Charge uctuation and charge-resolved entanglement in a monitored quantum circuit with U1symmetry

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Charge ﬂuctuation and charge-resolved entanglement in a monitored quantum circuit

with U(1) symmetry

Hisanori Oshima1and Yohei Fuji1

1Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan

(Dated: January 31, 2023)

We study a (1+1)-dimensional quantum circuit consisting of Haar-random unitary gates and pro-

jective measurements, both of which conserve a total U(1) charge and thus have U(1) symmetry. In

addition to a measurement-induced entanglement transition between a volume-law and an area-law

entangled phase, we ﬁnd a phase transition between two phases characterized by bipartite charge

ﬂuctuation growing with the subsystem size or staying constant. At this charge-ﬂuctuation tran-

sition, steady-state quantities obtained by evolving an initial state with a deﬁnitive total charge

exhibit critical scaling behaviors akin to Tomonaga-Luttinger-liquid theory for equilibrium critical

quantum systems with U(1) symmetry, such as logarithmic scaling of bipartite charge ﬂuctuation,

power-law decay of charge correlation functions, and logarithmic scaling of charge-resolved entan-

glement whose coeﬃcient becomes a universal quadratic function in a ﬂux parameter. These critical

features, however, do not persist below the transition in contrast to a recent prediction based on

replica ﬁeld theory and mapping to a classical statistical mechanical model.

CONTENTS

I. Introduction 1

II. Model 2

III. Numerical results 4

A. Entanglement transition 4

1. Entanglement entropy 4

2. Two-site mutual information and squared

correlation functions 5

3. Bipartite and tripartite mutual

information 5

4. Scaling behaviors at entanglement

transition 6

B. Charge ﬂuctuation 7

1. Bipartite charge ﬂuctuation 7

2. TLL-like scaling behaviors 10

C. Charge-resolved entanglement 11

IV. Discussion 13

Acknowledgments 14

A. Inﬁnite-temperature average 14

B. Numerical results for other ﬁllings 16

1. ¯n= 1/4 ﬁlling 16

2. ¯n= 1/6 ﬁlling 18

C. Scaling analysis 18

References 20

I. INTRODUCTION

Quantum many-body systems evolved under repeated

measurements have recently been shown to harbor a rich

variety of phases and phase transitions that have no coun-

terparts in equilibrium [1,2]. The unitary dynamics for

typical thermalizing systems causes linear growth of bi-

partite entanglement entropy, whose saturation value af-

ter a long time scales extensively with the volume of the

subsystem. Projective measurements of local operators,

on the other hand, disentangle local degrees of freedom

from the rest of the system and suppress the growth

of the entanglement. The competition between unitary

time evolution and repeated measurements leads to an

entanglement transition between a volume-law and an

area-law entangled phase [3–5]. Surprisingly, the entan-

glement transition in (1+1) dimensions exhibits emer-

gent conformal invariance, akin to equilibrium critical

phenomena, which is not captured by physical quantities

linear in the density matrix but revealed in character-

istic behaviors of nonlinear quantities, such as logarith-

mic scaling of the entanglement entropy and algebraic

decays of squared correlation functions [6]. While such

measurement-induced phase transitions (MIPTs) have

been intensively studied in unitary-measurement-hybrid

quantum circuits [7–22], they are expected to occur in

a diverse range of monitored quantum systems, such as

measurement-only quantum circuits [23–27], free fermion

systems [28–33], interacting systems subject to continu-

ous monitoring [34–39], and long-range interacting sys-

tems [40–46].

However, universal properties of generic MIPTs re-

main to be well understood, except for certain models

that can be mapped to classical percolation problems

[47,48]. As indicated by the emergent conformal invari-

ance, the MIPTs appear to share common features with

equilibrium phase transitions, for which symmetry plays

an indispensable role in understanding their universal-

ity classes. In fact, hybrid quantum circuits with two

competing measurements that preserve a global Z2sym-

metry have been shown to possess two distinct area-law

phases separated by an entanglement transition [19,23]

and thus bear a strong resemblance with the Ising model

arXiv:2210.16009v3 [cond-mat.dis-nn] 30 Jan 2023

in equilibrium (see also Refs. [14,24,25,49–51] for re-

lated studies). It has also been argued that, although

measurement outcomes are intrinsically random, trans-

lation symmetry of the corresponding statistical ensem-

ble in combination with a global symmetry, such as an

SU(2) spin rotation symmetry, gives rise to super-area-

law entanglement [49], in analogy with the Lieb-Schultz-

Mattis theorem for ground states of quantum many-body

systems [52]. Furthermore, it has been shown that inter-

play between global symmetry and dynamically gener-

ated symmetry acting on a replica space leads to a variety

of exotic measurement-induced phases [53].

Recently, it has been predicted that hybrid quantum

circuits with U(1) symmetry, or equivalently particle-

number conservation, undergo a novel type of MIPT dis-

tinguished from the entanglement transition as the mea-

surement rate is increased [54–56]. For a hybrid quantum

circuit consisting of charged qubits and neutral qudits

with dlevels, mapping to a classical statistical mechan-

ical model, which becomes analytically tractable in the

limit of large d, has been employed to show the pres-

ence of charge-sharpening transition within the volume-

law phase of entanglement [54,55]. For the d= 1

case, which reduces to the monitored Haar-random cir-

cuit with U(1) symmetry, Ref. [54] has numerically shown

that the charge-sharpening transition can be dynami-

cally characterized when the initial state mixes diﬀer-

ent charge sectors; an ancilla probe or charge variance

can be used to quantify a time duration required for

the initial state to collapse into a single charge sector,

which grows linearly with the system size in a charge

fuzzy phase below the transition whereas sublinearly in

a charge sharp phase above the transition. In Ref. [55],

the statistical mechanical model in the d→ ∞ limit

has been studied by both numerical and ﬁeld-theoretical

approaches to show that the charge-sharpening tran-

sition is of Berezinskii-Kosterlitz-Thouless (BKT) type

and the charge fuzzy phase below the transition exhibits

critical steady-state properties described by Tomonaga-

Luttinger-liquid (TLL) theory.

In this paper, we numerically investigate TLL-like

critical phenomena emerging from the monitored Haar-

random circuit with U(1) symmetry. While this model

has already been studied in Ref. [54], we exclusively focus

on static, steady-state properties obtained by evolving

an initial state within a single charge sector at a given

ﬁlling fraction. Besides the entanglement transition be-

tween the volume-law and area-law phase, we identify an-

other phase transition, dubbed charge-ﬂuctuation transi-

tion, which separates two phases where bipartite charge

ﬂuctuation grows with the subsystem size below the tran-

sition whereas stays constant above the transition. In

the vicinity of the charge-ﬂuctuation transition, we ﬁnd

that bipartite charge ﬂuctuation, (unsquared) charge cor-

relation functions, and charge-resolved entanglement all

exhibit scaling behaviors peculiar to critical systems de-

scribed by TLL theory. While one may think that the

charge-ﬂuctuation transition coincides with the charge-

sharpening transition dynamically located in Ref. [54],

as the former also exists slightly below the entanglement

transition, we cannot ﬁnd clear signatures of the BKT-

type universality at the charge-ﬂuctuation transition or

an extended critical phase described by TLL theory be-

low the transition as predicted from mapping to the clas-

sical statistical mechanical model in Ref. [55]. Our re-

sults thus call for more careful studies on universal prop-

erties of measurement-induced criticality in the presence

of U(1) symmetry.

The rest of this paper is organized as follows. In

Sec. II, we describe our monitored quantum circuit with

U(1) symmetry and simulation protocol. In Sec. III, we

present our numerical results with particular focus on

the half-ﬁlling case and discuss the presence of an entan-

glement transition and a charge-ﬂuctuation transition.

We then analyze scaling properties of various steady-

state quantities at and below the transitions. Numeri-

cal results for other ﬁlling fractions are provided in Ap-

pendix B. We conclude in Sec. IV with discussions and

future directions.

II. MODEL

We consider a (1+1)-dimensional [(1+1)D] hybrid

quantum circuit consisting of local unitary gates and in-

terspersed local projective measurements, both of which

preserve a global U(1) symmetry [54], as schematically

shown in Fig. 1. The system is deﬁned on a one-

dimensional chain of qubits qi∈ {0,1}with the length L

where i∈[1, L] denotes the site index. We impose the

periodic boundary condition such that qL+1 ≡q1. We

introduce a charge operator niacting on a local Hilbert

space labeled by site ias ni|qii=qi|qii, which can be

written as

ni=Ii−Zi

2,(1)

in terms of the Pauli operator Ziand the identity oper-

ator Ii. We then deﬁne the total charge operator by

ntot =

i=1

ni,(2)

which is conserved during time evolution by the hybrid

quantum circuits. Since qican also be interpreted as the

local particle number for a boson, we use the terms of

charge and particle number interchangeably throughout

this paper.

The local unitary gate is a 4 ×4 unitary matrix acting

on two neighboring sites and is chosen to take the block-

diagonal form,

Ui,i+1 =



U1×1

U2×2

U1×1

,(3)

FIG. 1. Schematic picture of the hybrid quantum cir-

cuit. The horizontal and vertical axis correspond to a spa-

tial and temporal direction, respectively. An initial N´eel

state |1010 · · · 10iis evolved by two-site Haar-random unitary

gates preserving the U(1) symmetry (blue rectangles) and by

projective measurements of qubits in the Pauli Zbasis (red

squares).

with Un×nbeing an n×nunitary matrix, such that it

commutes with the total charge operator ntot in Eq. (2).

These unitary gates are arranged in a brick-wall fashion

in spacetime (see Fig. 1). Thus, in the absence of mea-

surements, a state |ψ(t)iat time tis evolved by



ψ(t)E= O

i∈St

Ui,i+1!|ψ(t)i,(4)

within a single time step. Here, St={1,3, ..., L −1}

for odd t > 0, and St={2,4, ..., L}for even t > 0.

At each time and for each link (i, i + 1), the unitary

matrices Un×nin Eq. (3) are independently drawn from

a Haar-random distribution, which can be generated by

following Ref. [57]: First, we create a random n×nma-

trix Msuch that each element follows a complex nor-

mal distribution (i.e., Mbelongs to the Ginibre ensem-

ble). We then apply the QR decomposition M=QR

to obtain a unitary matrix Qand an upper triangu-

lar matrix R. By multiplying the diagonal matrix Λ =

diag(R11/|R11|,··· , Rnn/|Rnn|) to Q, we ﬁnally obtain

Q0=QΛ whose distribution is given by the Haar measure

on U(n).

At every time step after applications of the local uni-

tary gates, each qubit is measured with probability p

in the Pauli Zbasis. The measurement outcome µ=

{+1,−1}for a state 

ψ(t)Eis obtained with the Born

probability

pµ=D˜

ψ(t)Pi,µ 

ψ(t)E,(5)

where we have deﬁned projectors onto the eigenstates of

Ziby

Pi,µ =Ii+µZi

2.(6)

According to the measurement outcome µ, the state is

updated after the measurement to be



ψ(t)E→

Pi,µ 

ψ(t)E



Pi,µ 

ψ(t)E



.(7)

When this process of local projective measurements runs

over all sites, the time evolution within a single time step

is completed and yields a state |ψ(t+ 1)i.

Since both unitary gates in Eq. (3) and projectors

in Eq. (6) commute with the total charge operator in

Eq. (2), if the initial state |ψ(0)iis an eigenstate of ntot

with eigenvalue N, the evolved state |ψ(t)iis kept an

eiganstate of ntot with the same eigenvalue. Indeed, we

only consider such initial states with ﬁxed Nin the fol-

lowing analysis. Speciﬁcally, for a given ﬁlling fraction

¯n=N/L with Ldivisible by N, we choose the initial

state to be a “N´eel state”, which is a product state formed

by alternating single |1i’s and 1/¯n−1 consecutive |0i’s:

|ψ(0)i=

N−1

n=0

Xn/¯n+1 |00 ···0i,(8)

where Xiis the Pauli Xoperator acting on a single qubit

as Xi|qii=|1−qii. For instance, we have |ψ(0)i=

|1010 ···10ifor ¯n= 1/2.

Starting from an initial pure state |ψ(0)i, we repeat

the above procedures of unitary evolution and projec-

tive measurement to obtain a pure state |ψ(t)iat time

t. Such a pure state is called the quantum trajectory

and speciﬁed by a given choice of unitary gates and mea-

surement positions and also by measurement outcomes.

Given the pure-state density matrix corresponding to a

trajectory ρ(t) = |ψ(t)ihψ(t)|at time t, any physical

quantity OA[ρ(t)], such as entanglement entropy or cor-

relation functions, supported on a spatial region A, is

computed after application of unitary gates and subse-

quent projective measurements. We then take an average

OA[ρ(t)] over diﬀerent trajectories, which are generated

for randomly drawn unitary gates and measurement po-

sitions and intrinsically random measurement outcomes.

We note that such a quantity averaged over diﬀerent tra-

jectories conditioned on measurement outcomes is gen-

erally diﬀerent from the unconditional average OA[ρ(t)]

calculated from a usually mixed, averaged density ma-

trix ρ(t); they coincide with each other only when OA(ρ)

is linear in ρ. In our circuit model, the averaged den-

sity matrix ρ(t) is expected to reach a unique, inﬁnite-

temperature mixed state ρ∝Iwithin a single charge

sector with total charge Nin the long-time limit t→ ∞,

irrespective of the measurement probability p. Thus,

MIPTs are revealed only in dynamics of the conditional

average of physical quantities OA[ρ(t)] nonlinear in ρ(t)

or, in other words, correlation among diﬀerent trajecto-

ries.

In addition to the average over trajectories as ex-

plained above, we also take a spatial average for phys-

ical quantities OA[ρ(t)]. Since our unitary gates are ar-

ranged in the brick-wall fashion, physical quantities av-

eraged over trajectories still exhibit even-odd eﬀects de-

pending on the choice of a region A. In order to suppress

this eﬀect, we further take an average of OA[ρ(t)] over

all translations of Afor each trajectory. Therefore, any

physical quantity OA[ρ(t)] shown in the following discus-

sions is the average over (i) translations of Aand (ii)

diﬀerent trajectories and is simply denoted by OA[ρ(t)]

hereafter.

III. NUMERICAL RESULTS

In this section, we show our numerical results for hy-

brid quantum circuits with a ﬁxed ﬁlling ¯n= 1/2. We

ﬁrst examine entanglement quantities to conﬁrm an en-

tanglement transition between an area-law and a volume-

law phase (Sec. III A). We then focus on charge ﬂuctu-

ation (Sec. III B) and charge-resolved entanglement en-

tropy (Sec. III C) to diagnose a charge-ﬂuctuation transi-

tion with TLL-like criticality peculiar to (1+1)D systems

with U(1) symmetry. We use the system sizes ranging

from L= 8 to 24 for entanglement quantities and those

from L= 8 to 26 for charge correlations. All physical

quantities shown in this section are averaged over 1000

trajectories. We have also performed similar numerical

analyses for ﬁlling fractions ¯n= 1/4 and ¯n= 1/6, whose

details are provided in Appendix B.

A. Entanglement transition

1. Entanglement entropy

We ﬁrst look at time evolution of the entanglement

entropy under bipartition of the system into a contigu-

ous region Aand its complement ¯

A. Given a pure-state

trajectory ρ(t) = |ψ(t)ihψ(t)|, the von Neumann entan-

glement entropy is deﬁned by

SA(t) = −TrA[ρA(t) ln ρA(t)],(9)

where ρA(t) is the reduced density matrix given by

ρA(t) = Tr ¯

A[ρ(t)]. In Fig. 2, we show time evolution

of the (trajectory averaged) von Neumann entropy un-

der a half cut (|A|=L/2) for L= 20 and for various

values of the measurement rate p. As the initial state

is a product state, the von Neumann entropy is initially

zero, but it grows in time by unitary dynamics and sat-

urates to a steady-state value for suﬃciently long time

0.5 1 2

t/L

SA(t)

p=0.0

p=0.04

p=0.08

p=0.12

p=0.16

p=0.2

p=0.24

p=0.28

FIG. 2. Time evolution of the von Neumann entanglement

entropy SA(t) for L= 20, |A|=L/2, and ¯n= 1/2. Error

bars indicate the standard errors on trajectory average.

2 4 6 8 10 12

|A|

p=0.0

p=0.02

p=0.04

p=0.06

p=0.08

p=0.1

p=0.12

p=0.14

p=0.16

p=0.18

p=0.2

p=0.22

p=0.24

p=0.26

p=0.28

p=0.3

FIG. 3. Steady-state value of the von Neumann entangle-

ment entropy SAas a function of the subsystem size |A|for

L= 24 and ¯n= 1/2.

t∼L. It is also clear that the von Neumann entropy

in the steady-state regime decreases as the measurement

rate pis increased.

In order to study the subsystem-size dependence of the

steady-state values of SA(t), we pick up t= 2L, which

is deep inside the steady-state regime for various values

of pand the system length L. Figure 3shows the von

Neumann entropy at t= 2Lfor L= 24 as functions of

the subsystem size |A|. When the measurement rate pis

suﬃciently small, SAincreases linearly in |A|and thus

exhibits a volume-law scaling. For p∼0.3, SAtakes a

constant value for large |A|and shows an area-law scal-

ing. We thus expect that a measurement-induced entan-

glement transition between a volume-law and an area-

law phase takes place at some p=pc, as observed in

(1 + 1)D monitored circuits with [54] or without U(1)

symmetry [3,4,6]. While the entanglement entropy is

expected to scale logarithmically with the subsystem size

due to emergent conformal invariance, directly resorting

to the scaling behavior of entanglement entropy does not

seem to be an accurate way for locating the transition

point p=pcfor small size systems. Instead, we con-

sider the two-site mutual information and the squares

of two-site correlations functions, whose peak positions

can be used as a rough indicator of the transition (see

Sec. III A 2). We also perform a scaling analysis for tri-

partite mutual information to more accurately estimate

pc(see Sec. III A 3).

In the rest of this section, we always focus on the

trajectory averages of steady-state quantities OA(t) at

t= 2L. We thus suppress the time dependence of OA(t)

and simply write it as OAhereafter.

2. Two-site mutual information and squared correlation

functions

We here focus on the von Neumann mutual information

between two subsystems Aand B, which is deﬁned by

I(A:B) = SA+SB−SA∪B,(10)

where SA,SB, and SA∪Bare the von Neumann entan-

glement entropies of the subsystem Aand Band their

disjoint union A∪B, respectively. The mutual infor-

mation gives an upper bound for correlation functions

through the inequality [58],

I(A:B)≥|hOAOBic|2

2kOAk2kOBk2.(11)

Here, OAand OBare arbitrary operators supported on

the subsystem Aand B, respectively, hOAOBicis the

connected correlation function,

hOAOBic= Tr(ρOAOB)−Tr(ρOA)Tr(ρOB),(12)

and kOkdenotes the operator norm of an operator O,

which is equivalent to the largest singular value of O.

Here, we focus on the mutual information and squared

correlation functions between two antipodal sites A={i}

and B={j}on a ring of the length L(i.e., |i−j|=L/2).

In Fig. 4(a), we plot the von Neumann mutual infor-

mation against the measurement rate pfor various sys-

tem sizes. It exhibits a broad peak around p∼0.15 as

observed in other monitored systems [6,34,43], indicat-

ing the presence of an entanglement transition; correla-

tions are enhanced by critical ﬂuctuation at the transi-

tion, whereas they diminish in the volume-law or area-law

phase. A similar peak can also be found for the square of

the connected correlation function for Pauli-Xoperators

between antipodal sites, hXiXji2

c, as shown in Fig. 4(b).

On the other hand, the squared correlation function for

charge operators, hninji2

c=hZiZji2

c/16, does not have a

peak for ﬁnite measurement rate p; it takes a maximum

at p= 0 and monotonically decreases with pas seen from

Fig. 4(c).

These qualitatively distinct behaviors of correlation

functions depending on the choice of Pauli operators

have also been observed in interacting boson systems

subject to continuous monitoring with charge conserva-

tion [34]. Such behaviors are not expected for hybrid

quantum circuits without symmetry and are indeed pe-

culiar to charge-conserving systems as studied here. In

the absence of measurements, a density matrix ρ(t) av-

eraged over random unitary gates reaches an inﬁnite-

temperature mixed state for suﬃciently long time. In

fact, each pure state |ψ(t)ievolved by application of ran-

dom unitary gates is already in a thermal pure state at

inﬁnite temperature, meaning that an expectation value

hψ(t)|O|ψ(t)iwell approximates the canonical ensemble

average of an operator Oat inﬁnite temperature. Evalu-

ated with respect to the inﬁnite-temperature mixed state

in a ﬁxed charge sector (N=L/2 in the present case), the

connected correlation function hXiXjicis zero whereas

hninjictakes a nonzero value,

hninjic=−1

4(L−1).(13)

It turns out that hninjichas a nonzero trajectory average

even in the presence of measurements, whose scaling be-

havior is studied in Sec. III B, whereas hXiXjicremains

zero. Thus, the trajectory average for the squared cor-

relation function hninji2

cis dominated by a nonzero con-

tribution from the inﬁnite-temperature value of hninjic,

leading to a monotonically decreasing behavior with a

peak at p= 0. In contrast, since the trajectory aver-

age of hXiXjicis zero, the trajectory average of hXiXji2

well captures a correlation among trajectories and peaks

around the entanglement transition. As detailed in Ap-

pendix A, the mutual information computed from corre-

lation functions for the inﬁnite-temperature mixed state

with total charge N=L/2 also has a ﬁnite value:

I(i:j) = 1

2ln 1−1

(L−1)2−1

2(L−1) ln 1−2

L,

(14)

which is in good agreement with a small peak at p=

0 for the numerically obtained mutual information in

Fig. 4(a).

3. Bipartite and tripartite mutual information

For a more accurate estimation of the transition point,

we can still use the von Neumann mutual information but

with a partition diﬀerent from that used in the previous

section. We here divide the system into four contigu-

ous subsystems A= [i, j), B= [j, k), C= [k, l), and

D= [l, i). For (1+1)D conformal ﬁeld theory (CFT),

the bipartite mutual information I(A:C) is associated

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Chargeuctuationandcharge-resolvedentanglementinamonitoredquantumcircuitwithU(1)symmetryHisanoriOshima1andYoheiFuji11DepartmentofAppliedPhysics,UniversityofTokyo,Tokyo113-8656,Japan(Dated:January31,2023)Westudya(1+1)-dimensionalquantumcircuitconsistingofHaar-randomunitarygatesandpro-jectivemeasuremen...

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Charge uctuation and charge-resolved entanglement in a monitored quantum circuit with U1symmetry

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