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# Critical phase and spin sharpening in SU(2)-symmetric monitored quantum
circuits

Shayan Majidy<EMAIL_ADDRESS>Perimeter Institute for Theoretical
Physics, Waterloo, Ontario N2L 2Y5, Canada  Institute for Quantum Computing,
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Utkarsh Agrawal
<EMAIL_ADDRESS>Kavli Institute for Theoretical Physics, University
of California, Santa Barbara, CA 93106, USA  Sarang Gopalakrishnan  Department
of Electrical and Computer Engineering, Princeton University, Princeton, NJ
08544, USA  Andrew C. Potter  Department of Physics and Astronomy, and Quantum
Matter Institute, University of British Columbia, Vancouver, BC, Canada V6T
1Z1  Romain Vasseur  Department of Physics, University of Massachusetts,
Amherst, MA 01003, USA  Nicole Yunger Halpern<EMAIL_ADDRESS>Joint Center
for Quantum Information and Computer Science, NIST and University of Maryland,
College Park, MD 20742, USA  Institute for Physical Science and Technology,
University of Maryland, College Park, MD 20742, USA

(August 28, 2024)

###### Abstract

Monitored quantum circuits exhibit entanglement transitions at certain
measurement rates. Such a transition separates phases characterized by how
much information an observer can learn from the measurement outcomes. We study
SU(2)-symmetric monitored quantum circuits, using exact numerics and a mapping
onto an effective statistical-mechanics model. Due to the symmetry’s non-
Abelian nature, measuring qubit pairs allows for nontrivial entanglement
scaling even in the measurement-only limit. We find a transition between a
volume-law entangled phase and a critical phase whose diffusive purification
dynamics emerge from the non-Abelian symmetry. Additionally, we numerically
identify a “spin-sharpening transition.” On one side is a phase in which the
measurements can efficiently identify the system’s total spin quantum number;
on the other side is a phase in which measurements cannot.

##  I Introduction

Traditionally, quantum many-body physicists have been limited to studying
closed systems in equilibrium. Thanks to the maturation of quantum simulators
[1], researchers can now prepare and control open quantum systems far from
equilibrium with high precision. Quantum simulators have helped answer
foundational questions about quantum entanglement and thermodynamics [2, 3].
Also, quantum simulators have the potential to solve real-world problems in,
e.g., materials science and chemistry [4, 3]. These advances have spurred the
development of tools for studying open quantum systems far from equilibrium
[5]. Monitored quantum circuits form one toolkit [6, 7]. A typical monitored
quantum circuit acts on a chain of $L$ qubits (spin-$1/2$ particles). The
circuit contains two-qubit unitary gates, after each layer of which every
qubit has a probability $p$ of being measured. Monitored circuits exhibit
measurement-induced phase transitions (MIPTs), due to the competition between
chaotic dynamics and measurements [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].

Initially, an MIPT was cast as a transition between phases characterized by
volume-law and area-law entanglement scaling [31, 32]. Equivalently, the
transition is a purification transition between a mixed phase and a pure phase
[8]. When the measurement rate $p$ is low, the chaotic dynamics scramble
information about the initial state. Local measurements cannot extract that
information in this mixed phase. An initially mixed state becomes pure,
conditionally on measurement outcomes, in a time $t_{\rm P}\sim\exp(L)$, with
$L$ the number of qubits. In contrast, at large $p$, the measurements can
distinguish different initial states efficiently. In this pure phase, an
initially mixed state purifies quickly, often at an $L$-independent rate [31].

Few properties restrict the simplest monitored circuits’ dynamics: unitarity
and locality. Monitored circuits can be enriched, though. Enhancements include
charge conservation [19, 20, 21, 22, 23, 24], measurements of particular
operators (such as generators of the toric-code stabilizer) [25, 26], and the
replacement of qubits with free fermions [27, 28, 29, 30, 33, 34].
U(1)-symmetric monitored circuits exhibit a charge-sharpening transition [21]
between a charge-fuzzy phase and a charge-sharp phase. These phases are
distinguished by how quickly measurements collapse superpositions of different
amounts of charge—how efficiently an observer can learn from local
measurements the amount of global charge in the system.

Noncommuting symmetry charges have spawned a growing subfield of quantum
thermodynamics [35, 36, 37]. Noncommutation of charges has been shown to
increase average entanglement [38], decrease entropy-production rates [39],
and necessitate modifications to the eigenstate thermalization hypothesis
(ETH) [40, 41]. Researchers have used trapped ions to bridge this subfield
from theory to experimental reality [42, 43, 44]. Motivated by this subfield’s
discoveries, we study monitored quantum circuits with three noncommuting
charges.

In this work, we explore monitored-random-circuit dynamics of one-dimensional
(1D) qubit chains with SU(2) symmetry. Equivalently, the circuits conserve
three noncommuting charges: the total spin angular momentum’s components.
First, we explore the purification dynamics of a spin chain initially
entangled with an ancilla spin. We identify a purification transition between
a mixed phase, in which the ancilla purifies over an exponential-in-$L$ time,
and a critical phase111For the purposes of this discussion, we classify
Goldstone phases with long-range order as critical. with scale-invariant
purification and entanglement growth. Above a critical measurement rate (at
$p>p_{\rm c}$), we observe an extended-in-$p$ critical phase in which the
purification time scales diffusively: $t_{\rm P}\sim L^{2}$. Second, we
examine the entanglement dynamics undergone by an initially unentangled state.
The purification transition doubles as an entanglement transition between
volume-law entanglement scaling, at $p<p_{\rm c}$, and subextensive
(logarithmic or small-power-law) scaling, at $p>p_{\rm c}$. The critical
entanglement dynamics $p>p_{\rm c}$—even in the measurement-only limit
($p=1$)—due to the local measurements’ noncommuting nature. In fact, a Lieb-
Shultz-Mattis-type anomaly precludes a simple area-law entangled regime [34],
as would arise when $p=1$, absent symmetries.

Observing the purification/entanglement transition experimentally would
require many instances of the same set of measurement outcomes. Such instances
occur with vanishing likelihood in the thermodynamic limit. This challenge is
the postselection problem. To evade this difficulty, we explore a “spin-
sharpening/learnability” transition. Denote by $s$ the total spin quantum
number. We examine whether the dynamics collapse an initial superposition of
states in different $s$ sectors. Unlike in the U(1)-symmetric problem, the
sectors generally cannot be shared by the (extensive) charges: our system’s
three charges, failing to commute, share only one sector. We identify a spin-
sharpening transition at a measurement rate $p=p_{\\#}$, which is numerically
indistinguishable from the entanglement-transition rate: $p_{\\#}\approx
p_{\rm c}$. In the spin-sharp phase ($p>p_{\\#}$), an observer can, in
principle, determine the system’s $s$ in a time scale $t\sim L^{2}$, with a
probability tending to unity as $L\rightarrow\infty$. In contrast, if
$p<p_{\\#}$, the information about $s$ is effectively hidden from the
observer, who requires an exponential-in-system-size time to determine $s$.
This “learning” perspective might be used to probe the transition
experimentally.

Finally, we interpret our results within an effective replica statistical-
mechanics model. We obtain the model by averaging over the gates and
measurement outcomes, building on previous results about asymmetric and
symmetric circuits [7]. This model casts dynamical properties of
SU(2)-symmetric monitored quantum circuits in terms of some effective
Hamiltonian’s low-energy properties. We interpret our numerical results in
terms of this effective-Hamiltonian model.

The rest of this paper is organized as follows. In Sec. II, we introduce
SU(2)-symmetric monitored quantum circuits. We present the
purification/entanglement transition in Sec. III and the spin-sharpening
transition in Sec. IV. Section V contains our statistical-mechanics mapping.
Section VI finishes with opportunities established by this work.

##  II Model: SU(2)-symmetric monitored circuits

Figure 1: SU(2)-symmetric monitored quantum circuits. $L$ qubits (circles) are
prepared in the state $\rho_{\rm i}$. Each “brick” in the brickwork circuit is
an SU(2)-symmetric unitary gate with a probability $1-p$ and is a two-qubit
projective measurement with a probability $p$. The circuit acts for some time
(some number of layers) before the final state, $\rho_{\rm f}$, is read out.
One brick illustrates which bonds have even (odd) indices.

Consider a brickwork circuit acting on a 1D chain of qubits, as depicted in
Fig. 1. The number $L$ of spins is even for convenience. Denote by
$\sigma_{j}^{(x,y,z)}$ the Pauli matrices acting on qubit $j$. The total spin
components $S^{(x,y,z)}=\frac{1}{2}\sum_{j=1}^{L}\sigma_{j}^{(x,y,z)}$
generate the algebra associated with a global SU(2) symmetry. We set $\hbar$
to 1. The spin-squared operator $\vec{S}^{2}$ has eigenvalues $s(s+1)$
labelled by the total spin quantum number $s$. We denote the eigenvalues of
$S^{(z)}$ by $m$, the two-qubit singlet state by $\ket{{\rm s}_{0}}$, and the
two-qubit eigenvalue-$m$ triplets by $\ket{{\rm t}_{m}}$.

Each brick is, with a probability $1-p$, a gate, or, with a probability $p$, a
projective measurement. The gates are chosen randomly from SU(2). The most
general such gate acting on spins $j$ and $j+1$, has the form

$\displaystyle\cos(\phi)\mathbb{I}-i\sin(\phi)\leavevmode\nobreak\
\mathrm{Sw}_{j,j+1},$ (1)

up to an irrelevant overall phase. $\mathrm{Sw}_{j,k}$ swaps the states of the
spins $j$ and $k$. We draw each gate’s parameter $\phi$ independently from the
uniform distribution on $[0,2\pi)$. Each measurement projects a two-qubit
state onto the singlet ($s=0$) or triplet ($s=1$) subspace (fusion channel).
Crucially, two measurements fail to commute when acting on overlapping spin
pairs. Crucially, the SU(2) symmetry precludes nontrivial single-qubit
measurements. One time step consists of a brick layer on even-index bonds and
a layer on odd-index bonds. In the even-index-bond layers, a brick connects
the first and $L^{{\rm th}}$ qubits, effecting periodic boundary conditions.

##  III Purification/entanglement transition

We first explore the model’s entanglement and purification dynamics. In Sec.
III A, we examine the purification dynamics of initially mixed states. The
spin chain’s state begins scrambled and entangled with an ancilla qubit $A$.
The ancilla’s entanglement decreases over time. In Sec. III B, we examine the
entanglement dynamics exhibited by initially short-range-entangled pure
states.

###  III A Purification time

Figure 2: The purification time reveals a $z{=}2$ phase. The entropy $S_{A}$
quantifies the ancilla qubit’s entanglement with the system. We plot
$\log(S_{A})$ for clarity, as $S_{A}$ decays exponentially. $t/L^{2}$ runs
along the $x$-axis to demonstrate the existence of a phase in which the system
purifies over a time scale $t_{\rm P}\sim L^{2}$. The curves’ collapsing at
$p>0.35$ evidences this phase. We used 30 000 samples when $L=8$ to $16$; 10
000 samples when $L=18$; and 1 500 samples when $L=20$. The $y$-axis’s lower
limit is $\log(10^{-3})\approx-6.91$.

We determine the purification time as follows [9]. Denote by
$\ket{s,m,\lambda_{0}}$ and $\ket{s,m,\lambda_{1}}$ two orthogonal states from
the same $(s,m)$ sector. The last index distinguishes degenerate states. We
entangle an ancilla qubit with the system’s $L$ qubits, forming the
$(L+1)$-qubit state

$\tilde{\ket{\psi_{\rm
i}}}=\tfrac{1}{\sqrt{2}}\left(\ket{0}_{A}\ket{s,m,\lambda_{0}}+\ket{1}_{A}\ket{s,m,\lambda_{1}}\right).$
(2)

The subscript $A$ distinguishes the ancilla from the system qubits. $A$ does
not undergo gates or measurements.

We choose two system states that have $s=1$ and $m=0$. In
$\ket{{s=1},{m=0},\lambda_{0}}$, qubits $1$ and $2$ are in the triplet
$\ket{{\rm t}_{0}}$; and the remaining pairs of qubits, in singlets $\ket{{\rm
s}_{0}}$. In $\ket{s=1,m=0,\lambda_{1}}$, qubits $3$ and $4$ are in $\ket{{\rm
t}_{0}}$, instead. These two system states are orthogonal, in the same
$\vec{S}^{2}$ sector, and in the same $S^{(z)}$ sector. However, one can
distinguish the states by measuring qubits $1$ and $2$. Such local
distinguishability is undesirable. Therefore, after preparing
$\tilde{\ket{\psi_{\rm i}}}$, we scramble the system: the system undergoes a
unitary-only ($p=0$) SU(2)-symmetric circuit for $L^{2}$ time steps. (The
$t_{\rm P}$ identified later in this subsection motivates the $L^{2}$.) The
scrambling encodes quantum information about the ancilla roughly uniformly in
many-body entanglement. This process prepares $\ket{\psi_{\rm i}}$.

$\ket{\psi_{\rm i}}$ undergoes $t=L^{2}$ time steps under monitored-random-
circuit dynamics with $p\geq 0$. Denote by
$\rho_{A}\coloneqq\Tr_{\bar{A}}(\outerproduct{\psi_{\rm f}}{\psi_{\rm f}})$
the final state of $A$. We calculate the final entanglement entropy between
$A$ and the system:

$S_{A}\coloneqq S(\rho_{A})\coloneqq-\Tr[\rho_{A}\log(\rho_{A})].$ (3)

(All logarithms are base-$e$.) We anticipate that the measurements will purify
the system at an exponential-in-$t$ rate: $S_{A}\sim e^{-t/t_{\rm P}(L)}$.
Therefore, we plot $\log(S_{A})$ in Fig. 2. Along the $x$-axes runs $t/L^{2}$.
At each $p>p_{\rm c}\approx 0.35$, the different-$L$ curves collapse. Hence
this phase purifies according to $S_{A}\sim{\rm e}^{-t/L^{2}}$ and so has a
dynamical critical exponent $z{=}2$. This $z$-value characterizes diffusive
scaling [45] and suggestively evokes ferromagnetic spin waves’ dynamics [46,
Ch. 33].

Figure 3: The purification time $t_{\rm P}$ grows exponentially with $L$ at
small $p$. This mixed-phase behavior characterizes $p=0.05$.

At lower measurement rates, $p\ll p_{\rm c}$, we observe a mixed phase. Figure
3 shows the purification time plotted against $L$, at several $p$ values. At
$p=0.05$, $t_{\rm P}\sim e^{L}$. At $p$ values between 0.05 and 0.35, the
scaling is unclear from the numerics; we cannot conclude to which phase this
intermediate regime belongs. Still, the exponential purification time
resembles that of asymmetric circuits [8]. We analyze this mixed phase
analytically in Sec. V, using a duality between monitored circuits and a
statistical-mechanics model.

###  III B Entanglement dynamics

To characterize the critical phase further, we explore an initially pure
state’s late-time bipartite entanglement entropy, $S_{\rm f}$. The
purification transition manifests as a qualitative change in the
$L$-dependence of $S_{\rm f}$ at $p=p_{\rm c}$.

Figure 4: The entanglement dynamics evidence no area-law phase. The bipartite
entanglement entropy reaches the long-time value $S_{\rm f}$. At $p=0$,
$S_{\rm f}$ is linear in $L$. As $p$ increases, $S_{\rm f}$ gradually becomes
logarithmic or power-law with a small exponent. When $L=10$ to $16$, we use 30
000 samples; when $L=18$ and $20$, we use 10 000.

We initialize the system in a short-range-entangled state $\ket{\psi_{\rm
i}}$, a tensor product of a triplet $\ket{{\rm t}_{0}}$ and $\tfrac{L-2}{2}$
singlets $\ket{{\rm s}_{0}}$. This choice’s details are unimportant. However,
we choose this state so that $\ket{\psi_{\rm i}}$ is in the same $s$ sector at
all system sizes $L$. The state undergoes monitored-circuit dynamics for
$L^{2}$ time steps. This time suffices for the entanglement entropy to reach a
steady value, regardless of the measurement rate, $p$. Figure 6 in Appendix A
illustrates this point at the extreme values $p=0,1$. We measure the bipartite
entanglement entropy, $S_{\rm f}$, between two equal-size halves of the chain.

Figure 4 shows the dependence of $S_{\rm f}$ on $L$ at different measurement
rates $p$. At $p=0$, we observe the volume-law phase common to monitored
circuits: $S_{\rm f}\sim L$. Figure 7 in Appendix A supports this claim more
precisely than does Fig. 4. At larger $p$ values,222According to [8], the
entanglement phase transition is equivalent to the purification transition.
Our system’s purification transition happens at $p_{\rm c}\approx 0.35$,
according to the previous subsection. This section’s numerics are consistent
with an entanglement transition at $p\approx 0.35$, but the transition’s exact
location is unclear. the entanglement scaling is less consistent with a linear
fit. Better fits are logarithmic and small-power-law $(S_{\rm
f}\sim\sqrt{L})$. One cannot definitively distinguish these behaviors at the
accessible system sizes, as detailed in Appendix A. The statistical-mechanics
model (Sec. V) provides stronger evidence for the absence of a volume-law
phase at large $p$.

Intuitively, the slow entanglement growth at large $p$, even in the
measurement-only ($p=1$) limit, arises from the noncommutativity of the
charges measured. Similar logarithmic entanglement growth has been observed
under measurement-only dynamics previously: Majorana fermions were subjected
to noncommuting measurements in [25].

Appendix B presents numerical results concerning the correlations between
local observables at different sites. The limitation on system size makes it
difficult to determine the functional form of the correlations’ decay with
distance. However, we find a qualitative change in how the correlations decay
at $p>p_{\rm c}$ and at $p<p_{\rm c}$.

##  IV Spin-sharpening transition

Having explored the purification dynamics within an $s$ sector, we explore the
purification of a superposition spread across $s$ sectors. We again entangle
the chain with an ancilla qubit. This time, the ancilla is in $\ket{0}$, and
the chain has a spin quantum number $s_{0}$, in superposition with the
ancilla’s being in $\ket{1}$ and the chain’s having $s_{1}$. The dynamics may
purify the ancilla in a given measurement trajectory. In this case, the
chain’s state has collapsed onto the $s_{0}$ (or $s_{1}$) sector.
Consequently, the measurement outcomes’ probability of being compatible with
the system’s having $s_{1}$ (or $s_{0}$) vanishes. An observer can therefore,
knowing the circuit, learn the spin quantum number by monitoring the
measurement outcomes.

Comparing spin sharpening with U(1)-charge sharpening is illuminating. One can
estimate as follows the total charge of qubits undergoing a U(1)-symmetric
hybrid circuit: Running the circuit, one obtains $ptL$ measurement outcomes
(0s and 1s), on average. Consider averaging the outcomes, multiplying by $L$,
and rounding to the nearest integer. If $t\sim L$, this procedure estimates
the charge accurately [47]. If the dynamics are SU(2)-symmetric on average (as
in Sec. II), sequential measurements fail to commute. Hence later measurements
render partially irrelevant the information obtained from earlier
measurements. An observer cannot obviously learn $s$ ever. Nevertheless, we
numerically identify a measurement-induced transition at a measurement rate
$p_{\\#}$. We call this transition a _spin-sharpening_ transition. It
separates regimes in which an observer can ($p>p_{\\#}$) and cannot
($p<p_{\\#}$) identify $s$ from the measurement outcomes, with a probability
tending to unity as the $L\rightarrow\infty$.

We diagnose the spin-sharpening transition using a similar procedure to the
one in Sec. III A. The difference is that, unlike in Eq. (2), we construct
$\tilde{\ket{\psi_{\rm i}}}$ from distinct $\vec{S}^{2}$ eigenspaces:

$\tilde{\ket{\psi_{\rm
i}}}=\tfrac{1}{\sqrt{2}}\left(\ket{0}_{A}\ket{s_{0},m,\lambda_{0}}+\ket{1}_{A}\ket{s_{1},m,\lambda_{1}}\right).$
(4)

We choose $m=0$, $s_{0}=1$, and $s_{1}=0$ for convenience: one can construct
such a $\tilde{\ket{\psi_{\rm i}}}$ by tensoring together singlets and an
$m=0$ triplet, regardless of $L$. After preparing $\tilde{\ket{\psi_{\rm
i}}}$, we scramble the system under a $p{=}0$ circuit for $L^{2}$ time steps,
as in Sec. III A. This procedure prepares a state $\ket{\psi_{\rm i}}$. Then,
we evolve the system under monitored-circuit dynamics with a fixed $p$.
Anticipating $z{=}2$ dynamical scaling, we evolve the system for $L^{2}$ time
steps. If the ancilla purifies after this short time, we say that the spin has
sharpened. We denote the final state by $\ket{\psi_{\rm f}}$.

Figure 5a shows the ancilla’s final entanglement entropy, $S_{A}$, plotted
against $p$. Different curves correspond to different system sizes $L$. The
curves cross at $p_{\\#}\approx 0.28$, suggesting that a spin-sharpening
transition occurs at $p_{\\#}$. Furthermore, Fig. 5b displays a finite-size
collapse. We used the scaling form $\log(S_{A})=(p-p_{\\#})L^{1/\nu}$, the
correlation length exponent $\nu=3.0$, and $p_{\\#}=0.28$.

Our tentative identification of a spin-sharpening transition at $p_{\\#}$ is
subject to at least two caveats. First, the crossing point drifts to larger
$p$ as $L$ increases (perhaps coalescing with the purification transition at
$p_{c}$ as $L\to\infty$). Second, the scaling ansatz we chose for the data
collapse in Fig. 5b may not be valid. The ansatz implies that the time scale
for a size-$L$ system to sharpen increases more quickly than $L^{2}$ for
$p<p_{\\#}$ and more slowly than $L^{2}$ for $p>p_{\\#}$. However, our data
for $p>p_{\\#}$ (see Fig. 8 in Appendix A) is compatible with a sharpening
time scale $\sim L^{2}$ deep in the critical phase. If the sharpening time
indeed scales as $L^{2}$ throughout the critical phase, the crossing in Fig.
5a must be a finite-size artifact. Precisely identifying $p_{\\#}$ and the
sharpening transition’s nature is outside the scope of this work, due to the
paucity of $L$ values accessible in exact computations. We defer a detailed
analysis of the spin-sharpening time scales to future work.

Figure 5: A spin-sharpening transition exists. The entropy $S_{A}$ quantifies
the ancilla qubit’s entanglement with the system. Different curves correspond
to different system sizes $L$. (a) The curves’ crossing at $p\approx 0.28$
indicates a phase transition. (b) We identify a finite-size collapse using
$\nu=3.0$ and $p_{\\#}=0.28$.

Finally, the spin-sharpening transition suggests a postselection-free means of
observing a measurement-induced transition experimentally [48, 47, 49, 50]:
identify whether an observer can learn $s$ from measurement outcomes. This
learning would require “decoders” for estimating $s$ from the outcomes. The
decoders’ accuracy, as a function of the measurement rate, would need to be
tested. In principle, one can learn $s$ most accurately via brute-force
decoding [47]. One would, upon running the circuit and obtaining the
measurement outcomes, simulate the circuit, postselected on the observed
outcomes and operating on a state in the $s_{0}$ sector. One would repeat the
simulation with an initial state in the $s_{1}$ sector. From each simulation,
one would infer the probability that $s_{0}$ (or $s_{1}$) had engendered the
observed outcomes.

However, this approach generically costs exponential-in-$L$ time (even if a
quantum computer performs the simulation, due to the postselection). Special
classes of monitored dynamics [48, 47, 49, 50] may allow for approximate
decoders that can be implemented efficiently on classical or quantum computers
without postselection. In this case, the transition’s nature will depend on
both the circuit and the decoder and may differ, in location or universality
class, from the spin-sharpening transition observed under optimal decoding. We
leave for future work the problem of designing efficient decoders for spin-
sharpening transitions.

##  V Effective-Hamiltonian description of the monitored dynamics

To complement the numerics, we derive an effective statistical-mechanics
model: a description of the monitored evolution as imaginary-time evolution
under an effective Hamiltonian acting on copies (replicas) of the system. In
the rest of this section, we describe the model. We elucidate its ground and
low-lying excited states in Sec. V A. Leveraging these results, we elucidate
the monitored circuit’s purification transition in Sec. V B. Section V C
explains the circuit’s lack of an area law.

In the statistical-mechanics model, measurement outcomes act as quenched
disorder for a quantum trajectory. A replica trick is needed to average
nonlinear quantities, such as entanglement, over trajectories [10, 11]. One
must average $Q$ replicas of the density matrix, $\rho^{\otimes Q}$. In the
replica limit, $Q\rightarrow 1$. More precisely, we want to calculate

$\displaystyle\overline{\rho^{(Q)}(t)}$ $\displaystyle=\sum_{\vec{m}}\int
dU\left(K_{\vec{m},U}^{\vphantom{\dagger}}\rho_{0}K_{\vec{m},U}^{\dagger}\right)^{\otimes
Q}\,,$ (5)

dependent on the evolution operator
$K_{\vec{m},U}\equiv\prod_{\ell=1}^{2t}P_{\ell,\vec{m}}U_{\ell}$. $U_{\ell}$
denotes the unitary implemented by circuit layer $\ell$. $P_{\ell,\vec{m}}$
denotes the projector onto the subspace associated with the list $\vec{m}$ of
outcomes yielded by the measurements at time step $i$. $\int dU$ denotes an
average over the SU(2)-symmetric gates (with the appropriate probability
measure).

An alteration to the circuit model will facilitate the analytics: we deform
the discrete-time, strong-measurement circuit dynamics into a continuous-time
version. We replace the gates with Hamiltonian evolutions over infinitesimal
time steps, and infinitesimally weak measurements replace the projective
measurements. We expect the continuous-time deformation to preserve the
purification/entanglement and charge-sharpening transitions’ universal scaling
properties. The reasons are analogous examples [47] and the system’s lack of
time-translation symmetry.

Here, we summarize the resulting Hamiltonian description. Appendix C contains
a detailed derivation. The effective Hamiltonian equals a sum of contributions
from the unitary dynamics and the weak measurements:
$H^{\mathrm{eff}}=H^{\mathrm{u}}+H^{\rm m}$. The terms are

$\displaystyle H^{\rm u}$
$\displaystyle=-J\sum_{i}\left[\sum_{a=1}^{Q}\left(\vec{S}^{a}_{i}\cdot\vec{S}^{a}_{i+1}-\vec{S}^{a*}_{i}\cdot\vec{S}^{a*}_{i+1}\right)\right]^{2}\;\text{and}$
(6) $\displaystyle H^{\rm m}$
$\displaystyle=\gamma\sum_{i}\sum_{a,b=1}^{2Q}\left(\vec{S}^{a}_{i}\cdot\vec{S}^{a}_{i+1}\right)\Pi_{a,b}\left(\vec{S}^{b}_{i}\cdot\vec{S}^{b}_{i+1}\right).$
(7)

The coupling constant $J$ encapsulates the unitary dynamics’ scrambling power;
and $\gamma$, the weak measurements’ strength. $P_{i,j}^{a}$ is the projector
onto the singlet sector of spins $i$ and $j$ in replica copy $a$. Equations
(6)–(7) have two kinds of summations over the replica index, $a$. When $a$
runs from 1 to $Q$, the summation is over forward copies of the replicas.
$a^{*}$ represents the corresponding backward copy. When $a$ runs from 1 to
$2Q$, the summation is over both the backward and forward copies. The
projector $\Pi_{a,b}=\delta_{ab}-\frac{1}{2Q}$ is onto inter-replica
fluctuation modes. The associated term in Eq. (7) is minimized when
$\vec{S}_{i}\cdot\vec{S}_{i+1}$ yields the same value, operating on any
replica, as operating on any other. If the measurements are projective, all
the replicas must yield the same measurement outcome. If the measurements are
weak, as above, this restriction is softened; a finite energy cost accompanies
inter-replica fluctuations in the measured operator $\vec{S}\cdot\vec{S}$.

The effective Hamiltonian has a left/right $S_{Q}\times S_{Q}$ symmetry:
$H^{\mathrm{eff}}$ remains invariant under permutations of the $Q$ forward
copies and permutations of the $Q$ backward copies. The monitored dynamics map
to imaginary-time evolution under $H^{\mathrm{eff}}$ (in the replica limit
$Q\to 1$). Thus, we must understand this Hamiltonian’s low-energy properties
to understand the monitored dynamics’ late-time properties.

###  V A Ground state and collective excitations at low measurement rates

We begin with a measurement-free model: $\gamma=0$ in Eq. (7). A ground state
is a configuration that, when acted on by
$\sum_{a=1}^{Q}\left(\vec{S}^{a}_{i}\cdot\vec{S}^{a}_{i+1}-\vec{S}^{a*}_{i}\cdot\vec{S}^{a*}_{i+1}\right)$
for any nearest-neighbor pair $(i,i+1)$, vanishes. Such a configuration is
achievable if and only if, for some pairing of $(a,b^{*})$,
$\vec{S}^{a}_{i}\cdot\vec{S}^{a}_{i+1}=\vec{S}^{b*}_{i}\cdot\vec{S}^{b*}_{i+1}$
for all $i$. The ground states thus can be labeled by all such pairings
$(a,b^{*})$. Furthermore, the ground states are represented by the elements
$\sigma$ of the permutation group $S_{Q}$ such that
$\vec{S}^{a}_{i}\cdot\vec{S}^{a}_{i+1}=\vec{S}^{\sigma(a)*}_{i}\cdot\vec{S}^{\sigma(a)*}_{i+1}$.
To satisfy this condition for all $i$, the interaction must be ferromagnetic,
precluding frustration. We show in Appendix C that the ground space of $H^{\rm
u}$ is that of an SU(4) ferromagnet. The ground states can thus be labeled as
$|\otimes_{i=1}^{L}\sigma\rrangle$. The permutation $\sigma\in S_{Q}$, and the
tensor product emphasizes the pairings’ uniformity across space.333As noted,
the ground space has a degeneracy labeled by the ground states of the SU(4)
ferromagnet. The label depends on the initial state. We drop the label from
our notation for simplicity, as the label does not impact the following
discussion. Importantly, the ground space breaks the discrete symmetry
$S_{Q}\times S_{Q}$ to $S_{Q}$.

We now briefly sketch the low-lying energy eigenstates.444The gaps between
these eigenstates’ energies and the ground-state energy vanishes in the
thermodynamic limit but remains nonzero at finite $L$. If $\gamma=0$, the
excitations over a symmetry-broken state $|\otimes_{i=1}^{L}\sigma\rrangle$
are described by $Q$ decoupled SU(4) ferromagnetic chains, each formed from
two SU(2) chains. Let us focus on one SU(4) chain. An SU(4) ferromagnet’s
Goldstone modes live on a 6-dimensional manifold. They result in three gapless
modes with energies vanishing as $L^{-z}$, wherein $z=2$. These gapless modes
are of two types: Two modes arise from fluctuations within single SU(2) spin
chains. The third mode arises from collective fluctuations of the two SU(2)
chains. In summary, $Q$ replicas lead to $2Q$ diffusive ($z{=}2$) modes
associated with fluctuations within single SU(2) chains, plus $Q$ diffusive
modes associated with inter-replica fluctuations. As noted above, measurements
affect only the inter-replica fluctuations and thus couple the $Q$ inter-
replica modes [Eq. (7)].

Consider increasing the measurement parameter $\gamma$ from 0. As in
U(1)-symmetric circuits [51], measurements gap out some inter-replica degrees
of freedom. Furthermore, the inter-replica gapless modes reduce to one
diffusive mode (corresponding to the fluctuations in the replicas’ average)
and $Q-1$ relativistic ballistic ($z{=}1$) modes, which describe inter-replica
fluctuations. These ballistic modes cause the Rényi entropies with indices
$n>1$ to grow ballistically in the presence of measurements [21, 51]. The
diffusive inter-replica modes are well-defined only for symmetry-broken states
whose forward and backward copies are paired explicitly. Thus, these modes are
expected to survive only in the replica-symmetry-broken phase (volume-law
phase). However, the $2Q$ intrachain-fluctuation SU(2) modes do not depend on
such pairings. Hence these modes are expected to exist at all measurement
strengths and so in the critical phase. As we discuss below, these surviving
gapless $z{=}2$ modes likely underlie two circuit behaviors that we observed:
the $L^{2}$ purification time scale and the absence of area-law entanglement.

###  V B Purification

Using the formalism above, we can understand the purification of an initially
maximally mixed state, $|\otimes^{L}_{i=1}e\rrangle$. The $e\in S_{Q}$ denotes
a permutation that pairs replica $a$ with $a^{*}$. At a late time $t$, the
density matrix’s trajectory-averaged purity $\Pi(t)$ is given by

$\displaystyle\Pi(t)=\lim_{Q\rightarrow
1}\frac{\llangle\otimes^{L}_{i=1}g|e^{-\beta
H^{\mathrm{eff}}}|\otimes^{L}_{i=1}e\rrangle}{\llangle\otimes^{L}_{i=1}e|e^{-\beta
H^{\mathrm{eff}}}|\otimes^{L}_{i=1}e\rrangle}\,.$ (8)

$g$ denotes the transposition that swaps replica 1 with 2* and 2 with 1* while
acting as the identity on the other replicas [10, 11]. The purification time
is when $\Pi(t)$ becomes $\mathcal{O}(1)$. In the absence of measurements,
$\gamma=0$ an initially maximally mixed state will fail to purify and will
have $\Pi(t)=\frac{1}{2^{L}}$ for all times. Indeed, at $\gamma=0$,
$|\otimes^{L}_{i=1}e\rrangle$ is a ground state of $H^{\mathrm{eff}}$ and has
vanishing energy. Thus, $\Pi(t)|_{\gamma=0}=\lim_{Q\to
1}\llangle{\otimes^{L}_{i=1}g}|\otimes^{L}_{i=1}e\rrangle=1/2^{L}$.
Excitations of the discrete-symmetry-broken phase are gapped domain-wall
configurations. Therefore, we expect this phase to be stable under the
strengthening of the weak measurements to low rates $\gamma$. For the system
to transition between different replica-symmetry-broken ground states, a
domain wall must tunnel across the entire system. Such transitions are thus
expected occur over an exponential-in-system-size time, which we identify as
the purification time: $t_{\rm P}\sim{\rm e}^{L}$. The SU(2) symmetry has
little bearing on the replica-symmetry breaking; essentially identical
arguments apply in the symmetry’s absence. This behavior contrasts with that
of lattice magnets that have continuous symmetries. There, a symmetry-broken
state can be deformed smoothly into another symmetry-broken state over a
$\mathrm{poly}(L)$ time scale. Interestingly, monitored free fermions have an
emergent, continuous inter-replica symmetry (as opposed to our discrete
$S_{Q}$ symmetry), resulting in linear purification times [52, 53, 54, 55].

The replica symmetry is restored at sufficiently high measurement strengths;
the argument for $t_{\rm P}\sim{\rm e}^{L}$ breaks down. Instead, the
purification time depends on the effective Hamiltonian’s energy gap. We
conjecture that this gap scales as $1/L^{2}$, due to the gapless modes
associated with the previous subsection’s $2Q$ $z{=}2$ modes. This gap scaling
results in a purification time $t_{\rm{P}}\sim L^{2}$.

###  V C Absence of area law under strong measurements

We can establish the absence of an area-law phase at any measurement rate by
adapting a Lieb-Shultz-Mattis-type-anomaly argument to the spin model in the
replica trick with $2Q$ copies, as first argued in [34]. (See also [56, 57],
which generalize this result to statistical symmetries.) Each of the $2Q$
copies has SU(2) symmetry, and the replica symmetry permutes the replicas.
Additionally, under averaging over the measurements and circuit elements, the
replica model has a $\mathbb{Z}$ lattice-translation symmetry. Overall, the
statistical-mechanics model’s symmetry group is $G=\mathbb{Z}\times\left[({\rm
SU}(2)^{\times Q}\rtimes S_{Q})\times({\rm SU}(2)^{\times Q}\rtimes
S_{Q})\rtimes\mathbb{Z}_{2}\right]$ [52]. Each site contains one (projective)
spin-$1/2$ representation of each replica factor of SU(2). Therefore, there is
a mixed anomaly between translation symmetry and the SU(2) spin-rotation
symmetry. This anomaly rules out the possibility of a featureless (short-
range-entangled, symmetry-preserving) ground state. Moreover, naïvely applying
the Mermin-Wagner theorem rules out spontaneous breaking of the SU(2) symmetry
(although subtle examples may violate this principle in the replica limit
[58]). Furthermore, we observe no tendency towards any spontaneous breaking of
the lattice-translation symmetry. These arguments suggest that no area-law
phase can arise, even in the measurement-only limit.

We can obtain further insight into the strong-measurement regime through our
mapping to an effective-Hamiltonian model. Each replica has an SU(2) symmetry,
which leads to gapless modes, as noted above, with ferromagnetic interactions
and thus $z{=}2$ dynamics. These conclusions are consistent with the critical
phase observed in our numerics.

Using the formalism above, one can calculate Rényi entropies of the reduced
density matrix of an interval A. Let $\rho$ denote any single-copy pure state;
and $|\rho\rrangle$, the $Q$-replica defined on $2Q$ copies of the Hilbert
space. We focus on the Rényi index $n=2$:

$\displaystyle e^{-S^{2}(\rho_{A})}=\lim_{Q\rightarrow
1}\frac{\llangle\otimes_{i=1}^{L}g^{A}_{i}|e^{-\beta
H^{\mathrm{eff}}}|\rho\rrangle}{\llangle\otimes_{i=1}^{L}e|e^{-\beta
H^{\mathrm{eff}}}|\rho\rrangle}\,.$ (9)

$g^{A}_{i}=e$ if $i$ does not belong to the interval $A$,
$g^{A}_{i}=(12)(34)...(2k{-}1\ 2k)$ if $i\in A$, and $Q=2k+1$. We define a
“twist” permutation $\tau$ such that
$\tau\left(\otimes_{i=1}^{L}g^{A}_{i}\right)=\otimes_{i=1}^{L}e$. Using
$\tau$, we can rewrite (9). Since the initial state is pure,
$\tau|\rho\rrangle=|\rho\rrangle$, and

$\displaystyle e^{-S^{2}(\rho_{A})}=\lim_{Q\rightarrow
1}\frac{\llangle\otimes_{i=1}^{L}e|\tau^{-1}e^{-\beta
H^{\mathrm{eff}}}\tau|\rho\rrangle}{\llangle\otimes_{i=1}^{L}e|e^{-\beta
H^{\mathrm{eff}}}|\rho\rrangle}\,.$ (10)

$\tau$, operating on $H^{\mathrm{eff}}$, introduces a twist operator at the
interval’s boundary, $\partial A$:
$\tau^{-1}e^{-\beta{H^{\mathrm{eff}}}}\tau\equiv T_{\partial A}\ e^{-\beta
H^{\mathrm{eff}}}$. Hence a size-$|A|$ interval’s Rényi-2 entropy is related
to the two-point correlator of the twist operator acting on sites separated by
a distance $|A|$. In the infrequent-measurement phase with spontaneously
broken replica symmetry, this correlator decays exponentially, leading to
volume-law Rényi entropies. Intuitively, domain walls in this discrete
ferromagnetic phase have finite line tensions. Hence creating a domain wall
costs an extensive (volume-law) amount of free energy. Under frequent
measurements, in the putative critical phase, the permutation degrees of
freedom are gapped. The twist operator should likelly, instead, couple to the
remaining low-energy SU(2) modes. Analyzing the critical phase’s nature,
requiring the effective Hamiltonian’s replica limit, presents a clear
challenge for future work.

##  VI Outlook

We studied the dynamics of monitored random circuits with SU(2) symmetry,
i.e., with three noncommuting charges: the total spin angular momentum’s
components. First, we numerically discovered a purification transition between
a mixed phase (at $p<p_{\rm c}\approx 0.35$) and a critical phase (at
$p>p_{\rm c}$). In the critical phase, the purification time scales as $t_{\rm
P}\sim L^{2}$. The purification transition doubles as an entanglement
transition, which separates volume-law (at $p<p_{\rm c}$) and subextensive
(logarithmic or small-power-law, at $p>p_{\rm c}$) entanglement scalings. Even
in the measurement-only limit (at $p=1$), the symmetry’s non-Abelian nature
enables nontrivial entanglement scaling. Additionally, we observed a spin-
sharpening transition across which there is a parametric change in the time at
which one can (in principle) learn the system’s total spin by monitoring
measurements.

Finally, we interpreted our results within an effective replica statistical-
mechanics model. The model supports the mixed-phase prediction that $t_{\rm
P}\sim e^{L}$. Also, the model also hints at a possible spin-wave mechanism
for the $t_{\rm P}\sim L^{2}$ dynamics in the critical phase. Furthermore, a
Lieb-Schultz-Mattis-type anomaly obstruction implies the absence of an area-
law phase. Instead, the entanglement should scale logarithmically with $L$ in
the critical phase, consistently with our numerics.

Our results open several opportunities for future work. One is to understand
the purification/entanglement and sharpening transitions analytically. Second,
one might leverage spin sharpening to observe an MIPT experimentally, avoiding
the postselection problem (Sec. I). The thermodynamics of noncommuting charges
have already been observed experimentally with trapped ions [42].
Superconducting qubits, quantum dots, and spinful fermionic atoms are natural
candidates, too [43, 44]. Finally, efficient classical and quantum spin-
sharpening decoders merit exploration.

###### Acknowledgements.

We thank Ehud Altman, Fergus Barratt, David Huse, Andreas Ludwig, and
Xiaoliang Qi for helpful discussions. We also thank Michael Gullans for
galvanizing this collaboration. This work received support from the National
Science Foundation under QLCI grant OMA-2120757 (N.Y.H.), the John Templeton
Foundation under award No. 62422 (N.Y.H.), the Air Force Office of Scientific
Research under Grant No. FA9550-21-1-0123 (R.V.), the Alfred P. Sloan
Foundation through Sloan Research Fellowships (A.C.P. and R.V.), National
Science Foundation under NSF Grants No. DMR-1653271 (S.G.), and the Vanier
C.G.S. (S.M.). This work was supported by the Simons Collaboration on Ultra-
Quantum Matter, which is a grant from the Simons Foundation (651440, U.A.)

##  Appendix A Additional numerics elucidating the entanglement dynamics and
spin sharpening

In Sec. III B, we claimed that $L^{2}$ time steps suffice for the bipartite
entanglement entropy, $S_{\rm f}$, to plateau. Figure 6 justifies this claim,
presenting $S_{\rm f}$ as a function of $\log(t)$ for $\leq L^{2}$ time steps
at the extreme values $p=0,1$. At both extrema, $S_{\rm f}$ stops changing (to
within minor fluctuations) by $L^{2}$ time steps.

Figure 6: The bipartite entanglement entropy saturates after $L^{2}$ time
steps. At the extreme $p$ values $p=0,1$, $S_{\rm f}$ quits changing (to
within minor fluctuations).

Section III B also discussed different fittings for $S_{\rm f}$ versus $L$.
Figure 7 presents three fittings [$L$, $\log(L)$, and $\sqrt{L}$] at each of
three measurement rates ($p=0$, $p=1$, and $p\approx p_{\rm c}$). At $p=0$,
the linear fit is the best. This observation is consistent with the existence
of a volume-law phase at $p=0$. At $p=0.35\approx p_{\rm c}$, it is unclear
which fit is most accurate. However, the two nonlinear fits are visibly best.
The $p=1$ fits resemble the $p=0.35$ ones.

Figure 7: Long-time bipartite entanglement entropy vs. system size. At $p=0$,
$S_{\rm f}\sim L$, signaling a volume law. At $p=1$, the entropy scales
logarithmically or as a small power law: $S_{\rm f}\sim\log(L)$, or $S_{\rm
f}\sim\sqrt{L}$.

Section IV claimed that our $p>p_{\\#}$ data are compatible with a sharpening
time scale $\sim L^{2}$ deep in the critical phase. Figure 8 justifies this
claim. We plot $\log(S_{A})$ against $t/L^{2}$ at various $p$ values. The
initial collapse occurs at $p>p_{\\#}$. The $L{=}8$ numerics deviate from the
collapse when $p\in[0.35,0.45]\cup[0.8,1]$. We suspect that these deviations
arise from finite-size effects.

Figure 8: In the critical phase, the numerics are consistent with a $\sim
L^{2}$ sharpening time scale. The entropy $S_{A}$ quantifies the ancilla
qubit’s entanglement with the system. We plot $\log(S_{A})$ for clarity, as
$S_{A}$ decays exponentially. $t/L^{2}$ runs along the $x$-axis to demonstrate
the numerics are consistent with a $\sim L^{2}$ sharpening time scale. We used
30 000 samples when $L=8$ to $14$; and 10 000 samples when $L=16$ to $L=18$.
The $y$-axis’s lower limit is $\log(10^{-3})\approx-6.91$.

##  Appendix B Mutual information

In Sec. III, we studied purification and entanglement dynamics. We complement
that numerical analysis by studying mutual information. To introduce the
mutual information, we consider a quantum system in a state $\ket{\psi}$. Let
$\mathcal{A}$ and $\mathcal{B}$ denote subsystems. The reduced state of
$\mathcal{A}$ is
$\rho_{\mathcal{A}}\coloneqq\tr_{\bar{\mathcal{A}}}(\outerproduct{\psi}{\psi})$.
The reduced states of $\mathcal{B}$ and $\mathcal{A}\mathcal{B}$ are defined
analogously. The mutual information between $\mathcal{A}$ and $\mathcal{B}$ is

$I(\mathcal{A}:\mathcal{B})\coloneqq
S(\rho_{\mathcal{A}})+S(\rho_{\mathcal{B}})-S(\rho_{\mathcal{AB}}).$ (B1)

The mutual information upper-bounds equal-time correlators between local
operators acting nontrivially on $A$ alone and on $B$ alone [59]. We denote by
$I^{(1)}_{j,k}$ the mutual information between sites $j$ and $k$. We denote by
$I^{(2)}_{j,k}$ the mutual information between the pair $(j,j+1)$ and the pair
$(k,k+1)$.

Figure 9: Mutual information between sites. The mutual information between (a)
sites $1$ and $j$ decays more quickly than between (b) sites $(1,2)$ and
$(j,j+1)$. The inset highlights how $I^{(2)}_{j,k}$ increases and then
decreases as $p$ grows, for some $j$. The errorbars represent one standard
deviation.

Figure 9 presents $I^{(1)}_{1,j}$ and $I^{(2)}_{1,j}$, plotted against $j$, at
$L=20$. $I^{(1)}_{1,j}$ grows with $p$ and rapidly decays with
$j$.555Throughout these numerics, the last layer of gates was applied on the
odd bonds (sites 1 and 2, sites 3 and 4, etc.), leading to larger
$I^{(1)}_{j,j+1}$ for odd $j$ than even $j$. At all $p$, $I^{(1)}_{j,k}$
decays rapidly over distances $|j-k|$ larger than a few sites. This result is
intuitive, since $I^{(1)}_{j,k}$ contains information about correlations
between individual spin components, whereas the measurements and unitaries
correlate spin-fusion channels (a property of two or more spins).

In comparison, $I^{(2)}_{j,k}$ decays more gradually with the distance $|j-k|$
at all $p>0$. For particular sites $j$ and $k$, $I^{(2)}_{j,k}$ may depend on
$p$ nonmonotonically (Fig. 9). However, the asymptotic decay rate
monotonically decreases as $p$ decreases. To explore this decay rate, we
examine the mutual information between antipodal pairs of sites:
$I^{(2)}_{1,L/2}$ (Fig. 10a). Given the limitations on system size, we cannot
convincingly determine the asymptotic decay’s functional form. A power-law
decay fits the data reasonably well (Fig. 10b). The fitted power $a$ gradually
decreases with $p$. Furthermore, $a$ changes qualitatively around $p_{\rm
c}=0.35$—from changing quickly with $p$, at $p<p_{\rm c}$, to drifting slowly
near $-2$, at $p>p_{\rm c}$. Given the small range of system sizes available,
exponential decay fits the data reasonably well, too; we cannot rule out this
behavior. Yet, given the other critical scaling behavior at $p>p_{\rm c}$, we
expect that power-law decay to be more natural in this regime. The data also
prohibit confident distinction between (i) one power at $p>p_{\rm c}$, with
drifts in the fitted exponent, due to finite-size corrections, and (ii)
continuously evolving power laws (as would arise in, say, a Luttinger liquid).

Figure 10: Mutual information at antipodal sites. We call sites $1$ and $L/2$
antipodal. (a) $\log(I^{(2)}_{1,L/2})$ is plotted against $\log(L)$ at several
$L$ values. Using the fit function $\log(I^{(2)}_{1,L/2})=a\log(L)+b$, we
identify the critical exponent $a$ in $I^{(2)}_{1,L/2}\sim L^{a}$. (b)
Plotting $a$ against $p$, we find that $I^{(2)}_{1,L/2}$ decays as a power law
in both phases, where $a$ seems to be drifting.

##  Appendix C Effective Hamiltonian

In this appendix, we map the monitored dynamics onto the imaginary-time
evolution of a replica effective Hamiltonian. Since the two-site gates are
sampled independently, the above average over $U$ factorizes over averages of
two-site SU(2)-symmetric gates. We parameterize the SU(2)-symmetric gates as

$\displaystyle U_{i,j}=P_{t}+e^{i\theta}P_{s}=e^{i\theta P_{s}}.$ (C1)

$P_{t}$ and $P_{s}$ denote projectors onto the triplet and singlet sectors of
the SU(2) symmetry on two qubits. $i$ and $j$ denote the qubits being acted
on. We will suppress the subscript $i$ and $j$ unless they are needed to avoid
confusion. This is similar Eq. (1), modulo a global phase factor. The variable
$\theta$ is sampled from a random distribution—for example, from a uniform
distribution between $0$ and $2\pi$. The $Q$ moment of the unitary gates is

$\displaystyle{\cal U}\equiv U^{\otimes
Q}\otimes\left(U^{\dagger}\right)^{\otimes
Q}=\mathrm{e}^{i\sum_{a}\theta\left(P^{(a)}_{s}-P^{(a^{*})}_{s}\right)}\,.$
(C2)

$P_{s}^{(a)}$ denotes the projector onto the singlet sector for replica index
$a$. The $*$ symbol implies that the operator operates on the conjugate
(backward) copy of the $a$ replica (with $a=1,2,\dots,Q$). We assume that
$\theta$ is sampled from a Gaussian distribution
$P(\theta)=\frac{1}{\sqrt{2\pi J}}e^{-\theta^{2}/(2J)}$, where $J$ is a large
constant controlling the unitary dynamics’ scrambling strength. Performing
average over $\theta$ yields

$\displaystyle\int d\theta\,P(\theta)\,{\cal U}$
$\displaystyle=\mathrm{exp}\left(-J\left\\{\sum_{a=1}^{Q}\left[P^{(a)}_{s}-P^{(a^{*})}_{s}\right]\right\\}^{2}\right)$
(C3) $\displaystyle\equiv\mathrm{exp}(-H^{\rm u}_{ij}).$

To model measurements in the continuum-time limit, we consider the weak-
measurement protocol of [51]. The action of the measurement of a local
operator $O$ on the replica density matrix is described as

$\displaystyle\sum_{m}P_{m}^{\otimes Q}\rho^{(Q)}P_{m}^{\otimes Q}$ (C4)
$\displaystyle\rightarrow\int
dm\,e^{-\gamma\sum_{a=1}^{Q}\left[(O^{a}-m)^{2}+(O^{a^{*}}-m)^{2}\right]}\rho^{(Q)}$
$\displaystyle=$ $\displaystyle\
\mathrm{exp}\left(-\gamma\sum_{a,b=1}^{2Q}O^{a}\Pi_{ab}O^{b}\right)\rho^{(Q)}$
(C5) $\displaystyle\equiv$ $\displaystyle\ \mathrm{exp}(-\gamma H^{\rm
m})\rho^{(Q)}.$

$m$ denotes the weak measurement outcome, and $\Pi_{a,b}=\delta_{a,b}-1/(2Q)$.
We have identified $(a^{*})$ with index $(Q+a)$ and $a=1,2,\dots,Q$. From now,
on this identification will be implicit whenever the replica index $a$ is
summed from $1$ to $2Q$. As in the main text, $\gamma$ denotes the weak-
measurement strength. For SU(2)-symmetric systems, we measure the operators
$O=\vec{S}_{i}\cdot\vec{S}_{j}$.

The expressions above are for averages of single 2-site unitary gates and
measurements. We combine these local averages and assume a Trotter
decomposition. The monitored evolution of the density matrix’s averaged
$Q^{\rm th}$ moment is given by imaginary-time evolution under an effective
Hamiltonian $H^{\mathrm{eff}}$:
$\overline{\rho^{(Q)}(t)}=e^{-tH^{\mathrm{eff}}}\rho_{0}^{(Q)}$. The effective
Hamiltonian decomposes as $H^{\mathrm{eff}}=\sum_{i}\left(H^{\rm
u}_{i,i+1}+H^{\rm m}_{i,i+1}\right)$, with

$\displaystyle H_{i,j}^{\rm
m}=\gamma\sum_{a,b}\left(\vec{S}^{a}_{i}\cdot\vec{S}^{a}_{j}\right)\Pi_{a,b}\left(\vec{S}^{b}_{i}\cdot\vec{S}^{b}_{j}\right).$
(C6)

The long-time properties of $\overline{\rho^{(Q)}(t)}$ are thus described by
low-temperature/ground-state properties of $H^{\mathrm{eff}}$. The Hamiltonian
has a $S_{Q}\times S_{Q}$ symmetry, corresponding to the global permutation
among the $Q$ forward replicas and $Q$ backward replicas.

A more illuminating way of understanding the structure of the Hamiltonian’s
ground states is to combine spin-$1/2$ particles at replica $a$ and
$\sigma(a)$, to form a fundamental representation of SU(4). The Hamiltonian’s
unitary part, in this identification, can be written as [the label $\sigma$
signifies that we have combined replicas $\bm{(}a,\sigma(a)\bm{)}$ to form an
SU(4) representation] $H^{\rm u}=\frac{J}{2}\left(H_{0}[\sigma]+V^{\rm
u}[\sigma]\right)$. The $H_{0}[\sigma]$ are $Q$ copies of the SU(4)
ferromagnet, and

$\displaystyle V^{\rm u}[\sigma]=$
$\displaystyle\sum_{i}\sum_{a<b}\left(\mathrm{Sw}^{a}_{i,i+1}-\mathrm{Sw}_{i,i+1}^{\sigma(a)*}\right)\left(\mathrm{Sw}^{b}_{i,i+1}-\mathrm{Sw}_{i,i+1}^{\sigma(b)^{*}}\right).$
(C7)

$\mathrm{Sw}_{i,j}^{a}$ denotes the SWAP operator between the spins at sites
$i$ and $j$ in replica index $a$:
$\mathrm{Sw}_{i,j}=1/2+2\vec{S}_{i}\cdot\vec{S}_{j}$. In terms of these SWAP
operators, the SU(4) ferromagnet is

$\displaystyle H_{0}[\sigma]=$
$\displaystyle\sum_{a=1}^{Q}\sum_{i}\left[1-\Big{(}\mathrm{Sw}^{a}_{i,i+1}\Big{)}\left(\mathrm{Sw}_{i,i+1}^{\sigma(a)*}\right)\right].$
(C8)

The measurement part of Hamiltonian also decomposes into two terms. One part
is the SU(4) ferromagnet, $H^{\rm
m}=\frac{\gamma}{Q}H_{0}[\sigma]-\frac{\gamma}{2Q}V^{\rm m}[\sigma]$, where

$\displaystyle V^{\rm m}[\sigma]=\sum_{i}\sum_{a\neq
b}\left(\mathrm{Sw}^{a}_{i,i+1}+\mathrm{Sw}^{\sigma(a)*}_{i,i+1}\right)\left(\mathrm{Sw}^{b}_{i,i+1}+\mathrm{Sw}^{\sigma(b)^{*}}_{i,i+1}\right).$
(C9)

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