of 28
PHYSICAL REVIEW B
110
, 155150 (2024)
Spontaneous strong symmetry breaking in open systems: Purification perspective
Pablo Sala
,
1
,
2
Sarang Gopalakrishnan
,
3
Masaki Oshikawa
,
4
and Yizhi You
5
1
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology
,
Pasadena, California 91125, USA
2
Walter Burke Institute for Theoretical Physics,
California Institute of Technology
, Pasadena, California 91125, USA
3
Department of Electrical and Computer Engineering,
Princeton University
, Princeton, New Jersey 08544, USA
4
Institute for Solid State Physics,
University of Tokyo
, Kashiwa, Chiba 277-8581, Japan
5
Department of Physics,
Northeastern University
, Boston, Massachusetts, 02115, USA
(Received 13 May 2024; revised 5 September 2024; accepted 30 September 2024; published 23 October 2024)
We explore the effect of decoherence on many-body mixed states from a purification perspective. Here,
quantum channels map to unitary transformations acting on a purified state defined within an extended Hilbert
space. First, we exploit this approach to analyze the phenomenon of spontaneous strong-to-weak symmetry
breaking (sw-SSB) triggered by strongly symmetric local quantum channels. We find that sw-SSB for mixed
states relates to proximity to a symmetry-protected topological (SPT) order in the purified state. Remarkably, the
measurement-induced long-range order in the purified SPT state, as characterized by the Rényi-2 correlator,
mirrors the ensuing long-range order in the mixed state due to sw-SSB. Moreover, we establish a 1-to-1
correspondence between various sw-SSB order parameters for the mixed state, and strange correlators in the
purification, which signify the SPT order. This purification perspective is further extended to explore intrinsic
mixed-state topological order and decoherent symmetry-protected topological phases.
DOI:
10.1103/PhysRevB.110.155150
I. INTRODUCTION
Classifying the ground states of gapped Hamiltonians is
one of the landmark achievements of many-body physics
[
1
3
]. A central idea behind this classification is that two
states are in the same phase if they are related by a finite-
depth local unitary (FDLU) quantum circuit—or equivalently
by a finite-time evolution under a local Hamiltonian [
2
,
4
8
]. Restricting the class of allowed evolutions, e.g., by im-
posing symmetries, generalizes this concept to encompass
“symmetry-protected” topological phases [
9
15
]. The Lieb-
Robinson theorem guarantees that two states that are related
by an FDLU circuit have the same asymptotic correlations and
entanglement structure at a large distance, and the same power
to encode quantum information [
8
,
16
22
]. The properties of a
quantum state that make it valuable for quantum information
processing tasks—e.g., the presence of non-Abelian anyons—
are robust properties that are present throughout a phase of
matter, and do not depend on fine-tuning the Hamiltonian
[
23
].
The control and manipulation of these highly entan-
gled states enable the preparation of resource states for
measurement-based quantum computing (MBQC) [
24
26
],
where measurements on bulk qubits of a resource state fa-
cilitate universal quantum computation. In recent years, there
has been a surge of progress in simulating quantum states
of matter with nontrivial entanglement on platforms sum-
marized as the noisy intermediate-scale quantum (NISQ)
technology [
27
], including simulating exotic quantum many-
body states such as topological order, spin liquids, conformal
field theory quantum critical points [
28
] and symmetry-
protected topological (SPT) states [
29
36
]. A quantum state
interacting with an environment can be understood as being
continuously
measured
by it, eventually becoming entangled
with the environment’s degrees of freedom. If the environ-
mental qubits are inaccessible or the measurement outcomes
are lost, then this effectively leads to the tracing out of the
environment’s qubits, transforming the original quantum state
into a mixed-state ensemble. It is natural to ask if mixed states
can be classified into distinct phases, separated, e.g., by their
ability to protect quantum information [
37
40
]. The threshold
theorem for quantum error correction guarantees the existence
of nontrivial mixed-state phases (such as the toric code subject
to weak enough noise), which preserve quantum coherence
[
26
,
41
43
]. However, the tools to characterize these phases
[
37
,
38
,
44
54
], as well as the transitions between them, re-
main largely undeveloped, despite some very recent progress
[
50
,
55
65
].
The example of the noisy toric code [
41
,
45
,
66
] illustrates
a fundamental difference between mixed-state and pure-state
phases. Suppose we start with one of the logical ground states
of the pristine toric code, and then subject it to local noise at
some rate
γ
, without performing any active error correction.
At sufficiently short times, the mixed state generated this
way will remain correctable, but at some
finite
time
t
(
γ
),
the density of errors will exceed the capacity of even the
optimal decoder to correct [
41
]: thus, logical information
will become irretrievable at some finite time. Such finite-
time transitions have no obvious pure-state analog. However,
they seem quite generic in the context of systems subject to
noise and
/
or measurements: a conceptually related finite-time
teleportation transition has been predicted in random quan-
tum circuits in two or more dimensions. These observations
challenge our conventional understanding of what a phase of
matter is in open systems [
39
,
50
,
55
,
56
,
58
,
60
,
61
,
63
,
67
72
],
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©2024 American Physical Society
SALA, GOPALAKRISHNAN, OSHIKAWA, AND YOU
PHYSICAL REVIEW B
110
, 155150 (2024)
and what properties are universal: in the pure-state context
it is assumed that finite-time evolution cannot change uni-
versal properties of a phase, but whether a mixed state is
error-correctable seems like a fundamental property. Even at a
technical level, the Lieb-Robinson bounds that ensure equiva-
lence of two pure states connected by a short-depth circuit also
apply to mixed states, and ensure that “conventional” diag-
nostics of order, such as correlation functions, cannot diverge
at mixed-state phase transitions. Thus, recently proposed di-
agnostics of mixed-state order include observables that are
nonlinear in the density matrix [
26
,
43
,
66
,
70
], metrics based
on whether the density matrix can be written as an ensemble
of trivial-phase pure states [
31
,
38
], and quantities such as
the conditional mutual information [
73
76
] (and the related
notion of measurement-induced entanglement [
40
,
77
]). How-
ever, the physical interpretation of these diagnostics, and their
relation to concepts of pure-state order, remains opaque.
The noisy toric code is an example where an ordered
phase appears to
lose
long-range correlation under a finite-
depth local quantum channel. The reverse phenomenon can
also occur, and has been termed “spontaneous strong sym-
metry breaking” (sw-SSB) [
43
,
55
] where the decoherence
effect triggers long-range ordering in mixed states. sw-SSB
occurs in product states [
78
] subject to finite-depth local
channels, so from the point of view of conventional correla-
tion functions a mixed state possessing sw-SSB will appear
trivial. Meanwhile, observables that are nonlinear in the den-
sity matrix, such as the Rényi-2 correlator and quantum
relative entropy, behave singularly [
43
,
46
,
61
]. Intuitively, an
open system is said to possess a strong symmetry when the
system-environment interaction does not exchange symme-
try charges—e.g., an open system of electrons coupled to
phonons has a strong fermion parity symmetry. A pertinent
question is why spontaneous strong symmetry breaking can be
triggered by decoherence and how to characterize such unique
long-range order in an open quantum system.
The goal of this work is to characterize spontaneous sw-
SSB from the perspective of purification, specifically by
considering the mixed-state density matrix as a partial trace of
a pure state in an extended Hilbert space. From this viewpoint,
quantum channels acting on the mixed-state ensemble are
equivalent to unitary operators acting on the purified state.
Our central finding is that mixed-state long-range order and
spontaneous strong-to-weak symmetry breaking, induced by
finite-depth quantum channels, can be mapped to SPT or-
der in the purified framework. This approach highlights why
mixed-state long-range order can manifest within a finite time:
an SPT phase transition from a trivial phase is achievable
through a symmetry-breaking finite-depth circuit. Particularly,
we demonstrate that sw-SSB in a mixed state implies that
its purification exhibits a nonvanishing strange correlator, a
hallmark of an SPT wave function. We link these strange
correlators to previously discussed observables, including
the fidelity correlator and type-2 strange correlator. Further-
more, we establish a correspondence between the mixed-state
Rényi-2 correlator and measurement-induced order in the cor-
responding purified state.
This work is organized as follows: In Sec.
III
,weintro-
duce strong-to-weak
Z
2
symmetry breaking in mixed states
triggered by local quantum channels in both one-dimensional
FIG. 1. Summary of results. Mapping between exotic mixed-
state phenomena and its purification description.
(1D) and two-dimensional (2D). These states can be mapped
to a SPT phase within the purification framework. We trace
the Rényi-2 correlator of sw-SSB back to the measurement-
induced long-range order observed in the purified SPT state.
Additionally, we find that while sw-SSB is unstable in 1D
when subject to a finite measurement rate quantum channel,
it becomes significantly more stable up to a certain threshold
in 2D and higher dimensions. In Sec.
IV
, the correspondence
between the averaged strange correlator in the purified state
and various sw-SSB observables in the mixed state is demon-
strated. In Sec.
V
, we broaden the scope of sw-SSB to include
a wider array of symmetry groups, such as higher-form sym-
metry, subsystem symmetry, and continuous groups. Finally,
in Secs.
VI
to
VII
, we explore mixed-state topological orders
and SPT phases from a purification perspective. Figure
1
summarizes the main results of this work in a comprehensive
table.
II. GENERAL FORMALISM
A. Mixed states, quantum channels, and purifications
We begin with a many-body pure state
|
ψ

, defined on
a Hilbert space of the system
H
. Under open-quantum dy-
namics, the system is coupled to an environment with Hilbert
space
A
, which (without loss of generality) can be assumed
to be initialized in a fixed pure state
|
0

. We will fol-
low the standard practice of referring to these environment
qubits (or more generally qudits) as ancillae. The global
evolution in
H
A
is then a unitary map
U
, such that
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U
(
|
ψ
⊗|
0

)
=|

ρ
∈
H
A
. To then describe dynamics of
the accessible degrees of freedom in the system, one traces
out
A
, arriving at the density matrix ˆ
ρ
D
Tr
A
(
|

ρ


ρ
|
).
We mainly focus on the cases in which the initial state
|
ψ

is a product state, the system and ancilla are defined on a
geometrically local space, and the operator
U
can be written
as a finite-depth, geometrically local unitary circuit.
The process we described above defines a local quantum
channel, i.e., a map
E
between density matrices on
H
.The
perspective where the channel is treated as a unitary in a
larger Hilbert space is called the Stinespring form of the
channel. The pure state
|

ρ
∈
H
A
is called a purification
of ˆ
ρ
D
. Stinespring forms and purifications are not unique,
as ˆ
ρ
D
is invariant if one replaces the map
U
above with a
map (
I
H
W
A
)
U
, where
W
A
is an arbitrary unitary that acts
only on the environment
A
[
79
]. Quantum channels take the
general form ˆ
ρ
D
=
E
ρ
)
=

m
K
m
ˆ
ρ
0
K
m
, where
{
K
m
}
is a set
of “Kraus operators” satisfying the trace-preserving condition

m
K
m
K
m
=
I
H
. Here the number of Kraus operators match
the local dimension of the ancillary space. As we saw before,
to derive the Kraus form from the Stinespring form, one traces
out over the ancillary space
A
, such that
K
m
=
0
|
U
|
m

with
{|
m
}
an orthonormal basis in
A
. When the Stinespring form
of a channel involves a local unitary
U
as specified above, all
the Kraus operators can be chosen to be local.
B. Weak and strong symmetries
Recall that for a pure state, a symmetric wave function
implies that the quantum state carries a conserved charge
with respect to the symmetry
G
, thereby rendering it invariant
under the symmetry transformation
U
g
,
U
g
|

=
e
i
θ
(
g
)
|


,
(1)
with
e
i
θ
(
g
)
being a global phase for all
g
G
For a mixed state, there are two notions of symmetry—
“weak” and “strong.” The concept of weak and strong
symmetries was first introduced for channels (or equivalently,
their continuous-time version, Lindblad master equations)
[
51
,
80
]. Similarly, as we will discuss below, mixed-state
density matrices can be symmetric under either type of sym-
metry transformation. We first explain them in the purification
(Stinespring) picture. In this picture, a weak symmetry exists
when the system-environment unitary
U
, and the initial state
of the environment
|
0

, are invariant under a symmetry trans-
formation that acts on
H
A
. Thus, for example, if a system
exchanges particles with the ancilla but the total number of
particles in the purified state defined on
H
A
is conserved,
then that corresponds to a weak
U
(1) symmetry. However,
a strong symmetry requires that the symmetry operator acts
only on
H
. That is, when the system interacts with the en-
vironment, there is no charge exchange between the system
and the ancilla. When a channel
E
is invariant under a strong
symmetry, it means that each Kraus operator commutes with
the symmetry operation, so [
K
m
,
U
g
]
=
0 for all
g
G
.
Instead of regarding these symmetries as properties of
quantum channels, we can instead treat them as properties
of a
mixed state
ˆ
ρ
, in particular, one that was arrived at by
applying
E
to a product state invariant under
G
[
43
,
49
,
55
,
61
].
The constraints on ˆ
ρ
are inherited from those on
E
. A weak
symmetry requires that the density matrix remains invariant
under the symmetry transformation
U
g
, acting on both the left
(ket) and right (bra) parts of the density matrix:
ˆ
ρ
=
U
g
ˆ
ρ
U
g
.
(2)
The weak symmetry transformation can be interpreted as
implementing the symmetry operation on both the ket and
bra spaces of the density matrix. Physically, it requires that
the density matrix be block-diagonal, with each block cor-
responding to a different charge under
G
. This is intuitive
since the system can exchange charge with the bath; however,
as the charge is conserved for the system and ancilla as a
whole, the reduced density matrix of the system can still be
block-diagonal in different charge sectors. For the special case
where the density matrix is the partition function in thermal
equilibrium ˆ
ρ
=
e
β
H
, the invariance of ˆ
ρ
under the weak
symmetry
G
implies that the Hamiltonian is
G
-symmetric.
Density matrices with strong symmetry, however, are those
with a
definite
value of the charge. When a quantum channel
has a strong symmetry, it preserves the invariance of density
matrices under the
U
g
symmetry transformation which acts
exclusively on either the left or the right part of the density
matrix:
e
i
θ
ˆ
ρ
=
U
g
ˆ
ρ,
(3)
with
e
i
θ
being a global phase for all
g
G
. If we diago-
nalize the density matrix as ˆ
ρ
=


p

|



|
, the strong
symmetry condition requires that all eigenvectors of the den-
sity matrix also be eigenstates of the symmetry
G
, such that
U
g
|

=
e
i
θ
|


, each carrying the same symmetry charge.
C. Strong-to-weak symmetry breaking (sw-SSB)
driven by finite-depth quantum channels
We begin by reviewing a concrete example of spontaneous
Z
2
sw-SSB, driven by local quantum channels, initially in-
troduced in Refs. [
43
,
50
,
81
]. Before proceeding, we briefly
revisit the correlation functions that can signal symmetry
breaking in thermal equilibrium ensembles. For canonical
ensembles ˆ
ρ
=
e
β
H
of systems with conserved charge
G
,
both the Hamiltonian and the thermal density matrix exhibit
invariance under the weak symmetry,
H
=
U
g
HU
g
,
ˆ
ρ
=
U
g
ˆ
ρ
U
g
.
(4)
With the symmetry operator acting from both the left and the
right sides of ˆ
ρ
, the spontaneous breaking of weak symme-
tries [
82
] is evidenced by the nonvanishing two-point function
Tr[ ˆ
ρ
O
x
O
y
] for a given charged operator
O
x
.
Now we move on to the strong symmetry condition. For
any mixed-state density matrix invariant under the strong
symmetry
U
g
, the charged operator
O
x
acting on the doubled
density matrix should vanish [
83
]:
Tr(
O
x
ˆ
ρ
O
x
ˆ
ρ
)
Tr( ˆ
ρ
2
)
=
0
.
(5)
Notably, while
O
x
ˆ
ρ
O
x
is “charged” under this strong sym-
metry, it remains neutral under the weak one. Prompted by
this observation, to define spontaneous sw-SSB, we aim to
identify correlation functions of operators charged under the
strong symmetry but neutral under the weak symmetry.
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A nonvanishing correlation function that can signal sw-
SSB is given by four-point functions:
C
II
(
x
,
y
)
tr(
O
x
O
y
ˆ
ρ
O
y
O
x
ˆ
ρ
)
tr(ˆ
ρ
2
)
.
(6)
For some charged operators
O
x
and
O
y
with
|
x
y
|→
,Eq.(
6
), also referred to as the Rényi-2 correlator in
Refs. [
43
,
55
] or the Edwards-Anderson correlator, is widely
used in systems with localization to characterize symme-
try breaking within disordered ensembles. As an immediate
sanity check, we observe that such strong-to-weak SSB is
distinctive in mixed states since, for pure states, the Rényi-2
correlator is merely the square of the conventional correlator.
The Rényi-2 correlator can be interpreted as the correla-
tion function for the paired of operators
O
x
and
O
x
, which
simultaneously “act” on the ket and bra spaces, namely, they
multiply the density matrix from both sides, thereby creat-
ing a strong symmetry charge at site
x
. References [
43
,
55
]
presented a straightforward illustration of sw-SSB by analyz-
ing the mixed state ˆ
ρ
+
=
(
I
+

i
X
i
)
/
2
L
ona1Dspin-1
/
2
chain of length
L
, which demonstrates a strong
Z
2
symmetry
generated by
X

i
X
i
. Here,
X
i
denotes the Pauli-
X
oper-
ator at each site
i
. To reach this mixed state through local
quantum channels, consider an initial state with all qubits
polarized in the
S
x
direction, expressed as
|
φ
0
=⊗
i
|→
i
.
To initiate the strong-to-weak symmetry breaking, one imple-
ments a quantum channel ˆ
ρ
0
ˆ
ρ
+
=
E
ρ
0
] with
E
=

i
E
i
that “measures” the spin bilinear term
Z
i
Z
i
+
1
at every bond
(
i
,
i
+
1) as
E
i
ρ
0
]
=
1
2
ˆ
ρ
0
+
1
2
Z
i
Z
i
+
1
ˆ
ρ
0
Z
i
Z
i
+
1
.
(7)
The resulting mixed state ˆ
ρ
+
spontaneously breaks a strong
Z
2
symmetry to a weak
Z
2
symmetry, evidenced by the non-
vanishing correlation outlined in Eq. (
6
):
Tr
(
Z
0
Z
i
ˆ
ρ
+
Z
0
Z
i
ˆ
ρ
+
)
Tr( ˆ
ρ
2
+
)
=
1
.
(8)
When expressed in the
Z
basis, ˆ
ρ
+
is essentially a convex sum
of GHZ states:
ˆ
ρ
+

s
(
|
s
+
X
|
s

)(

s
|+
s
|
X
)
,
(9)
where
s
is a bit string in the
Z
basis.
III. MIXED-STATE SW-SSB FROM SPT PURIFICATION
In this section, we explore the strong-to-weak symmetry
breaking in mixed states triggered by local quantum channels,
from the perspective of purification. Just as all mixed states
can be viewed as subsystems of a pure state (denoted as the
purification state
) in an extended Hilbert space, local quan-
tum channels can equivalently be described by local unitary
operators acting on the purification state. While the combined
ancilla and system qubits remain in a pure state after the
unitary operations, the system’s density matrix, ˆ
ρ
, obtained
by tracing out the ancilla, generally represents a mixed state.
As the initial purified state (encompassing both the system
and the ancilla) lacks long-range order, its quantum correla-
tions remain short-ranged within the Lieb-Robinson bound
after applying finite-depth local unitaries. Consequently, we
FIG. 2. One-dimensional sw-SSB and purification. (a) The uni-
tary operator entangles the system (blue) with the ancilla (red). The
unitary gates consist of a cluster of three-body gates, each acting on
two adjacent system qubits and one ancilla situated between them.
(b) The unitaries fix the total
S
z
parity of the two system spins on
each link and the intervening ancilla spin, ensuring
Z
i
̃
Z
i
Z
i
+
1
=
1. (c)
Such a unitary in the extended Hilbert space is equivalent to quantum
channels that measure
Z
i
Z
i
+
1
on adjacent spins.
do not expect the system’s density matrix, after tracing out
the ancilla, to exhibit any long-range order measurable by
physical observables linear in ˆ
ρ
. Meanwhile, Refs. [
43
,
55
,
81
],
as summarized in Eq. (
9
), provides a straightforward yet illus-
trative example of how a local quantum channel could induce
strong symmetry breaking, exemplified by the Rényi-2 corre-
lator in Eq. (
8
). A pertinent question arises: If the mixed state
ˆ
ρ
undergoes spontaneous strong-to-weak symmetry breaking
induced by quantum channels, then what happens to its pu-
rification states under unitary evolution? Additionally, what
characteristics must the purification state possess to allow
sw-SSB and long-range Rényi-2 correlation for the system as
a mixed state, after tracing out the ancilla?
A. One-dimensional sw-SSB: Purification from SPT state
To set the stage, we focus on the purified wave function
within the extended Hilbert space, which encompasses both
system and ancilla qubits. Although we will eventually trace
out the ancilla qubits to obtain the mixed-state density matrix,
this approach aims to explore the sw-SSB of mixed states from
the perspective of purification and reveal the correspondence
between conditional long-range correlation in the purified
state and the long-range order manifested via the Rényi-2
correlator of the mixed state.
The system of interest encompasses a 1D spin system,
initialized as
|
φ
0
=⊗
i
|→
i
on the sites, along with a chain
of ancillae that resides on the links between the system’s spins,
as depicted in Fig.
2
. The initial state of the ancillae is a tensor
product of spins, polarized in the
S
z
direction, expressed as
|
φ
A
0
=⊗

i
,
j

|
̃
↑
i
,
j
. The unitary operator
U
has the following
form:
U
=

i
U
i
,
i
+
1
,
U
i
,
i
+
1
=
(1
+
Z
i
Z
i
+
1
)
2
̃
I
i
,
i
+
1
+
(1
Z
i
Z
i
+
1
)
2
i
̃
Y
i
,
i
+
1
.
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The unitary operator involves a cluster of three-body gates,
each acting on two adjacent system qubits and one ancilla
situated between them, as illustrated in Fig.
2
. Here,
Z
i
refers
to the operator acting on the system qubits, while
̃
Y
i
,
i
+
1
acts
on the ancilla. Such a unitary operation exhibits a
decorated
domain wall
feature. When the system spins at the (
i
,
i
+
1)
link are aligned parallel in the
z
direction, the (
i
,
i
+
1) an-
cilla on the link remains in the
|
̃
↑
i
,
i
+
1
state. Conversely, if
the system spins at the (
i
,
i
+
1) link are opposite in the
z
direction, then the ancilla at (
i
,
i
+
1) is flipped to the
|
̃
↓
i
,
i
+
1
state. After applying the local unitary, the resultant wave
function in the extended Hilbert space precisely reflects a SPT
state,
|

SPT
=
U

sites
i
|→
i

bonds (
i
,
i
+
1)
|
̃
↑
(
i
,
i
+
1)
,
(11)
which corresponds to the ground state of a stabilizer Hamilto-
nian,
H
=−

i
(
Z
i
̃
Z
i
,
i
+
1
Z
i
+
1
+
̃
X
i
1
,
i
X
i
̃
X
i
,
i
+
1
)
.
(12)
This Hamiltonian comprises two stabilizer operators. The
Z
i
̃
Z
i
,
i
+
1
Z
i
+
1
stabilizer ensures that the
S
z
parity in the three-
body clusters is even, while the
̃
X
i
1
,
1
X
i
̃
X
i
,
i
+
1
stabilizer
generates resonance between distinct spin patterns that share
the same
S
z
parity across all clusters. The stabilizer Hamil-
tonian, along with the wave function, respects the following
symmetry:
Z
A
2
:

i
̃
Z
i
,
i
+
1
,
Z
S
2
:

i
X
i
.
(13)
The
Z
A
2
symmetry acts on the ancilla, while
Z
S
2
acts on the
system. Remarkably, the unitary operator applied in Eq. (
10
)
preserves the
Z
S
2
symmetry while breaking the
Z
A
2
symmetry.
This outcome is expected, as no local unitary path can connect
a tensor product state with an SPT state while preserving
both symmetries. While the SPT wave function in Eq. (
11
)
is short-range correlated, it carries hidden quantum correla-
tions that can be characterized by a nonvanishing string-order
parameter:

O
s
=

SPT
|
Z
i

n

a
=
0
̃
Z
i
+
a
,
i
+
1
+
a

Z
i
+
n
+
1
|

SPT
=
1
.
(14)
This string order reveals the conditional mutual information
shared between the system and the ancilla [
84
86
]. When
the ancilla qubits are fixed in a specific charge sector (e.g.,
̃
Z
i
+
a
,
i
+
1
+
a
=
1), long-range mutual information between the
system qubits is established. Hence, two-point correlations
among the system spins, denoted as

Z
i
Z
i
+
n
+
1

, are deter-
mined by the
S
z
parity charge of the ancilla on the string

n
a
=
0
(
̃
Z
i
+
a
,
i
+
1
+
a
) in between. Therefore, if we project all an-
cillae into the
|
̃
↑
state, then the system would transition into a
spontaneously symmetry-breaking “cat state” with long-range
order (LRO) for the two-point correlation

Z
i
Z
i
+
n
+
1
=
1.
Once we trace out the ancilla and obtain the system’s
density matrix ˆ
ρ
=
Tr
ancilla
|

SPT


SPT
|
, it indeed reproduces
the mixed state introduced in Eq. (
9
). This process averages
out the even versus odd
S
z
charge of the ancilla on the string.
Consequently, the resultant two-point correlation vanishes,
as indicated by Tr[ˆ
ρ
Z
i
Z
i
+
n
+
1
]
=

SPT
|
Z
i
Z
i
+
n
+
1
|

SPT
=
0.
However, if we examine the Rényi-2 correlator introduced
in Eq. (
8
), based on two copies of the density ma-
trix, it remains nonvanishing at large distances indicating
sw-SSB.
To provide a physical interpretation of the Rényi-2 corre-
lator, we duplicate our entire Hilbert space by creating two
identical copies of the SPT state, denoted
|

SPT

1
and
|

SPT

2
.
Next, we perform a Schmidt decomposition between the an-
cilla and the system for each copy of the SPT wave function
as
|

SPT

1
=

α
λ
α
|
α

1
s
|
̃
α

1
a
,
|

SPT

2
=

α
λ
α
|
α

2
s
|
̃
α

2
a
.
(15)
Here,
|
α

s
denotes the Schmidt basis for the system, while
|
̃
α

a
refers to the Schmidt basis for the ancilla degrees of
freedom. These two identical copies,
|

SPT

1
⊗|

SPT

2
, carry
a nonvanishing string order, which is essentially the product of
the strings for each copy:
O
1
s
O
2
s
=
Z
1
i
Z
2
i

n

a
=
0
̃
Z
1
i
+
a
,
i
+
1
+
a
̃
Z
2
i
+
a
,
i
+
1
+
a

Z
1
i
+
n
+
1
Z
2
i
+
n
+
1
.
(16)
We now project ancilla from the first and second copy onto a
symmetric EPR pair, hence forcing their alignment in the
S
z
direction:
ˆ
P
i
,
i
+
1
=
1
2
(
|
̃
1
̃
2
+|
̃
1
̃
2

)(

̃
1
̃
2
|+
̃
1
̃
2
|
)
(
i
,
i
+
1)
.
(17)
This projection is implemented for every ancilla pair at bond
(
i
,
i
+
1). The normalized wave function, after this projection,
is given by
|


pp

i
ˆ
P
i
,
i
+
1


α
λ
α
|
α

1
s
|
̃
α

1
a


α
λ
α
|
α

2
s
|
̃
α

2
a

→|


pp
=
1


α
|
λ
α
|
4

α
|
λ
α
|
2
|
α

1
s
|
α

2
s
.
(18)
In the last step, we omit the ancilla qubits as they form a tensor
product of EPR pairs between two copies and are decoupled
to the system qubits. The projection
ˆ
P
i
,
i
+
1
on the two an-
cilla copies results in the charge string (

n
a
=
0
̃
Z
1
i
+
a
̃
Z
2
i
+
a
) being
uniformly even throughout the postprojected wave function.
Consequently, the postprojection state
|


pp
exhibits long-
range order in the four-point correlation function:


|
pp
Z
1
i
Z
2
i
Z
1
i
+
n
+
1
Z
2
i
+
n
+
1
|


pp
=
1
.
(19)
Alternatively, we could trace out the ancilla from the
purified state in Eq. (
11
) obtaining the density matrix ˆ
ρ
=

α
|
λ
α
|
2
|
α

α
|
. Tracing out the ancilla effectively involves
enforcing the ancilla in both the ket and bra spaces being
identical. By interpreting the bra as a duplicate copy and
applying the Choi-Jamiokowski mapping to ˆ
ρ
such that ˆ
ρ

α
|
λ
α
|
2
|
α
|
α

, the ancilla tracing procedure becomes anal-
ogous to the projection operation in Eq. (
18
). Drawing from
this analogy, the Rényi-2 correlator precisely corresponds to
the four-point correlation of the postprojected wave function
155150-5
SALA, GOPALAKRISHNAN, OSHIKAWA, AND YOU
PHYSICAL REVIEW B
110
, 155150 (2024)
FIG. 3. Rényi-II correlator and EPR-induced long range order.
(a) Taking two copies of the SPT state
|

SPT

1
⊗|

SPT

2
and pro-
jecting the ancilla (indicated by red squares) from both copies onto a
symmetric EPR pair, the postprojection state of the system (indicated
by blue squares) displays a long-range correlation characterized by
the four-point correlator. (b) The tensor representation of the Rényi-2
correlator measures the four-point correlator acting on the double-
density matrix. When we trace out the legs of the ancilla (tensor legs
contraction), it effectively projects the ancilla in the bra and ket space
onto the EPR pair.
(see Appendix
A
for additional details of the derivation):
Tr
(
Z
0
Z
i
ˆ
ρ
Z
0
Z
i
ˆ
ρ
)
Tr( ˆ
ρ
2
)
=

|
pp
Z
1
0
Z
2
0
Z
1
i
Z
2
i
|


pp
.
(20)
This result demonstrates that the Rényi-2 correlator can be
interpreted as the correlation function of the postprojection
state
|


pp
. If we conceptualize the density matrix from the
tensor perspective as illustrated in Fig.
3
, then tracing over the
density matrix entails connecting the ancilla’s legs between
the ket and bra spaces. This process is analogous to taking two
copies of the SPT state
|

SPT

1
⊗|

SPT

2
and projecting the
ancilla from both copies onto a symmetric EPR pair. In this
context, we consider the second copy of the wave function
in its complex conjugate form to mirror the wave function
in the bra space. This correspondence implies that measuring
any operators on the duplicated density matrix is equivalent to
duplicating the purified state as
|

SPT

1
⊗|

SPT

2
, followed
by an EPR projective measurement to ensure the ancilla in
both copies are identical. While local unitary operators acting
on the system and ancilla (with its duplicate) cannot induce
long-range order (LRO), an additional projective measure-
ment can facilitate this enhancement. This phenomenon of
measurement-induced long-range order was originally pro-
posed in Refs. [
40
,
84
,
87
93
] as a shortcut to create the
long-range entangled state. The emergence of long-range or-
der in the Rényi-2 correlator is attributed to the nonvanishing
measurement-induced long-range order in its purified state.
By projecting the ancilla charges on the string into the even
sector, the postprojection state manifests long-range order.
B. General quantum channel in 1D
We now delve into a broader scenario of quantum channels
with a tunable error rate
p
. These can be interpreted as the
system being measured at a given rate but the outcomes of
the measurement not being recorded, effectively resulting in
dephasing noise. This type of decoherence channel can be
represented as follows:
ˆ
ρ
D
=
E
ρ
0
]
,
E
=

i
,
i
+
1
E
i
,
i
+
1
,
E
i
ρ
0
]
=
(1
p
ρ
0
+
pZ
i
Z
i
+
1
ˆ
ρ
0
Z
i
Z
i
+
1
.
(21)
The quantum channel
E
preserves the strong
Z
2
symmetry.
When
p
=
1
/
2, the decoherence channel becomes a pure mea-
surement channel, leading to the mixed state
ρ
+
introduced
in Eq. (
9
) that breaks the strong
Z
2
symmetry spontaneously.
A pertinent question arises: Can spontaneous strong-to-weak
symmetry breaking occur for
p
<
1
2
? More specifically, is it
possible to continuously change the error rate and trigger a
phase transition?
Our previous discussion highlighted that the long-range
order in the mixed state’s Rényi-2 correlator is inherited from
the measurement-induced long-range order of the purified
SPT state in the enlarged Hilbert space. Therefore, it would
be beneficial to examine the corresponding unitaries acting on
the enlarged Hilbert space, defined as follows:
U
(
θ
)
=

i
U
i
,
i
+
1
(
θ
)
,
U
i
,
i
+
1
(
θ
)
=
(1
+
Z
i
Z
i
+
1
)
2
(cos
θ
̃
I
i
,
i
+
1
+
i
sin
θ
̃
Y
i
,
i
+
1
)
+
(1
Z
i
Z
i
+
1
)
2
(sin
θ
̃
I
i
,
i
+
1
+
i
cos
θ
̃
Y
i
,
i
+
1
)
,
|

SPT
(
θ
)
=
U
(
θ
)

i
⊗|→
i
⊗|
̃
↑
i
,
i
+
1
.
(22)
When
θ
=
0, the gate simplifies to the unitary introduced in
Eq. (
10
), which corresponds to the measurement-only chan-
nel. At
θ
=
π/
4, the unitary only rotates the ancilla qubits,
leaving the system qubits untouched, ensuring that the system
remains in a pure state after tracing out the ancilla. Upon
applying
U
(
θ
), the purified state

SPT
(
θ
) in the extended
Hilbert space retains
Z
A
2
symmetry (act on the ancillae) in
Eq. (
13
) only when
θ
=
0. Returning to the quantum channel
perspective, after applying this unitary and tracing out the
ancilla, we return to the general quantum channel in Eq. (
21
),
with an error rate
p
=
1
sin(2
θ
)
2
.
For
θ
within the interval (0
,π/
4], the wave function

SPT
(
θ
) breaks
Z
A
2
symmetry, leading to the immediate dis-
appearance of the measurement-induced long-range order
[
84
,
91
]. As a result, the EPR-projected wave function defined
as Eq. (
19
) decays exponentially for finite values of
θ
, causing
the Rényi-2 correlator to exhibit only short-range correlations.
In Appendix
A1
, we demonstrate that the Rényi-2 correlator
for
p
<
1
/
2 maps to the correlation function of the 1D Ising
model at finite temperature, which lacks long-range order.
155150-6
SPONTANEOUS STRONG SYMMETRY BREAKING IN OPEN ...
PHYSICAL REVIEW B
110
, 155150 (2024)
FIG. 4. Two-dimensional sw-SSB and purification from 1-form
SPT. (a) The unitary gates comprise a cluster of three-body gates act-
ing on both the x-link and y-link. (b) Such a unitary in the extended
Hilbert space is tantamount to quantum channels that measure the
spin bilinears on the link.
Consequently, sw-SSB in 1D is fragile and can only occur
in pure measurement channels at
p
=
1
/
2. In our next sec-
tion, we will show that sw-SSB can be more robust in a 2D
quantum channel, with a phase transition occurring at a finite
measurement rate.
The correspondence between local quantum channels in
the mixed state and local unitary operations in the purified
state provides new insights into our exploration of sponta-
neous strong-to-weak symmetry breaking: (i) The presence
of mixed-state sw-SSB, detectable via the Rényi-2 correla-
tor, stems from the conditional mutual information shared
between the system and the ancillae in the SPT state [
77
]. This
raises a compelling question: Can all mixed-state long-range
orders be purified as an SPT wave function? (ii) If a quantum
channel induces an sw-SSB transition in a mixed state, then
what occurs in its purified state during the transition? In what
follows, we will adopt a comprehensive approach and demon-
strate how various examples of sw-SSB can be mapped to SPT
through purifications.
C. Two-dimensional sw-SSB: Purification from
1-form SPT state
We now proceed to investigate a concrete example of sw-
SSB on a 2D square lattice. As before, we begin our analysis
with the purified state defined in the extended Hilbert space.
The purified state comprises the system qubits located at the
vertices, initialized as
|
φ
0
=⊗
i
|→
i
, together with ancilla
qubits living on the links. The initial state of these ancillae
|
φ
A
0
=⊗|
̃
↑
, is a tensor product of spins polarized in the
S
z
direction. To entangle these components, we apply a set
of three-body unitaries on the links of the square lattice [see
Fig.
4(a)
] as follows:
U
=

i
U
x
i
U
y
i
,
U
x
i
=
(1
+
Z
i
Z
i
+
ˆ
x
)
2
̃
I
i
+
ˆ
x
2
+
(1
Z
i
Z
i
+
ˆ
x
)
2
i
̃
Y
i
+
ˆ
x
2
,
U
y
i
=
(1
+
Z
i
Z
i
+
ˆ
y
)
2
̃
I
i
+
ˆ
y
2
+
(1
Z
i
Z
i
+
ˆ
y
)
2
i
̃
Y
i
+
ˆ
y
2
,
|

SPT
=
U

i

|→
i
⊗|
̃
↑
i
+
ˆ
x
2
⊗|
̃
↑
i
+
ˆ
y
2

.
(23)
In this unitary,
Z
denotes the operator acting on the system
qubits at the vertex, while
̃
Y
operates on the ancilla located on
the links. The symbols ˆ
x
and ˆ
y
represent the unit vectors in the
square lattice. This three-body unitary operation enforces the
cluster of three spins on each link to have an even
S
z
parity:
Z
i
̃
Z
i
+
ˆ
y
2
Z
i
+
ˆ
y
=
1
,
Z
i
̃
Z
i
+
ˆ
x
2
Z
i
+
ˆ
x
=
1
.
(24)
The unitary gates in Eq. (
23
) preserve the
Z
2
symmetry of
the system qubits, denoted by
X
=

i
X
i
. After applying the
unitary, the resultant wave function
|

SPT

represents an SPT
state and is the ground state of the stabilizer Hamiltonian:
H
=−

b
̃
Z
b

v
e
b
Z
v

s
X
s

b
v
s
̃
X
b
.
(25)
Here
̃
Z
b
represents the ancilla qubit that resides on link
b
,
and
v
e
b
refers to the two vertex ends of the link where the
system qubits are located. In the second term,
X
s
acts on the
system qubits at the vertex, while
a
v
s
spans the four links
connecting to the vertex. This Hamiltonian is known as the
parent Hamiltonian for the higher-form SPT protected by a
1-form
Z
A
2
and a 0-form
Z
S
2
symmetry [
94
]:
Z
A
2
:

i
γ
̃
Z
i
,
Z
S
2
:

i
X
i
,
(26)
where
Z
A
2
represents the 1-form symmetry that acts on the
ancillae, with
γ
denoting any closed loop along the links.
Z
S
2
is the 0-form
Z
2
symmetry (global symmetry) acting on
the system qubits. Importantly, the unitary operator we apply
preserves the
Z
S
2
symmetry while simultaneously breaking the
Z
A
2
symmetry. This higher-form SPT state can be detected by
a nonvanishing string-order parameter [
26
,
42
,
94
]:

O
s
=

j
l
Z
j


i
l
̃
Z
i

.
(27)
Here,
l
represents an arbitary open string along the link, and
l
refers to the two endpoints of the string (on the vertex).
The string order contains a product of the ancilla qubits
̃
Z
i
along the string, decorated with two system qubits
Z
j
situated
at the two ends of the string [see Fig.
5(a)
]. The presence
of a nonvanishing string order indicates that the two distant
system qubits share conditional mutual information mediated
by the ancilla. The two-point correlation

Z
i
Z
j

is influenced
by the 1-form charge of the ancilla located on the open string
connecting them. As a result, projecting all ancillae into the
|
̃
↑
state leads the system to evolve into a cat state with
long-range order [
84
,
88
,
90
,
91
,
94
].
Upon tracing over the ancilla from
|

SPT

, we obtain a den-
sity matrix reminiscent of ˆ
ρ
+
=
(
I
+

i
X
i
)
/
2
L
. The quantum
channel corresponding to the unitaries in Eq. (
23
)is
ˆ
ρ
D
=
E
ρ
0
]
,
E
=

i
E
x
i
E
y
i
,
E
x
i
ρ
0
]
=
1
2
ˆ
ρ
0
+
1
2
Z
i
Z
i
+
ˆ
x
ˆ
ρ
0
Z
i
Z
i
+
ˆ
x
,
(28)
E
y
i
ρ
0
]
=
1
2
ˆ
ρ
0
+
1
2
Z
i
Z
i
+
ˆ
y
ˆ
ρ
0
Z
i
Z
i
+
ˆ
y
.
155150-7
SALA, GOPALAKRISHNAN, OSHIKAWA, AND YOU
PHYSICAL REVIEW B
110
, 155150 (2024)
FIG. 5. Rényi-II correlator and EPR-induced long range order
in 2D. (a) The 1-form SPT exhibits a string order, where the two-
point correlation (light blue dots)

Z
(
x
)
Z
(
y
)

of the system qubits is
influenced by the charge string of the ancilla (green string)

a
̃
Z
a
that interconnects them. (b) Taking two copies of the SPT state
|

SPT

1
⊗|

SPT

2
and projecting the ancilla from both copies onto
a symmetric EPR pair (illustrated by the red bond) results in long-
range order (LRO) in the four-point correlator (represented by light
blue dots).
This quantum channel preserves the strong
Z
2
symmetry,
defined as
X
=

i
X
i
, which indeed corresponds to the
Z
S
2
symmetry defined for the purified state in Eq. (
26
).
By tracing out the ancilla qubits on the links, the mixed-
state density matrix exhibits spontaneous breaking of the
strong
Z
2
symmetry that can be characterized by the Rényi-2
correlator, which operates on the doubled density matrix as
defined in Eq. (
8
). As discussed in the previous section, this
corresponds to the duplicate two-point correlation functions
evaluated on the EPR-projected state
|
ψ

pp
:
tr(
Z
i
Z
j
ˆ
ρ
Z
i
Z
j
ˆ
ρ
)
tr(ˆ
ρ
2
)
=

|
pp
Z
1
i
Z
2
i
Z
1
j
Z
2
j
|


pp
.
(29)
Now we consider the general decoherence channel with a
finite measurement rate:
ˆ
ρ
D
=
E
ρ
0
]
,
E
=

i
E
x
i
E
y
i
,
E
x
i
ρ
0
]
=
(1
p
ρ
0
+
pZ
i
Z
i
+
ˆ
x
ˆ
ρ
0
Z
i
Z
i
+
ˆ
x
,
(30)
E
y
i
ρ
0
]
=
(1
p
ρ
0
+
pZ
i
Z
i
+
ˆ
y
ˆ
ρ
0
Z
i
Z
i
+
ˆ
y
,
which maintains strong
Z
2
symmetry for any value of
p
.
Can spontaneous strong symmetry breaking now occur at
a finite rate
p
<
1
/
2? To investigate the impact of this general
quantum channel, we return to the corresponding unitary op-
erators acting on the enlarged Hilbert space. As in the 1D case
we define a set of three-body unitary gates that operate on the
links:
U
(
θ
)
=

i
U
x
i
(
θ
)
U
y
i
(
θ
)
,
U
x
i
(
θ
)
=
(1
+
Z
i
Z
i
+
ˆ
x
)
2

cos
θ
̃
I
i
+
ˆ
x
2
+
i
sin
θ
̃
Y
i
+
ˆ
x
2

+
(1
Z
i
Z
i
+
ˆ
x
)
2

sin
θ
̃
I
i
+
ˆ
x
2
+
i
cos
θ
̃
Y
i
+
ˆ
x
2

,
U
y
i
(
θ
)
=
(1
+
Z
i
Z
i
+
ˆ
y
)
2

cos
θ
̃
I
i
+
ˆ
y
2
+
i
sin
θ
̃
Y
i
+
ˆ
y
2

+
(1
Z
i
Z
i
+
ˆ
y
)
2

sin
θ
̃
I
i
+
ˆ
y
2
+
i
cos
θ
̃
Y
i
+
ˆ
y
2

,
|

SPT
(
θ
)
=
U
(
θ
)

i
|→
i
⊗|
̃
↑
i
+
ˆ
x
2
⊗|
̃
↑
i
+
ˆ
y
2
,
(31)
where
θ
controls the measurement rate with
p
=
1
sin(2
θ
)
2
.
When
θ
=
0, the gate simplifies to the unitary introduced in
Eq. (
23
), initiating a quantum channel of pure measurement
with
p
=
1
2
[see Fig.
4(b)
]. At
θ
=
π
4
, the unitary only rotates
the ancilla qubits, so the system and ancilla remain unentan-
gled.
Upon applying
U
(
θ
), the resulting wave function
|

SPT
(
θ
)

retains the
Z
A
2
1-form symmetry only at
θ
=
0.
For other values of
θ
, the wave function loses its 1-form
symmetry, causing the string order defined in Eq. (
17
)to
vanish [
94
]. Despite this, several significant characteristics
of the higher-form SPT persist even after the explicit sym-
metry breaking. As discussed in Refs. [
94
97
], higher-form
symmetries can manifest phenomena distinctly different from
those of conventional 0-form global symmetry. For example,
the ground-state degeneracy resulting from the spontaneous
symmetry breaking (SSB) of a 1-form symmetry (interpreted
as topological degeneracy) remains robust against the explicit
breaking of the 1-form symmetry. Similarly, if we weakly
break the 1-form symmetry in an SPT state, then the edge
mode continues to be gapless up to a certain threshold [
69
].
These unique attributes arise because 1-form symmetries,
though not present at the ultraviolet level, can emerge at the in-
frared level in a gapped system with a finite correlation length
[
98
]. In our subsequent discussion, we will show that the
purified state
|

SPT
(
θ
)

with weak 1-form symmetry breaking
(up to a certain threshold), exhibits a nonvanishing strange
correlator [
20
]. This strange correlator precisely captures the
mixed-state long-range order resulting from sw-SSB.
As is delineated in Appendix
A1
, the Rényi-2 correlator
is mapped to the thermal two-point spin-spin correlator of
the 2D classical Ising model [as per Eq. (
A15
)] at an inverse
temperature of 2
β
with tanh(
β
)
=
p
/
(1
p
). Consequently,
there exists an extended region above a critical error rate
p
c
=
1
2
(1

2
1)
0
.
178 [
38
,
45
] (equivalently, below
a critical
θ
c
with sin(2
θ
c
)
=

2
1), where the Rényi-
2 correlator remains finite. This represents a spontaneous
strong-to-weak symmetry-breaking transition triggered by lo-
cal quantum channels in 2D.
IV. AVERAGED STRANGE CORRELATOR IN
PURIFIED STATE
At this point, we have demonstrated that the mixed-state
density matrix with strong-to-weak symmetry breaking in-
duced by local quantum channels can be purified into an
SPT wave function. In this purification framework, the sys-
tem’s qubits are entangled with the ancilla through local
unitary gates. They exhibit conditional long-range mutual in-
formation, contingent on the ancilla’s projection (or relatedly,
conditional long-range order as measured by two-point corre-
lators). The Rényi-2 correlator, pertinent to sw-SSB, can be
155150-8