{Q}$ that sort the $Q$ replicas into groups of the same charge. Then we can
write:

$\displaystyle\mathbb{E}_{U}\left[U^{Q}\otimes U^{*Q}\right]=\sum_{n_{1\dots
R}:\sum_{\alpha}n_{\alpha}=Q}\sum_{\sigma,\tau\in\frac{S_{Q}}{S_{n_{1}}\times\dots
S_{n_{R}}}}W_{\sigma,\tau}\bigotimes_{\alpha}\mathbb{E}_{U}\left(U_{\alpha}^{n_{\alpha}}P_{\alpha}^{n_{\alpha}}\otimes
U^{*n_{\alpha}}_{\alpha}P^{*n_{\alpha}}_{\alpha}\right)W_{\sigma,\tau}^{\dagger}$
(45)

where $W_{\sigma,\tau}$ is the unitary acting on
$\mathcal{H}^{q}\otimes\mathcal{H}^{*q}$ that permutes the $Q$ copies of $U$
by $\sigma$ and the $Q$ copies of $U^{*}$ by $\tau$, and the permutation
elements $\sigma,\tau$ range over the quotient group
$\frac{S_{Q}}{S_{n_{1}}\times\dots S_{n_{R}}}$ to avoid over counting
equivalent permutations that cycle identical replicas with the same charge-
sector. Here we have labeled projectors acting on $\mathcal{H}^{*}$ with a ∗
simply for readability, and this mark carries no mathematical content.

The Haar average of each charge-sector-group is:
$\mathbb{E}_{U}\left(U_{\alpha}^{n_{\alpha}}\otimes
U^{*n_{\alpha}}_{\alpha}\right)=\sum_{\sigma_{\alpha},\tau_{\alpha}\in
S_{n_{\alpha}}}\text{Wg}_{Dd^{2}}(\sigma_{\alpha}^{-1}\tau_{\alpha};n_{\alpha})|\sigma_{\alpha}\rangle\rangle\langle\langle\tau_{\alpha}|$,
where Wg is the Weingarten function, $D_{\alpha}$ is the number of states in
the charge-sector $\alpha$, and $|\sigma\rangle\rangle$ denotes the operator
which permutes the input legs of $U$ by $\sigma$, and contracts them with the
corresponding legs of $U^{*}$ (and similarly for $\langle\langle\sigma|$
acting on the output legs). For our purposes, we will only need the large-d
limit:

$\displaystyle\lim_{d\rightarrow\infty}\text{Wg}_{d}(\sigma^{-1}\tau;Q)\sim\frac{1}{d^{Q}}\delta_{\sigma,\tau}.$
(46)

The sum over the total-charge sector permutations $\sigma_{\alpha}$, can now
be combined with the quotient-group permutations $\sigma,\tau$ to yield a
simpler sum over $S_{Q}$ permutations:

$\displaystyle\mathbb{E}_{U}\left[U^{Q}\otimes
U^{*Q}\right]=\frac{1}{d^{2Q}}\sum_{\alpha_{1}\dots\alpha_{Q}}\prod_{i}D_{\alpha_{i}}^{-1}\sum_{\sigma\in
S_{Q}}P_{\alpha_{1}}\otimes\dots P_{\alpha_{Q}}\otimes
P^{*}_{\alpha_{\sigma(1)}}\otimes\dots
P^{*}_{\alpha_{\sigma(Q)}}|\sigma\rangle\rangle\langle\langle\sigma|P_{\alpha_{1}}\otimes\dots
P_{\alpha_{Q}}\otimes P^{*}_{\alpha_{\sigma(1)}}\otimes\dots
P^{*}_{\alpha_{\sigma(Q)}}.$ (47)

Note that, the two sets of projectors are partly redundant since
$P^{2}_{\alpha}=P_{\alpha}$, and since
$P_{\alpha}\otimes\mathbbm{1}|\sigma\rangle\rangle=\mathbbm{1}\otimes
P_{\alpha_{\sigma(i)}}|\sigma\rangle\rangle$, but are written in this way to
emphasize that charge is separately conserved in each replica, and also that
the charge-sector labels on the $\mathcal{H}^{*}$ spaces are related to those
in the $\mathcal{H}$ spaces by the permutation element $\sigma$.

Though complicated in appearance, Eq. 47 has a simple interpretation: each
gate becomes a vertex in the statistical mechanics model labeled by i) a
permutation element $\sigma\in S_{Q}$, and ii) $Q$-different total charge-
sector labels (those of the conjugate copies are related by permutation),
which can be conveniently associated with the $Q$ different individual link
charges. After averaging, the indices of the gate input and output are
unrelated, except by total charge conservation in each replica. Hence any
input charge configuration can be transferred with equal weight $\sim
1/D_{\alpha}$ to any outgoing charge configuration with the same total charge
$\alpha$. The remaining rules for domain wall and measurement constraints
closely parallel those of the U(1) model described above.

##### Example: $\mathbb{Z}_{2}$ Symmetric Monitored Circuits

As an example, consider a random monitored circuit ensemble with symmetry
group $G=\mathbb{Z}_{2}$, consisting of qubits with $\mathbb{Z}_{2}$ symmetry
charge $\mathfrak{q}_{i}=\frac{1+\sigma^{x}_{i}}{2}\in\\{0,1\\}$, and charge
addition rule $\mathcal{Q}=\sum_{i}\mathfrak{q}_{i}~{}\text{mod}~{}2$, (each
accompanied by large dimension qudits that transform trivially under the
symmetry). The effective statistical mechanics model in the
$d\rightarrow\infty$ limit is an “8-vertex” model, which has an additional two
vertices compared to the U(1)/6-vertex case that respectively correspond to
pair creation and annihilation of charges, and differs also in that all 8
vertices come with weight $v=1/2$.

##### Example: $\mathbb{Z}_{N}$ Symmetric Monitored Circuits

As a second example, we can consider models with symmetry group
$G=\mathbb{Z}_{N}$ for general $N$ consisting of “quNits” with charge basis
states $\\{|0\rangle,\dots|N-1\rangle\\}$ having symmetry charge
$\mathfrak{q}=0\dots N-1$ (again, each accompanied by large-d qudits). The
resulting statistical mechanics model would be an $N^{3}$-vertex model, with
$N$ different groups of vertices corresponding to the $N$-different total
charges, and each group has $N$ different ways to apportion the incoming
charge between the two input legs, and $N$ different ways to apportion it
between the output legs, each weighted by a factor of $v=1/N$.

From these examples, one can readily generalize to arbitrary finite Abelian
groups for which $G$ can be written as a product of different $\mathbb{Z}_{N}$
factors.

###  B.2 Dual entangling phases classical and quantum orders

In principle, the statistical mechanics models sketched above can be simulated
by sign-problem-free Monte Carlo for arbitrary Abelian symmetry groups and
dimensions. Though we have only performed systematic numerics on the U$(1)$
symmetric models in one-dimension, we hypothesize that the resulting physics
and phase diagram is similar for other Abelian symmetry groups and higher
dimensions. Assuming this hypothesis holds, then standard duality
transformations would relate these simple symmetric monitored random circuit
models, to those with more complex types of phases including symmetry-
protected topological (SPT) phases, and discrete gauge theories.

In the following, we denote a system with $n$ spatial dimensions and one time
dimension as $(n+1)d$, where $d$ should not be confused with the dimension of
the qudits. In all the examples we discuss here, there will only be a single
type of measurement. Generalizing the statistical mechanics framework to
multiple types of incompatible measurements remains an open challenge for
future work, and to date has only been done for Clifford circuit models Nahum
and Skinner (2020); Lavasani  _et al._ (2020); Ippoliti  _et al._ (2020); Sang
and Hsieh (2020). The $1+1d$ examples with $\mathbb{Z}_{2}$ symmetries were
extensively explored in Ref. Bao  _et al._ (2021) from a related but different
perspective, and many aspects of the physics we discuss below echos that work
(though we do not consider possible phases arising from additional replica
permutation symmetries as it is not clear whether these can be realized in the
physical replica limit).

####  B.2.1 $1+1d$: Symmetry Protected Topology (SPT)

In $1+1d$, discrete symmetry groups of the form
$\mathbb{Z}_{m}\times\mathbb{Z}_{n}$ have non-trivial projective
representations and can protect topological phases Chen  _et al._ (2013, 2012)
when $m$ and $n$ have a nontrivial greatest common divisor. These SPT phases
can be mapped (at the level of local operators) onto trivial paramagnets by a
“decorated domain wall” mapping Chen  _et al._ (2014), which attaches
conjugate operators that add (remove) a $\mathbb{Z}_{n}$ charge to the left
(right) of the $\mathbb{Z}_{m}$ symmetry generator. For example, with
$G=\mathbb{Z}^{A}_{2}\times\mathbb{Z}^{B}_{2}$ represented on a dimerized
chain of spins-1/2s with the two $\mathbb{Z}_{2}$ factors acting on the A and
B sublattices respectively, the duality mapping is generated by unitary
transformation
$U_{\text{dual}}=e^{-i\pi/4\sum_{i}\sigma^{z}_{i}\left(\sigma^{z}_{i-1}+\sigma^{z}_{i+1}\right)}$
(which commutes with the symmetry, but is not generated by a symmetric
Hamiltonian). $U_{\text{dual}}$ interchanges the stabilizers of a trivial
paramagnet with those of the cluster state SPT:
$\sigma^{x}_{i}\rightarrow\sigma^{z}_{i-1}\sigma^{x}_{i}\sigma^{z}_{i+1}$.

How would the charge- sharp and fuzzy phases of the $\mathbb{Z}_{2}$-symmetry
circuit model transform under this duality? To answer this question, note that
the total charge in a region of length $\ell$ maps onto the string-order
parameter of length $\ell$ for the SPT phase. For example, in the
$G=\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}$ example, the
$\mathbb{Z}_{2}^{A}$ interval-charge maps onto
$\sigma^{z}_{2i-1}\left(\prod_{i=1}^{\ell}\sigma^{x}_{2i}\right)\sigma^{z}_{2(i+\ell)+1}$.
In the charge sharp phase, the interval charge at fixed time in a single
trajectory has non-zero expectation value that is asymptotically independent
of $\ell$ (i.e. charge fluctuations satisfy an area law). Correspondingly, in
the dual model, there is a long-range string-order parameter in each
trajectory. We note that, while the string order has a non-vanishing
expectation value at any fixed time slice in the dual-charge-sharp phase, this
expectation value will vary from time slice to time slice due to intermediate
evolution causing short-range charge fluctuations across the end of the
string.

While it is perhaps not surprising that in the area law phase with SPT order
can be stabilized at large measurement rates (in this dual mapping the
original charge measurements become projections onto the stabilizers of the
SPT phase), the possibility of a volume law entangled phase with SPT order in
a fixed trajectory does not have any precedent in equilibrium, and represents
a fundamentally new type of phenomena that relies essentially on both non-
equilibrium and non-unitary evolution.

####  B.2.2 $1+1d$: Spontaneous Symmetry Breaking

By a closely related reasoning to the section above, applying standard Kramers
Wannier (KW) duality (which exchanges the paramagnetic and spontaneous
symmetry broken phases) to a $1+1d$ $\mathbb{Z}_{N}$ symmetric random circuit
model relates a charge-sharp phase to a dual phase with long-range symmetry
breaking order in a given trajectory. I.e. the KW duality exchanges sharp and
fuzzy phases. E.g. for a $\mathbb{Z}_{2}$ symmetric circuit duality (KW)
mapping exchanges
$\sigma_{i}^{x}\leftrightarrow\sigma^{z}_{i}\sigma^{z}_{i+1}$.

####  B.2.3 $2+1d$: Discrete Gauge Theories

As a final application, we consider KW dualities in $2+1d$ systems, which map
models with global symmetries onto dual “pure” gauge theories that have gauge
magnetic-flux degrees of freedom but lack dynamical electrical charge. In this
mapping, the original site degrees of freedom, which live on vertices of a
square lattice and transform under a $\mathbb{Z}_{N}$ symmetry, map to dual
link variables residing on a dual lattice whose sites are centered on
plaquettes of the original lattice. Symmetry domain wall operators map onto
$\mathbb{Z}_{N}$ electric field operators living on links, and local symmetry
generators map onto the $\mathbb{Z}_{N}$ flux through the dual plaquette. We
define the duality mapping to act trivially on the large-d qudits, which in
the dual lattice now reside at the center of plaquettes.

This duality mapping interchanges symmetry charge in the original variables
with gauge magnetic flux in the dual variables. Hence, the charge-sharp and
fuzzy phases respectively map onto confined and deconfined phases of the gauge
theory. Namely, in the charge-sharp phase, the total charge in a region
$\Sigma$, $\mathcal{Q}_{\Sigma}$, has fluctuations only from local quantum
fluctuations near the boundary, $\partial\Sigma$. In the dual mapping,
$\mathcal{Q}_{\Sigma}$, transforms into a Wilson loop around the boundary,
which we write schematically as
$W_{\Sigma}=e^{i\oint_{\partial\Sigma}\boldsymbol{A}\cdot d\boldsymbol{\ell}}$
(where $e^{iA}$ is the conjugate variable to the $\mathbb{Z}_{N}$ electric
field, and all continuum notations should be appropriately interpreted as
lattice sums). Thus in a given trajectory at late times, the dual of the
charge- sharp and fuzzy phases are characterized by area- or volume- law
scaling of Wilson loops respectively:

$\displaystyle\lim_{t\rightarrow\infty}\left[|\langle\psi_{\bf
m}(t)|W_{\Sigma}|\psi_{\bf
m}(t)\rangle|\right]\sim\begin{cases}e^{-|\partial\Sigma|}&\text{dual-charge-
sharp}\\\ e^{-|\Sigma|}&\text{dual-charge-fuzzy}\end{cases},$ (48)

i.e. correspond to confined and deconfined phases respectively. Note here, as
for spin-glass-like order parameters, it is important to take absolute values
before averaging to obtain the area law scaling.

Again, the prospect of a volume-law entangled, but trajectory-deconfined gauge
theory represents a new possibility not present in equilibrium settings or
closed-system dynamics. We leave establishing the (non)existence of this
phase, and the study of potential critical properties of putative
unconventional volume-law entangled confinement transitions as a loose end for
future study. Other potentially interesting generalizations include both gauge
theories with electrically charged matter, which are not dual to systems with
global symmetries, and continuous $U(1)$ gauge invariant random circuit
dynamics, for which the duality omits monopole instantons that are known to
confine equilibrium gauge theories in $2+1d$, but perhaps have a different
fate in monitored random circuit dynamics.

##  Appendix C Additional numerical results

###  C.1 Entanglement dynamics in the statistical mechanics model

Figure 12: Entanglement dynamics in the statistical mechanics model. (a) Plot
of $S_{1}$, $S_{2}$, $S_{\infty}$, and $-\ln P_{0}$ vs $\sqrt{t}$ for $p=0$
obtained using the statistical model with a fixed vertical minimal cut. We
clearly see different growth of $S_{1}$ and $S_{\infty}$, $S_{2}$. The curve
of $-\ln P_{0}$ exactly overlaps the $S_{\infty}$ curve as argued in the main
text. (b) Plot showing $[-\ln P_{0}]$ vs $\sqrt{t}$ for various $p$, and
$L=12$. We find that this quantity grows linearly with time for any non-zero
$p$. We also plot the average $-\ln[P_{0}]$ for various $p$ (dashed curves),
which grows as $\sqrt{t}$ independently of $p$.

In this appendix, we present additional results on the entanglement dynamics
obtained from the statistical mechanics model.

We start with the $p=0$ case, and analyze how the argument for $\sqrt{t}$
growth translates to the statistical model language. We are working in the
regime where $L\gg t$, so the minimal cut runs along the time direction. For
simplicity of the argument, we assume that the cut does not fluctuate and is
exactly vertical, that is, the cut passes through the same link at all times
(we checked that our results are independent from averaging over fluctuations
of the minimal cut). Using Eq. (20), the Rényi entanglement entropies are
related to the classical Rényi entropies for charge configurations along the
minimal cut. Let us denote this distribution by $P_{\rm DW}$, with
$S_{\infty}=\ln p_{\text{max}}$, where $p_{\text{max}}$ is the maximum of
$P_{\rm DW}$. We find that $p_{\text{max}}$ is given by $P_{\rm
DW}(0...0)\equiv P_{0}$. $P_{0}$ is the probability for all charges on the
vertical cut to be equal to $0$ (equivalently, we could have also considered
$P_{\rm DW}(1...1)\equiv P_{1}$). $P_{0}$ describes the part of the dynamics
where there are no exchange of charge across the cut and is therefore
dominated by dead regions in the initial state, which we know to be the source
of the dominant contribution in the Schmidt values. As we discussed in Sec.
II, if the initial state has a dead region of size $\sqrt{t}$ centered at the
entanglement cut, charges cannot diffuse to the cut until times of order $t$,
so the configurational entropy of charge along the vertical minimal cut will
remain zero. It follows that $P_{0}\geq\exp(-\sqrt{Dt})$ and therefore that
$S_{n}\leq\sqrt{Dt}$ for $n>1$. Meanwhile, typical components of the initial
wavefunction give rise to essentially random charge configurations on the
minimal cut. Owing to the greater multiplicity of typical configurations they
dominate $S_{1}$, which grows linearly in $t$. These expectations are borne
out in Fig. 12(a).

We now turn to $p>0$. For $p>0$, the charge on measured sites in a given
trajectory is constrained to match the measurement outcome. As we noted in
Sec. II, this suffices to eliminate dead regions in typical trajectories. To
capture the effect of dead regions on the growth of $S_{\infty}$ for $p>0$, we
calculate $-\ln P_{0}$ as a proxy for entanglement entropies. For numerical
convenience, we make two simplifying assumptions: (1) we ignore fluctuations
of the minimal cut, (2) we do not perform measurements on links adjacent to
the cut (this avoids numerically expensive postselection procedures as the
trajectories with non-zero $P_{0}$ quickly become rare as we increase $p$).

We plot $[-\ln P_{0}]$ and $-\ln[P_{0}]$ vs $\sqrt{t}$ in Fig. 12(b). We find
that the quantity $-\ln[P_{0}]$ grows as $\sqrt{t}$ for all $p$. This is
expected because we are averaging over the trajectories before calculating the
(pseudo)entropy; this is same as in the $p=0$ case where the unitary evolution
can be seen as equivalent to doing the sum over all trajectories. We find that
at low $p$, $[-\ln P_{0}]$ stays closer to the $\sqrt{t}$ growth for longer
times. At higher $p$ it diverges significantly and crosses over to linear
growth. Though $[-\ln P_{0}]$ is not exactly equal to the Rényi entropy, this
transition from $\sqrt{t}$ to linear growth $\sim t$ is a generic phenomenon
for all quantities where the survival of dead regions becomes a rare
occurrence due to measurements, consistent with the general argument in Sec.
II, and the results of Sec. V.

###  C.2 Charge sharpening dynamics in the fuzzy phase and near the charge-
sharpening transition

Figure 13: Dynamics of charge sharpening in qubit chains. Fraction of
trajectories with $\delta\mathcal{Q}^{2}<\epsilon$ with $\epsilon=10^{-3}$ at
$p=0.085$, inside the fuzzy phase.  Figure 14: Dynamics of charge sharpening
in the statistical model. Fraction of trajectories with
$\delta\mathcal{Q}^{2}<\epsilon$ with $\epsilon=10^{-2}$ at $p=0.24$, inside
the fuzzy phase. Inset: Different threshold $\epsilon=10^{-10}$, showing a
similar scaling of the charge sharpening over a timescale $t\sim L$.

In this appendix, we present numerical evidence that the charge sharpens on a
time scale $t_{\\#}\sim L$, in agreement with the general argument of Sec. II.
We plot the fraction $N_{0}$ of trajectories with
$\delta\mathcal{Q}^{2}<\epsilon$ versus $t\sim L$, both in the qubit chain
numerics (Fig. 13) and in the statistical mechanics model (Fig. 14). We
observe a clear crossover when $N_{0}$ lifts off from zero on a time scale
scaling linearly with $L$, as expected. Note that this sharpening time scale
$t_{\\#}\sim L/p$ is much smaller than the purification time scale
$t_{\pi}\sim{\rm e}^{L}$ Gullans and Huse (2020a).

Figure 15: Charge variance distribution in the qubit model. Distribution of
the charge variance at $t/L=4$ (a) in the charge fuzzy phase $p=0.05$, (b)
near the critical point $p=0.10$, and (c) in the charge sharp phase $p=0.14$.

We now turn to the critical dynamics near the charge sharpening phase
transition in qubit chains. Before embarking into numerical details, we
summarize the physics of the critical dynamics obtained from the simulations.

For generic initial states that mix multiple charge sectors, the charge
sharpening happens in two stages: the measurement first sharpens the charge
from multiple sectors to two consecutive sectors $(N,N+1)$ The measurement
then further collapses the superposition of the two sectors $(N,N+1)$ to a
unique charge (either $N$ or $N+1$). The two stages are separated by the
crossover time $t_{\\#}\sim L/p$. At much later times $t\gg t_{\\#}$ we find
$N_{0}\rightarrow 1$ in a critical manner that we now turn to.

The numerical simulations of qubit chains suggest that the second stage is
governed by charge-sharpening criticality. In the long-time limit, $t\gg
t_{\\#}$, we find that the universal scaling law for the critical dynamics is
an exponential function,

$O(t,p)\sim
A_{O}(x)e^{-t/\xi_{t}(x)},\,\,\mathrm{where}\,\,x\equiv(p-p_{\\#})L^{1/\nu_{\\#}},$
(49)

$L$ is the system size, and $O$ is an observable sensitive to the criticality
(e.g. $N_{0}$ and $S_{1,\mathcal{Q}})$. The universal decay rate $\xi_{t}$ is
a time scale that is different from the crossover time $t_{\\#}$. Due to the
space-time symmetry, the time scale $\xi_{t}$ follows a scaling law,

$\xi_{t}(x)/L=B(x).$ (50)

Both universal scaling functions $A_{O}(x)$ and $B(x)$ are smooth in the
critical regime. Right at the transition point, we obtain $\xi_{t}\approx
0.5L/p_{\\#}$.

We now show our numerical evidence to support the above physical picture. We
present the charge variance distribution $\delta\mathcal{Q}^{2}$ in each phase
and the vicinity of the charge sharpening critical point in Fig. 15 to provide
additional clarity on the nature of the charge sharpening dynamics. More
specifically, Fig. 15(a) is deep in the charge fuzzy phase characteristic of
the first stage of dynamics $t\ll t_{\\#}$. It reveals a wide charge
distribution, indicating that the quantum state at this stage is charge fuzzy