# Quantum criticality under decoherence or weak measurement

Jong Yeon Lee  Kavli Institute for Theoretical Physics, University of
California, Santa Barbara, CA 93106  Chao-Ming Jian  Department of Physics,
Cornell University, Ithaca, New York 14853  Cenke Xu  Department of Physics,
University of California, Santa Barbara, CA 93106

###### Abstract

Decoherence inevitably happens when a quantum state is exposed to its
environment, which can affect quantum critical points (QCP) in a nontrivial
way. As was pointed out in recent literature on $(1+1)d$ conformal field
theory (CFT) [1], the effect of weak measurement can be mathematically mapped
to the problem of boundary CFT. In this work, we focus on the $(2+1)d$ QCPs,
whose boundary and defect effects have attracted enormous theoretical and
numerical interests very recently. We focus on decoherence caused by weak
measurements with and without postselecting the measurement outcomes. Our main
results are: (1) for an $\mathrm{O}(N)$ Wilson-Fisher QCP under weak
measurement with postselection, an observer would in general observe two
different types of boundary/defect criticality with very different behaviors
from the well-known Wilson-Fisher fixed points; in particular, it is possible
to observe the recently proposed exotic “extraordinary-log” correlation. (2)
An extra quantum phase transition can be driven by decoherence, if we consider
quantities nonlinear with the decohered density matrix, such as the Renyi
entropy. We demonstrate the connection between this transition to the
information-theoretic transition driven by an error in the toric code model.
(3) When there is no postselection, though correlation functions between local
operators remain the same as the undecohered pure state, nonlocal operators
such as the “disorder operator” would have qualitatively distinct behaviors;
and we also show that the decoherence can lead to confinement.

##  I Introduction

When a quantum state is exposed to an environment, it is being constantly
probed and “measured”, forming entanglement with the degrees of freedom in the
environment. If one is ignorant of the environment and the measurement outcome
is lost, it amounts to tracing out the environment’s degrees of freedom and
the original pure quantum state becomes a mixed state, which results in the
loss of coherent quantum information. This process is referred to as quantum
decoherence, and it is the bridge between the quantum mechanics that governs
the microscopic nature, and our classical macroscopic world [2]. More
generally, if a quantum state is weakly measured and the measurement outcome
is still accessible, one can consider the effect of postselecting the
measurement outcome and study process that is more general than decoherence.
In recent years there has been a surge of progress in simulating quantum
states of matter with nontrivial entanglement on platforms summarized as the
noisy intermediate-scale quantum (NISQ) technology [3], including simulating
exotic quantum many-body states such as topological order, spin liquids, and
symmetry protected topological (SPT) states [4, 5, 6, 7, 8], which have long
been discussed in condensed-matter physics. In these platforms, decoherence
can happen due to various reasons, which motivated recent inspection of the
fate of SPT states under decoherence [9, 10, 11, 12] (related studies
motivated from other contexts were also conducted [13, 14]).

Quantum criticality represents another class of quantum many-body states with
peculiar and universal entanglement. Recently a class of $(1+1)d$ conformal
field theory (CFT), i.e., the Luttinger liquid under weak measurements has
been studied, and it was pointed out that the effect of weak measurements can
be mathematically mapped to the problem of the boundary of the CFT [1], which
is a subject that was studied extensively in the past [15]. The same trick was
used in the recent study of SPT states under decoherence [12], which exploited
the observation that the wave function of the SPT states can be mapped to the
partition function of the boundary states of the system after a space-time
rotation [16, 17].

In this work, we focus on quantum critical points in $(2+1)d$. The connection
between the decohered QCPs (or QCPs under weak measurement) in the bulk and
the boundary criticality still holds, but the boundary criticality of $(2+1)d$
QCPs is a subject that has only been carefully studied very recently, and it
has attracted enormous interests from both the theoretical and numerical
communities [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. It has been
understood since long back that there exists an ordinary boundary condition of
a $(2+1)d$ Wilson-Fisher QCP (or a $3d$ classical Wilson-Fisher critical
point), where the Landau order parameter $\phi$ has a scaling dimension
$\Delta^{b}_{\phi}>1$, which is far greater than the bulk scaling dimension of
the order parameter (which is slightly greater than $1/2$). Only recently it
has become clear that at the boundary of an $\mathrm{O}(N)$ Wilson-Fisher
critical point, in addition to the well-known ordinary boundary criticality,
there is a so-called “extraordinary-log” boundary criticality, meaning the
correlation function of the order parameter at the boundary reads

$\displaystyle\langle\phi({\bm{0}})\phi({\bm{x}})\rangle\sim\frac{1}{\left(\ln|{\bm{x}}|\right)^{q}},$
(1)

where $q$ depends on $N$. This peculiar scaling was proposed theoretically
[25, 26] and recently confirmed numerically in Monte Carlo simulations [28,
29].

Our current work will bridge these two directions that are under active
studying, and we will demonstrate that a $(2+1)d$ QCP under decoherence or
weak measurements naturally exhibits the peculiar boundary criticality studied
recently. This work is organized as follows:

In section II, we develop the general formalism of analyzing decohered quantum
critical states, including the general connection between the decohered or
weakly measured bulk state and the boundary/defect criticality.

In section III we discuss the $(2+1)d$ $\mathrm{O}(N)$ Wilson-Fisher quantum
critical points under decoherence or weak measurements, and in section III.1
we focus on the quantities linear with the density matrix, which can be
observed directly in experiments. We demonstrate that weak measurements
generally render the observed quantities rather different from the bulk QCP
(with certain postselection that preserves the $\mathrm{O}(N)$ symmetry in the
mixed state ensemble), and we may observe the exotic extraordinary-log
correlation mentioned above.

In section III.2 we discuss quantities nonlinear with the density matrix,
which reveals a lot more structures of the mixed state density matrix of the
critical state under decoherence. In particular, we discuss a quantum
information phase transition that can be diagnosed through the 2nd Renyi
entropy of the decohered system, along with other correlations defined in this
section.

Here we would like to point out an important difference between the physical
scenarios to be considered in section III.1 and III.2. In section III.1 we
discuss physics under weak measurements on quantities such as energy density,
and we allow a postselection on the measurement outcomes. Section III.2
considers quantities nonlinear with the density matrix where postselection is
not needed in this scenario; hence it can genuinely correspond to physics
under decoherence due to coupling to the environment.

In section III.4 we demonstrate that there is an explicit lattice model with
an information transition driven by the strength of decoherence (or weak
measurement) analogous to the decoherence-driven transition in section III.2.
We also show that the transition in this lattice model is dual to an
information transition in the toric code model, which is related to the error
threshold of the toric code [31].

If we forbid postselection, local correlation functions of the (locally)
decohered density matrix would remain largely unchanged from the undecohered
pure state. In recent years the nonlocal disorder operator has become a very
important diagnosis of quantum states of matter. In section IV we demonstrate
that the nonlocal disorder operator can still have qualitatively different
behavior from the undecohered pure state density matrix, even if there is no
postselection. In particular, we show that in several examples, decoherence
can lead to confinement.

Figure 1:  Euclidean spacetime diagram. The expectation value of operators
amounts to evaluating the path-integral in the imaginary space-time with a
defect inserted at the $d$-dimensional slab between $\tau=0^{+}$ and
$\tau=\beta^{-}$. Microscopically, the defect (non-trivial $S^{\textrm{int}}$)
is captured by the summation over Kraus operators for a local decoherence
channel. No postselection corresponds to $P$ being all possible labels of
$\\{m_{i}\\}$. Since $\sum_{m}K_{m}^{\dagger}K_{m}^{\vphantom{\dagger}}=1$, if
the inserted operator $O(\phi)$ is the product of local operators, its
expectation value (correlation function) would only acquire a constant
correction. With postselection, the set $P$ is constrained, and the effect of
the summation of $P$ corresponds to a non-trivial defect inserted at $\tau=0$
(or equivalently $\tau=\beta$), and connections to recently studied
boundary/defect criticality of $(2+1)d$ QCP can be made.

##  II general formalism

In order to discuss quantum states under decoherence, one approach is to first
explicitly derive the ground-state wave function $|\Psi\rangle$ in either
exactly soluble lattice models or effective field theories, then construct the
density matrix $\hat{\rho}^{D}$ under decoherence [12]. This approach relies
on an explicit derivation of the ground-state wave function. The ground-state
wave function can be derived for gapped states such as SPT phases [9, 16, 17],
and also gapless phases with a Gaussian (free boson) Lagrangian, such as the
$(1+1)d$ Luttinger liquid [1], or the Rokhsar-Kivelson (RK) point in $(2+1)d$
models [32]. But for general interacting theories, such as systems near the
Wilson-Fisher quantum critical points, deriving the ground-state wave function
is cumbersome. In this section, we follow a more general procedure to study
interacting systems under decoherence.

Let us prepare an interacting quantum state which is the ground state of a
Hamiltonian, whose pure state density matrix is given as $\hat{\rho}_{0}$.
After the quantum state is prepared, we turn off the Hamiltonian and expose
the system to _local_ decoherence. For example, for a lattice model of qubits,
a decohered density matrix may be represented as

$\displaystyle\hat{\rho}^{D}=\mathcal{E}[\hat{\rho}_{0}],\ \
\mathcal{E}=\prod_{{\bm{x}}}{\cal E}_{{\bm{x}}},$ $\displaystyle{\cal
E}_{{\bm{x}}}[\hat{\rho}_{0}]=(1-p)\hat{\rho}_{0}+pZ_{{\bm{x}}}\hat{\rho}_{0}Z_{{\bm{x}}}.$
(2)

where $\hat{\rho}^{D}$ describes a mixed state (ensemble) and ${\cal E}$ is
given as the composition of local decoherence channels ${\cal E}_{\bm{x}}$.
One interpretation of this decoherence channel is that, there is a certain
probability for the environment to measure qubits in the $Z$ basis (weak
measurements) at each location ${\bm{x}}$, and the measurement outcomes are
“lost”, which amounts to the dephasing noise in the study of quantum circuits.
A generic decoherence channel maps a density matrix $\hat{\rho}_{0}$ into
$\hat{\rho}^{D}=\sum_{m}K_{m}\hat{\rho}_{0}K^{\dagger}_{m}$, where
$\\{K_{m}\\}$ is a set of Kraus operators satisfying the condition
$\sum_{m}K^{\dagger}_{m}K_{m}=\openone$. If we interpret a decoherence channel
to be induced by weak measurements, we can consider a postselection followed
by the channel where the resulting density matrix is given as

$\displaystyle\hat{\rho}^{D}_{P}\equiv\frac{{\cal
P}[\hat{\rho}^{D}]}{\mathrm{tr}\\{{\cal P}[\hat{\rho}^{D}]\\}},\quad{\cal
P}[\hat{\rho}^{D}]\equiv\sum_{m\in P}K_{m}\hat{\rho}_{0}K^{\dagger}_{m}$ (3)

where ${\cal P}$ is a generalized projection onto a subset of measurement
outcomes $P$, i.e., postselection.

In this work, we will mostly study systems at or close to a QCP, hence we will
use a coarse-grained continuous space rather than a lattice model. In the
coarse-grained formalism, the density matrix of a pure state is given by the
following imaginary time path-integral

$\displaystyle[\hat{\rho}_{0}]_{\phi_{1}({\bm{x}}),\phi_{2}({\bm{x}})}=\langle\phi_{1}({\bm{x}})|\Psi\rangle\langle\Psi|\phi_{2}({\bm{x}})\rangle$
$\displaystyle\qquad\sim\lim_{\beta\rightarrow\infty}\int_{\begin{subarray}{c}\phi({\bm{x}},0)=\phi_{1}({\bm{x}})\\\
\phi({\bm{x}},\beta)=\phi_{2}({\bm{x}})\end{subarray}}D\phi({\bm{x}},\tau)\exp(-{\cal
S}),$ (4)

where ${\cal S}=\int_{0}^{\beta}d\tau d{\bm{x}}\ {\cal L}(\phi)$ is the bulk
action of the system and ${\bm{x}}=(x_{1},...,x_{d})$ is the spatial
coordinate. Following Ref. 1, 12, a class of decoherence problem in the
coarse-grained continuous space can be converted into the following imaginary-
time path-integral:

$\displaystyle[\hat{\rho}^{D}]_{\phi_{1}({\bm{x}}),\phi_{2}({\bm{x}})}\sim\lim_{\beta\rightarrow\infty}\int_{\begin{subarray}{c}\phi({\bm{x}},0)=\phi_{1}({\bm{x}})\\\
\phi({\bm{x}},\beta)=\phi_{2}({\bm{x}})\end{subarray}}D\phi({\bm{x}},\tau)\exp\left(-{\cal
S}-{\cal S}^{\mathrm{int}}\right);$ (5) (6) (7) $\displaystyle{\cal
S}=\int_{0}^{\beta}d\tau d{\bm{x}}\ {\cal L}(\phi),\ \ \ {\cal
S}^{\mathrm{int}}=\int d{\bm{x}}\ {\cal
L}^{\mathrm{int}}(\phi({\bm{x}},0),\phi({\bm{x}},\beta)).$ (8)

The effect of decoherence or postselection is captured by an extra interaction
term ${\cal L}^{\mathrm{int}}$ in the Lagrangian, and ${\cal
L}^{\mathrm{int}}$ has no temporal integral, i.e., it is a “long range”
interaction in the Euclidean temporal direction, between fields at $\tau=0$
and $\tau=\beta$. The coupling constant in ${\cal L}^{\mathrm{int}}$ is
controlled by the strength of decoherence (or strength of weak measurement)
$p$. A natural choice of ${\cal L}^{\mathrm{int}}$ favors configurations with
$\phi({\bm{x}},0)\sim\phi({\bm{x}},\beta)$, meaning it enhances the weight of
the diagonal components of the density matrix, which drives the system into a
mixed state density matrix.

The most important factor that determines the form of ${\cal
L}^{\mathrm{int}}$ is still its symmetry. Let us label the symmetry of the
original Lagrangian ${\cal L}$ as $G$, and assume that the original pure state
without decoherence is a symmetric state under $G$. Then there are two types
of symmetry constraints on ${\cal L}^{\mathrm{int}}$. If the environment is
weakly “measuring” quantities that are invariant under $G$, then ${\cal
L}^{\mathrm{int}}$ must be invariant under a “doubled” symmetry
transformation, i.e., it is invariant under symmetry transformation $G$ on
$\phi({\bm{x}},0)$ and $\phi({\bm{x}},\beta)$ separately. In the language of
Ref. 12, this is the case that preserves the doubled $G^{u}\times G^{l}$
symmetry, where $G^{u}$ and $G^{l}$ correspond to the upper and lower symmetry
in the formalism of doubled Hilbert space using the Choi-Jamiolkowski
isomorphism [33, 34], which maps any density matrix to a pure ket-state in the
doubled Hilbert space. However, if the environment is measuring quantities
that carry a nontrivial representation of $G$, but eventually we sum over the
measurement outcomes within the same representation of $G$ with equal weight,
meaning the symmetry $G$ is broken in each quantum trajectory but still
preserved in an average sense, then ${\cal L}^{\mathrm{int}}$ is only
invariant under symmetry $G$, which is the diagonal subgroup of $G^{u}\times
G^{l}$, and it corresponds to a simultaneous transformation of
$\phi({\bm{x}},0)$ and $\phi({\bm{x}},\beta)$.

In Eq. (II), if the original pure state is invariant under a $\mathbb{Z}_{2}$
symmetry action $\prod_{\bm{x}}X_{\bm{x}}$ then the pure state density matrix
$\hat{\rho}_{0}$ is invariant under a doubled
$\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ symmetry, where
$\mathbb{Z}_{2}^{u}$ and $\mathbb{Z}_{2}^{l}$ correspond to the left and right
operations of the $\mathbb{Z}_{2}$ symmetry. But since the decoherence is
caused by weakly measuring the Ising variable which carries a nontrivial
representation of $\mathbb{Z}_{2}$, the decohered density matrix
$\hat{\rho}^{D}$ is no longer invariant under separate $\mathbb{Z}_{2}^{u}$ or
$\mathbb{Z}_{2}^{l}$ action, it is instead only invariant under the
simultaneous action of both $\mathbb{Z}_{2}^{u}$ and $\mathbb{Z}_{2}^{l}$,
hence $\hat{\rho}^{D}$ is invariant under a diagonal $\mathbb{Z}_{2}$
symmetry. Here the $\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ symmetry is
explicitly broken down to $\mathbb{Z}_{2}$ by the decoherence; later we will
also discuss an example where the decoherence preserves the doubled symmetry
but the doubled symmetry can be spontaneously broken down to the diagonal
$\mathbb{Z}_{2}$.

Now suppose we would like to compute the expectation value of a certain
quantity $O(\hat{\phi})$ that is a composite of $\hat{\phi}$. The expectation
value is linear with the density matrix, and it can be evaluated in the path
integral form:

$\displaystyle\mathrm{tr}\\{\hat{\rho}^{D}O(\hat{\phi})\\}\sim\int_{\phi({\bm{x}},0)=\phi({\bm{x}},\beta)}D\phi({\bm{x}},\tau)$
(9) (10) $\displaystyle\times O(\phi_{\tau=0})\exp\left(-{\cal S}-{\cal
S}^{\mathrm{int}}\right),$ (11)

where we have identified the field configurations $\phi({\bm{x}},0)$ and
$\phi({\bm{x}},\beta)$ due to the trace. If there is no postselection at all,
one can easily show that $\mathrm{tr}\\{\hat{\rho}^{D}O(\hat{\phi})\\}$ and
$\mathrm{tr}\\{\hat{\rho}_{0}O(\hat{\phi})\\}$ have essentially the same
behavior (except for some local corrections) as long as $O(\hat{\phi})$ is a
product of local operators 111Note that $\mathrm{tr}\\{{\cal
E}[\rho]O\\}=\mathrm{tr}\\{\rho{\cal E}^{*}[O]\\}$, where if ${\cal
E}[\rho]=\sum_{i}K_{i}\rho K_{i}^{\dagger}$ then ${\cal
E}^{*}[O]=\sum_{i}K^{\dagger}_{i}\rho K_{i}$. Accordingly, the correlation
function only acquires a local correction. , as pictorially illustrated in
Fig. 1. For example, the correlation function
$\mathrm{tr}\\{\hat{\rho}^{D}\hat{\phi}({\bm{x}}_{1})\hat{\phi}({\bm{x}}_{2})\\}$
should have the same scaling in space as
$\mathrm{tr}\\{\hat{\rho}_{0}\hat{\phi}({\bm{x}}_{1})\hat{\phi}({\bm{x}}_{2})\\}$.
In the field theory language, this corresponds to ${\cal S}^{\textrm{int}}$
being trivial when $\phi({\bm{x}},0)=\phi({\bm{x}},\beta)$.

But if there is some weak postselection by ${\cal P}$ even on quantities that
are singlet under $G$,
$\mathrm{tr}\\{\hat{\rho}^{D}_{P}\hat{\phi}({\bm{x}}_{1})\hat{\phi}({\bm{x}}_{2})\\}$
can be very different from
$\mathrm{tr}\\{\hat{\rho}_{0}\hat{\phi}({\bm{x}}_{1})\hat{\phi}({\bm{x}}_{2})\\}$.
With postselection, ${\cal S}^{\mathrm{int}}$ remains non-trivial even at
$\phi({\bm{x}},0)=\phi({\bm{x}},\beta)$, which effectively corresponds to the
insertion of defects at the slab $\tau=0$ (or $\tau=\beta$) in the path-
integral in the $(d+1)-$dimensional Euclidean space-time. This is where we can
make a connection to all the recent studies on boundary criticality for
systems with $d=2$ [36], and the desired decohered correlation functions
become the correlation functions restricted on the plane-like defect.

Figure 2:  Euclidean spacetime diagram and its space-time rotated version. The
evaluation of the second Renyi entropy of $\hat{\rho}^{D}$ becomes a path-
integral in the Euclidean space-time with an interaction between fields at
$\tau=0$ and $\tau=\beta/2$. In the space-time rotated picture, such an
interaction corresponds to the long-range interaction between fields at $y=0$
and $y=L/2$, where $L=\beta$. Note that the first coordinate of
${\bm{x}}=(x,y)$ is not shown in the diagram.

The expectation value of an operator is linear with the density matrix, which
is directly related to experimental observables. But a lot of information of
the quantum system is encoded in quantities that are nonlinear with the
density matrix. The most famous example of such is the von Neumann entropy of
the density matrix. In this work, we will evaluate quantities such as the 2nd
Renyi entropy of $\hat{\rho}^{D}$, which is an analogue of the von Neumann
entropy, and it also provides an approximate characterization 222Formally, we
can evaluate the $n$-th Renyi entropy, and takes the limit $n\rightarrow 1$ to
obtain the behavior of the von Neumann entropy. of the amount of quantum
information lost due to entangling with the environment:

$\displaystyle S^{(2)}$
$\displaystyle=-\log\mathrm{tr}\\{(\hat{\rho}^{D})^{2}\\}$
$\displaystyle\sim-\log\lim_{\beta\rightarrow\infty}\int
D\phi({\bm{x}},\tau)\,\times$ $\displaystyle\,\,\exp\left(-{\cal S}-\int
d{\bm{x}}\ {\cal
L}^{\mathrm{int}}(\phi({\bm{x}},0),\phi({\bm{x}},\beta/2))\right).$ (12)

This calculation is schematically shown in Fig. 2, which amounts to evaluating
the partition function and free energy of the system with an interaction
between fields at imaginary time $\tau=0$ and $\tau=\beta/2$. Or we can also
rotate the space-time (assuming there is a Lorentz symmetry), then the problem
becomes evaluating the partition function of the system with nonlocal
interaction in space, but constant in time.

Other quantities of interest include
$\mathrm{tr}\\{(\hat{\rho}^{D})^{2}O(\hat{\phi})\\}$ or
$\mathrm{tr}\\{\hat{\rho}^{D}O(\hat{\phi})\hat{\rho}^{D}O(\hat{\phi})\\}$. We
will show that these quantities nonlinear with $\hat{\rho}^{D}$ would reveal
some novel quantum phase transitions. In the example we will discuss in the
next section, the decoherence respects the doubled
$\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ symmetry; but when we increase
the decoherence strength, the 2nd Renyi entropy of the system (which captures
the information loss to the environment) may encounter singularity at a
critical decoherence strength, which corresponds to the spontaneous symmetry
breaking from $\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ to the diagonal
$\mathbb{Z}_{2}$.

##  III Decohered Wilson-Fisher critical point

In this section, we study two different scenarios for the system under weak
measurements: In Sec. III.1, we postselect a set of measurement outcomes and
show that the resulting correlation functions linear in the density matrix
$\hat{\rho}^{D}_{P}$ exhibit novel behaviors. In Sec. III.2, we average over
all measurement outcomes, which corresponds to genuine decoherence; although
correlation functions linear in $\hat{\rho}^{D}$ would exhibit an ordinary
Wilson-Fisher behavior in this case, we show that quantities non-linear in
$\hat{\rho}^{D}$ still exhibit novel behaviors including the extraordinary-log
criticality and information-theoretic phase transition.

###  III.1 Quantities linear with $\hat{\rho}^{D}_{P}$

First, we discuss critical physics under postselection as elaborated in Eq.
(3). The critical system we consider is the O($N$) Wilson-Fisher fixed point
in $(2+1)d$, where ${\bm{x}}=(x,y)$ is a two-dimensional spatial coordinate.
Under postselection onto a set of measurement outcomes $P$, we would like to
consider quantities linear with $\hat{\rho}^{D}_{P}$, for example the
correlation function
$\mathrm{tr}\\{\hat{\rho}^{D}_{P}\hat{{\bm{\phi}}}(0)\cdot\hat{{\bm{\phi}}}({\bm{x}})\\}$.
The evaluation of quantities such as
$\mathrm{tr}\\{\hat{\rho}^{D}_{P}O(\hat{{\bm{\phi}}}({\bm{x}}))\\}$ can be
performed by evaluating path-integral Eq. (11). Since we would like to keep at
least the diagonal $\mathrm{O}(N)$ symmetry, the postselected measurement
outcomes $P$ should be invariant under $\mathrm{O}(N)$ symmetry. Under that
condition, the corresponding ${\cal S}^{\mathrm{int}}$ becomes a function of
the magnitude of the order parameter $|{\bm{\phi}}|$ at $\tau=0$, which is
always equal to $|{\bm{\phi}}|$ at $\tau=\beta$ due to the trace. The simplest
term (and most relevant term) of ${\cal S}^{\mathrm{int}}$ is

$\displaystyle{\cal S}^{\mathrm{int}}=\int dxdy\
\varepsilon|{\bm{\phi}}({\bm{x}},0)|^{2}.$ (13)

The term ${\cal S}^{\mathrm{int}}$ can be interpreted as weakly measuring the
energy density or the order parameter ${\bm{\phi}}$ of the system, followed by
postselection.

Although there is only one simple term in ${\cal S}^{\mathrm{int}}$, the
physical consequence is already rather nontrivial. ${\cal S}^{\mathrm{int}}$
is a $2d$ interface in a $(2+1)d$ cylindrical space-time with extra mass
$\varepsilon$ for the order parameter, and obviously $\varepsilon$ is always a
relevant perturbation, as the scaling dimension of $|{\bm{\phi}}|^{2}$ is
always smaller than 2 at the $\mathrm{O}(N)$ Wilson-Fisher fixed point. When
$\varepsilon>0$, ${\cal S}^{\mathrm{int}}$ suppresses the fields at $\tau=0$,
and “cuts” the connection between $\tau=0^{+}$ and $\tau=\beta^{-}$, i.e.
$\tau=0^{+}$ and $\tau=\beta^{-}$ become two boundaries with ordinary boundary
condition. In this case, the correlation function
$\mathrm{tr}\\{\hat{\rho}^{D}_{P}\
\hat{{\bm{\phi}}}({\bm{0}})\cdot\hat{{\bm{\phi}}}({\bm{x}})\\}$ scales as [38]

$\displaystyle\mathrm{tr}\\{\hat{\rho}^{D}_{P}\
\hat{{\bm{\phi}}}({\bm{0}})\cdot\hat{{\bm{\phi}}}({\bm{x}})\\}\sim\frac{1}{|{\bm{x}}|^{2\Delta^{b}_{\bm{\phi}}}},$
(14)

where

$\displaystyle\Delta^{b}_{{\bm{\phi}}}=1+\frac{2}{3N}+O\left(\frac{1}{N^{2}}\right)$
(15)

is the boundary scaling dimension of the order parameter ${\bm{\phi}}$ to the
first order expansion of $1/N$. Note that the value of
$\Delta^{b}_{{\bm{\phi}}}$ is far larger than the bulk scaling dimension of
${\bm{\phi}}$.

When $\varepsilon<0$, the latest progress [26, 36] of the boundary criticality
indicates that ${\cal S}^{\mathrm{int}}$ will drive the interface $\tau=0$
into an extraordinary-log criticality, which implies that the correlation
function becomes

$\displaystyle\mathrm{tr}\\{\hat{\rho}^{D}_{P}\,\hat{{\bm{\phi}}}({\bm{0}})\cdot\hat{{\bm{\phi}}}({\bm{x}})\\}\sim\frac{1}{(\ln|{\bm{x}}|)^{q}}.$
(16)

Please note that, for a single exposed boundary against the vacuum, the
extraordinary-log criticality exists only for $N<N_{c}$, with the critical
$N_{c}$ estimated around $N_{c}\sim 5$ [26]; but for an interface defect
inserted in space-time, the theoretical prediction is that
$N_{c}\rightarrow\infty$ [36].

Figure 3:  Correlation functions in the doubled Hilbert space. The red and
blue lines represent the upper and lower Hilbert spaces respectively in the
doubled Hilbert space (See Sec. III.4).

###  III.2 Quantities nonlinear with $\hat{\rho}^{D}_{P}$

Now we evaluate quantities nonlinear with $\hat{\rho}^{D}$. Although our
formalism can be straightforwardly generalized for any higher orders of
$\hat{\rho}^{D}$, here we focus on the quantities that are quadratic in
$\hat{\rho}^{D}$. The quantities of interest include the 2nd Renyi entropy
$S^{(2)}$, the correlation function $C^{(2)}({\bm{x}})$, the “crossed”
correlation function $C^{(2)}_{X}({\bm{x}})$, and the “crossed-Ising”
correlation function $C^{(2)}_{XI}({\bm{x}})$ defined as follows:

$\displaystyle S^{(2)}=$
$\displaystyle-\log\mathrm{tr}\\{(\hat{\rho}^{D})^{2}\\},$ $\displaystyle
C^{(2)}({\bm{x}})\sim$
$\displaystyle\,\,\mathrm{tr}\\{(\hat{\rho}^{D})^{2}\hat{{\bm{\phi}}}({\bm{0}})\cdot\hat{{\bm{\phi}}}({\bm{x}})\\},$
$\displaystyle C^{(2)}_{X}({\bm{x}})\sim$
$\displaystyle\sum_{a}\mathrm{tr}\\{\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\rho}^{D}\hat{\phi}_{a}({\bm{x}})\\}$
$\displaystyle C^{(2)}_{XI}({\bm{x}})\sim$ $\displaystyle\sum_{a\neq
b}\mathrm{tr}\\{\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\phi}_{b}({\bm{x}})\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\phi}_{b}({\bm{x}})\\}.$
(17)

The expression of the correlation functions above need to be divided by the
_purity_ of the decohered density matrix $\mathrm{tr}\\{(\rho^{D})^{2}\\}$ to
be properly normalized. We note that these correlation functions are
distinguished by their representation under
$\mathrm{O}(N)^{u}\times\mathrm{O}(N)^{l}$ symmetry, which becomes clear in
the doubled Hilbert space as in Fig. 3.

As an example, we consider the following interaction term ${\cal
S}^{\mathrm{int}}$ in Eq. II:

$\displaystyle{\cal S}^{\mathrm{int}}$ $\displaystyle=\int dxdy\
\Big{[}W\left(|{\bm{\phi}}({\bm{x}},0)|,\
|{\bm{\phi}}({\bm{x}},\beta/2)|\right)$
$\displaystyle\qquad-\,w\left({\bm{\phi}}({\bm{x}},0)\cdot{\bm{\phi}}({\bm{x}},\beta/2)\right)^{2}\Big{]}.$
(18)

Here $W$ is still a function of the magnitude of the order parameter at
$\tau=0$ and $\tau=\beta/2$. The two terms in Eq. 18 correspond to two
different types of weak measurements (decoherence) channels. The first term
still corresponds to weakly measuring the energy density of the system, which
preserves the doubled symmetry of the system, resulting in the coupling

$\displaystyle\int dxdy\ W\left(|{\bm{\phi}}({\bm{x}},0)|,\
|{\bm{\phi}}({\bm{x}},\beta/2)|\right)$ $\displaystyle~{}~{}~{}~{}~{}~{}=\int
dxdy\
\frac{\varepsilon}{2}(|{\bm{\phi}}({\bm{x}},0)|^{2}+|{\bm{\phi}}({\bm{x}},\beta/2)|^{2})+...$
(19)

where the “…” part includes quartic and higher-order terms in
$|{\bm{\phi}}({\bm{x}},0)|$ and $|{\bm{\phi}}({\bm{x}},\beta/2)|$, such as
$|{\bm{\phi}}({\bm{x}},0)|^{2}|{\bm{\phi}}({\bm{x}},\beta/2)|^{2}$. The second
$w$ term can be rewritten as

$\displaystyle-w\left({\bm{\phi}}({\bm{x}},0)\cdot{\bm{\phi}}({\bm{x}},\beta/2)\right)^{2}\sim$
$\displaystyle\qquad-w\sum_{a,b=1}^{N}\left(Q_{ab}({\bm{x}},0)Q_{ab}({\bm{x}},\beta/2)\right)+\cdots,$
(20)

where $Q_{ab}=\phi^{a}\phi^{b}-\frac{1}{N}\delta_{ab}|{\bm{\phi}}|^{2}$ is the
traceless rank-2 symmetric tensor of the O($N$) vector ${\bm{\phi}}$, and the
ellipsis are terms that only depend on $|{\bm{\phi}}|$ and can be absorbed
into $W$. The $w$ term can be interpreted as a weak measurement on the tensor
$Q_{ab}$, and eventually all the measurement outcomes are summed with equal
weight. The $w$ term breaks the $\mathrm{SO}(N)^{u}\times\mathrm{SO}(N)^{l}$
down to the diagonal $\mathrm{SO}(N)$, but still preserves the
$\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ symmetry, where $\mathbb{Z}_{2}$
corresponds to changing the sign of ${\bm{\phi}}$.

One can perform the space-time rotation in the $(y,\tau)$ plane, so the two
interfaces at $\tau=0,\beta/2$ become spatial interfaces at $y=0$ and $L/2$,
with $L=\beta$. The interaction term then becomes

$\displaystyle{\cal S}^{\mathrm{int}}$ $\displaystyle=$ $\displaystyle\int
d\tau dx\ W\left(|{\bm{\phi}}(x,\tau)|_{y=0},\
|{\bm{\phi}}(x,\tau)|_{y=L/2}\right)$ (23)
$\displaystyle-\,w\left({\bm{\phi}}(x,\tau)_{y=0}\cdot{\bm{\phi}}(x,\tau)_{y=L/2}\right)^{2}.$

The first term $W$ will either explicitly include an extra mass term
$\varepsilon|{\bm{\phi}}|^{2}$ at the interface $y=0$ and $y=L/2$, or generate
the mass term through renormalization group flow. Then depending on the sign
of $\varepsilon$, there can be three possible scenarios:

(1) If $\varepsilon>0$, the extra mass term $\varepsilon|{\bm{\phi}}|^{2}$ at
the interface $y=0$ and $y=L/2$ is a relevant perturbation. The role of this
extra mass term is to “cut” the system into two halves: the region from
$y\in(0,L/2)$ and region $y\in(L/2,L\sim 0)$. The relevant mass term will make
the two interfaces at $y=0$ and $y=L/2$ both at ordinary boundary criticality.

In this scenario, $w$ is obviously an irrelevant perturbation since both
interfaces $y=0$ and $y=L/2$ have ordinary boundary criticality, and the
scaling dimension of ${\bm{\phi}}$ is greater than 1 at the interfaces. Hence
the directions of ${\bm{\phi}}$ at the two interfaces are pretty much
uncorrelated to each other. Then we expect the correlation function, the
crossed-Ising correlation, and the crossed-correlation functions to behave as

$\displaystyle C^{(2)}$ $\displaystyle\sim$
$\displaystyle\mathrm{tr}\\{(\hat{\rho}^{D})^{2}\hat{{\bm{\phi}}}({\bm{0}})\cdot\hat{{\bm{\phi}}}({\bm{x}})\\}\sim\frac{1}{|{\bm{x}}|^{2\Delta^{b}_{{\bm{\phi}}}}},$
(24) $\displaystyle C^{(2)}_{XI}$ $\displaystyle\sim$
$\displaystyle\sum_{a\neq
b}\mathrm{tr}\\{\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\phi}_{b}({\bm{x}})\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\phi}_{b}({\bm{x}})\\}\sim
0,$ (26) $\displaystyle C^{(2)}_{X}$ $\displaystyle\sim$
$\displaystyle\sum_{a}\mathrm{tr}\\{\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\rho}^{D}\hat{\phi}_{a}({\bm{x}})\\}\sim
0.$ (28)

For example, the crossed-correlation $C^{(2)}_{X}$ corresponds to the
correlation function between order parameters at the two different interfaces;
since $w$ is irrelevant when $\varepsilon>0$, the directions of order
parameters at the two interfaces are uncorrelated, hence $C^{(2)}_{X}$ should
vanish in the limit $\beta,L\rightarrow\infty$. Another way to perceive these
results is that, since $w$ is irrelevant, the system should have a full
$\mathrm{SO}(N)^{u}\times\mathrm{SO}(N)^{l}$ symmetry in the infrared, but the
correlation function $C^{(2)}_{X}$ and $C^{(2)}_{XI}$ break the
$\mathrm{SO}(N)^{u}\times\mathrm{SO}(N)^{l}$ symmetry, hence they must vanish.

(2) In the case with $\varepsilon<0$, the extra local mass term $\varepsilon$
is still a relevant perturbation, and it flows to the extraordinary-log
criticality. Then the $w$ term (we assume $w>0$) is a very relevant
perturbation, and it will flow to $w\rightarrow\infty$.

The fate of the system with strong $w$ can be perceived through a mean field
decoupling of ${\cal L}^{\mathrm{int}}$. We first recognize that the $w$ term
is analogous to the coupling between two sets of Néel order parameters in the
$J_{1}-J_{2}$ Heisenberg model on the square lattice, which has applications
in the context of frustrated magnets and iron-pnictides superconductors [39,
40, 41, 42, 43]. Guided by the previous studies in these contexts, the most
natural mean field decoupling of the $w$ term is

$\displaystyle-w\left({\bm{\phi}}(x,\tau)_{y=0}\cdot{\bm{\phi}}(x,\tau)_{y=L/2}\right)^{2}\sim$
(29) (30)
$\displaystyle\quad-2w\Phi(x,\tau)\left({\bm{\phi}}(x,\tau)_{y=0}\cdot{\bm{\phi}}(x,\tau)_{y=L/2}\right)$
(31) (32) $\displaystyle\quad+w\Phi(x,\tau)^{2}.$ (33)

Here we have introduced an Ising order parameter
$\Phi(x,\tau)\sim\left({\bm{\phi}}(x,\tau)_{y=0}\cdot{\bm{\phi}}(x,\tau)_{y=L/2}\right)$.
The order parameter $\Phi$ is analogous to the nematic order parameter in the
$J_{1}-J_{2}$ Heisenberg model on the square lattice, and the phase with large
$w$ is likely a phase with condensation of $\Phi$, which spontaneously breaks
$\mathbb{Z}^{u}_{2}\times\mathbb{Z}^{l}_{2}$ down to the diagonal
$\mathbb{Z}_{2}$.

The condensation of $\Phi$ will “pin” $\hat{{\bm{\phi}}}_{y=0}$ and
$\hat{{\bm{\phi}}}_{y=L/2}$ along the parallel direction. With a nonzero
condensate of $\Phi$, the three correlation functions mentioned above behave
like

$\displaystyle C^{(2)}$ $\displaystyle\sim$
$\displaystyle\mathrm{tr}\\{(\hat{\rho}^{D})^{2}\hat{{\bm{\phi}}}({\bm{0}})\cdot\hat{{\bm{\phi}}}({\bm{x}})\\}\sim\frac{1}{(\ln|{\bm{x}}|)^{q}},$
(34) $\displaystyle C^{(2)}_{XI}$ $\displaystyle\sim$
$\displaystyle\sum_{a\neq
b}\mathrm{tr}\\{\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\phi}_{b}({\bm{x}})\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\phi}_{b}({\bm{x}})\\}$
(36) $\displaystyle\sim$ $\displaystyle\mathrm{Const},$ (38) $\displaystyle
C^{(2)}_{X}$ $\displaystyle\sim$
$\displaystyle\sum_{a}\mathrm{tr}\\{\hat{\rho}^{D}\hat{\phi}_{a}({\bm{0}})\hat{\rho}^{D}\hat{\phi}_{a}({\bm{x}})\\}\sim\frac{1}{(\ln|{\bm{x}}|)^{q}}.$
(40)

The crossed-Ising correlation saturates to a nonzero constant in the limit of
large $|{\bm{x}}|$ due to the condensate of $\Phi$, which also leads to an
extraordinary-log correlation of crossed-correlation function. When $w$ flows
to infinity and $\Phi$ condenses, there is only one $\mathrm{O}(N)$ symmetry
in the infrared; hence all these correlations can be nonzero.

(3) Now if $\varepsilon=0$, with large but finite $N$, the scaling dimension
of the traceless rank-2 symmetric tensor $Q_{ab}$ is $\Delta=1+32/(3\pi^{2}N)$
to the leading order of $1/N$ expansion. This means that $w$ is weakly
irrelevant with scaling dimension $[w]=-64/(3\pi^{2}N)$. Then the beta
function of $w$ should be

$\displaystyle\beta(w)=\frac{dw}{d\ln l}=-\frac{A}{N}w+Cw^{2},$ (41)

where $A=64/(3\pi^{2})$. The constant $C$ can be extracted through some OPE
calculation, or simply a one-loop calculation in the large$-N$ limit. Since
$C$ is positive, there is a fixed point at finite $w_{\ast}$, beyond which $w$
will flow strongly, and the order parameter $\Phi$ defined above will
condense.

###  III.3 The phase diagram

Figure 4:  Phase Diagram. The global phase diagram of decohered Wilson-Fisher
critical point in terms of quantities nonlinear in $\hat{\rho}^{D}$.

In phase diagram Fig. 4 we summarize the results related to
$(\hat{\rho}^{D})^{2}$ discussed in this section. The order-disorder
transition of $\Phi$ should extend to the region with $\varepsilon>0$, where
there is a competition between $\varepsilon$ which drives the interfaces to
the ordinary boundary criticality and $w$ which drives the crossed-Ising
transition. In this phase diagram, the critical $w_{c}$ is a function of
$\varepsilon$:
$w_{c}(\varepsilon)-w_{c}(0)\sim\varepsilon^{\Delta_{w}/\Delta_{\varepsilon}}$,
and $\Delta_{w,\varepsilon}$ are the scaling dimensions of parameter $w$ and
$\varepsilon$.

Since
$\Phi({\bm{x}})\sim{\bm{\phi}}({\bm{x}},0)\cdot{\bm{\phi}}({\bm{x}},\beta/2)$
is the crossed-Ising order parameter, the crossed-Ising correlation function
should show a transition from short-range to correlation while increasing $w$.
If we consider $\mathrm{tr}\\{(\hat{\rho}^{D})^{2}\\}$ as the partition
function, and the 2nd Renyi entropy
$S^{(2)}=-\log\mathrm{tr}\\{(\hat{\rho}^{D})^{2}\\}$ as the free energy, the
nature of the transition at $w=w_{c}$ and $\varepsilon>0$ should belong to a
$2d$ classical Ising universality class. Without coupling to other modes with
nonlocal correlations in the infrared, the order-disorder transition of an
Ising order parameter in the $2d$ space should belong to the $2d$ Ising
universality class, and here we should consider the perturbation of the
coupling $w$ on top of the $2d$ Ising transition. In fact, if we start with a
$2d$ classical Ising transition of order parameter $\Phi$, the coupling
$-w\Phi({\bm{x}})\left({\bm{\phi}}({\bm{x}},0)\cdot{\bm{\phi}}({\bm{x}},\beta/2)\right)$
is irrelevant knowing the fact that the scaling dimension
$\Delta^{b}_{{\bm{\phi}}}>1$ for the ordinary boundary criticality at
$\varepsilon>0$. Hence at $w=w_{c}$ and $\varepsilon>0$, the crossed-Ising
correlation function should scale as

$\displaystyle C^{(2)}_{XI}({\bm{r}})\sim\frac{1}{|{\bm{r}}|^{1/4}}.$ (42)

Also, when we increase $w$ across $w_{c}$, the 2nd Renyi entropy
$S^{(2)}=-\log\mathrm{tr}\\{(\hat{\rho}^{D})^{2}\\}$ should have the same
singularity as the free energy of the classical $2d$ Ising model.

Figure 5:  Choi Isomorphism and Duality. Schematic diagrams illustrate how
decohered mixed states map into the coupled bilayer system under the Choi-
Jamiolkowski isomorphism. Through the Kramers-Wannier duality,
$\mathbb{Z}_{2}$ paramagnet under symmetric decoherence (left) maps into the
$\mathbb{Z}_{2}$ toric code under dephasing noise (right). Accordingly, their
phase diagrams in the doubled Hilbert space is also dual to each other. At
$p>p_{c}^{(2)}$, both sides are characterized by non-vanishing correlation
functions of operators, which corresponds to (left) the formation of mean-
field for $Z_{v,u}Z_{v,l}$ operator or (right) condensation of a pair of
$e$-anyons $e_{u}e_{l}$.

###  III.4 Lattice Model and Doubled Hilbert Space

We have shown that a decoherence channel whose Kraus operators are symmetric
under $O(N)$ can drive an Ising-type phase transition that spontaneously
breaks the $\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}\subset O(N)_{u}\times
O(N)_{l}$ symmetry down to the diagonal $\mathbb{Z}_{2}$ symmetry for
quantities nonlinear in the density matrix. However, most physical quantities
are linear in the density matrix, and this calls for a proper physical
interpretation of these quantities. As we will see, our crossed-Ising
transition is closely related to a better known information transition in the
context of topological surface codes.

In the following, we use another model to illustrate the essential physics of
such a spontaneous symmetry breaking (SSB) from
$\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ to the diagonal $\mathbb{Z}_{2}$.
Instead of starting with a state at quantum criticality, we consider a
concrete lattice model with a trivially disordered state
$|\Omega_{0}\rangle=|+\rangle^{\otimes N}$ on a $L\times L$ square lattice
with qubits defined on vertices. We consider the following quantum channel as
a symmetric local decoherence model:

$\displaystyle{\cal E}_{e=(v,v^{\prime})}:\rho$
$\displaystyle\rightarrow(1-p)\rho+pZ_{v}Z_{v^{\prime}}\rho
Z_{v}Z_{v^{\prime}},$ (43)

where “$e$” labels the edge between the nearest-neighbor pair of sites $v$ and
$v^{\prime}$. The decoherence channel is given as the composition of local
channels, ${\cal E}=\prod_{e}{\cal E}_{e}$. To proceed further, we apply the
Choi-Jamiolkowski isomorphism to map the density matrix into the pure state
and decoherence channel into the operator in the doubled Hilbert space. Under
this mapping, the Choi state of the pure state density matrix
$\rho_{0}=|\Omega_{0}\rangle\langle\Omega_{0}|$ is denoted as
$\|\rho_{0}\rangle\rangle\equiv|\Omega_{0}\rangle|\Omega_{0}\rangle$, which is
nothing but a vectorized density matrix, and the Choi operator of the
decoherence channel is given as

$\displaystyle\hat{{\cal E}}=\prod_{e=(v,v^{\prime})}(1-2p)^{1/2}e^{\tau
Z_{v,u}Z_{v^{\prime},u}Z_{v,l}Z_{v^{\prime},l}}$ (44)

where $\tanh\tau=p/(1-p)$. As illustrated in Fig. 5, the isomorphism
effectively maps the density matrix into the pure state in the bilayer system.
Now, when the system is subject to the decoherence, we can show that
$\|\rho^{D}\rangle\rangle=\hat{{\cal E}}\|\rho_{0}\rangle\rangle$ is the
ground-state wavefunction of the following _local_ Hamiltonian in the doubled-
Hilbert space [9, 12]:

$\displaystyle H_{\textrm{tot}}$ $\displaystyle=H_{u}+H_{l}+H_{\textrm{int}}$
$\displaystyle H_{u(l)}$ $\displaystyle=-\cosh 2\tau\sum_{v}X_{v,u(l)}$
$\displaystyle H_{\textrm{int}}$
$\displaystyle=\sum_{v}\cosh^{4}2\tau\prod_{v^{\prime}\in
v}(1-Z_{v,u}Z_{v^{\prime},u}Z_{v,l}Z_{v^{\prime},l}\tanh 2\tau),$ (45)

where $v^{\prime}\in v$ means that $v^{\prime}$ is a nearest-neighbor site of
$v$. At $p=\tau=0$, it reduces into two decoupled disordered states. At $p>0$,
the decoherence gives rise to the (local) coupling between two layers that may
induce a phase transition into an SSB phase that breaks the
$\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ into the diagonal
$\mathbb{Z}_{2}$ symmetry. The (unnormalized) groundstate can be written as

$\displaystyle\hat{{\cal
E}}\|\rho_{0}\rangle\rangle=(1-p)^{2N_{v}}\sum_{\bm{l}}(\tanh\tau)^{\absolutevalue{{\bm{l}}}}|\partial{\bm{l}}\rangle\otimes|\partial{\bm{l}}\rangle$
(46)

where the summation is taken over all string configurations ${\bm{l}}$ on the
edges of the square lattice,
$|\partial{\bm{l}}\rangle\equiv\prod_{v\in\partial{\bm{l}}}Z_{v}|\Omega_{0}\rangle$,
and $N_{v}=L^{2}$ is the number of vertices in the square lattice. We stress
that $\hat{{\cal E}}\|\rho_{0}\rangle\rangle$ is normalized in the sense that
its corresponding density matrix $\rho^{D}$ (under the Choi-Jamiolkowski
isomorphism) is normalized with $\mathrm{tr}\\{\rho^{D}\\}=1$ in the single
Hilbert space while it is unnormalized as a state in the doubled Hilbert
space. Then, it is straightforward to show that the norm of the wavefunction
$\hat{{\cal E}}\|\rho_{0}\rangle\rangle$, i.e., the purity of the density
matrix $\mathrm{tr}\\{(\rho^{D})^{2}\\}$ is equivalent to the partition
function of the 2d Ising model at the temperature $\beta=2\tau$, as explicitly
derived in Appendix. A.6. Accordingly, at $\beta=0.441$, which corresponds to
$p^{(2)}_{c}=0.178$, the wavefunction in the doubled Hilbert space undergoes
the transition of 2d Ising universality. Here the superscript in $p$ implies
that the transition happens in the quantity that involves the product of two
density matrices.

The SSB transition between the
$\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$-symmetric phase and the phase
with only the diagonal $\mathbb{Z}_{2}$ symmetry in the doubled Hilbert space
is captured by the susceptibility $\chi$ to an external symmetry-breaking
field coupled to $O=\sum_{v}Z_{v,u}Z_{v,l}$ (in the doubled Hilbert space):

$\displaystyle\chi\equiv\frac{1}{L^{2}}\Big{(}\langle\langle
O^{2}\rangle\rangle-\langle\langle O\rangle\rangle^{2}\Big{)}$
$\displaystyle\,=\frac{1}{L^{2}}\bigg{(}\frac{\langle\langle\rho^{D}\|O^{2}\|\rho^{D}\rangle\rangle}{\langle\langle\rho^{D}\|\rho^{D}\rangle\rangle}-\frac{\langle\langle\rho^{D}\|O\|\rho^{D}\rangle\rangle^{2}}{\langle\langle\rho^{D}\|\rho^{D}\rangle\rangle^{2}}\bigg{)}$
$\displaystyle\,=\sum_{v,v^{\prime}}\frac{\mathrm{tr}\\{\rho^{D}Z_{v}Z_{v^{\prime}}\rho^{D}Z_{v}Z_{v^{\prime}}\\}}{L^{2}\mathrm{tr}\\{(\rho^{D})^{2}\\}}-\left(\frac{\sum_{v}\mathrm{tr}\\{\rho^{D}Z_{v}\rho^{D}Z_{v}\\}}{L\,\mathrm{tr}\\{(\rho^{D})^{2}\\}}\right)^{2}$
(47)

which is given by the summation over connected correlation functions of the
order parameter $Z_{v,u}Z_{v,l}$ of this SSB transition in the doubled Hilbert
space, related to the crossed-Ising correlation function $C^{(2)}_{XI}$ in Eq.
(34). $\chi$ is closely related to the crossed-Ising correlation functions in
Eq. (34). $\chi$ diverges as $|p-p_{c}^{(2)}|^{-7/4}$ at the SSB transition.
The exponent $7/4$ is the signature of the 2d Ising universality class of this
SSB transition. Note that $\langle\langle O\rangle\rangle$ in the definition
of $\chi$ is the order parameter that takes a non-zero value in the symmetry-
broken phase and should be evaluated with $\delta=0^{+}$ after the
thermodynamical limit is taken. We elaborate more on this in the next section.

In the doubled Hilbert space, the definition of $\chi$ and the associated
notion of the external symmetry-breaking field are standard for generic
condensed-matter systems with global symmetries. In Sec. III.5, we explain
their physical meaning at the level of the density matrix $\rho^{D}$.

###  III.5 Correlation Functions and Physical Meanings

In the previous sections, we have defined several correlation functions or
physical quantities which are nonlinear in decohered density matrices.
Accordingly, these quantities are not directly accessible from experiments,
raising questions about what they really mean physically. In order to connect
these quantities to experiments, we consider the signatures of the transition
in the probability distribution of measurement outcomes. More precisely, we
will show that $\chi$ in Eq. (III.4) corresponds to the sensitivity of the
decohered mixed state against small perturbations. This, in turn, is related
to the amount of information that can be obtained from measuring the mixed
state.

To proceed, we define the notion of distance between two different density
matrices as the following:

$\displaystyle
D_{\textrm{J}}(\rho,\sigma)\equiv\mathrm{tr}(\rho\log\rho-\rho\log\sigma+\sigma\log\sigma-\sigma\log\rho).$
(48)

This quantity is the quantum generalization of the Jeffreys divergence [44]
(or symmetrized quantum relative entropy), which quantifies the distance
between two density matrices. It satisfies good properties to be a valid
metric between mixed states: $(i)$ non-negative, $(ii)$ vanishing if and only
if$\rho=\sigma$, and $(iii)$ monotonically decreasing under the application of
quantum channels.

In order to discuss the “sensitivity” of the decohered density matrix, we
define an infinitesimal variation of the original density matrix under
symmetry breaking “perturbation”, defined as the following quantum channel:

$\displaystyle{\cal M}_{\delta,v}:\rho\rightarrow(1-\delta)\rho+\delta
Z_{v}\rho Z_{v}$ $\displaystyle\rho_{\delta}\equiv{\cal
M}_{\delta}[\rho],\quad{\cal M}_{\delta}\equiv\prod_{v}{\cal M}_{\delta,v}.$
(49)

which amounts to the application of weak measurement in the $Z$ basis. Hence,
$\delta$ also serves as a symmetry-breaking field in the doubled Hilbert
space. Then, $D(\delta)\equiv D_{\textrm{J}}(\rho^{D},\rho^{D}_{\delta})$
quantifies the difference between the unperturbed and perturbed decohered
density matrices. Accordingly, how fast the distance changes with $\delta$
tells how sensitive the decohered system is against weak measurement in the
$Z$ basis (or how informative the $Z$ measurement is). This is captured by the
second derivative of the distance, which is related to the (classical) Fisher
information as ${\cal
F}\equiv\partial_{\delta}^{2}D(\delta)\big{|}_{\delta=0}$ [45].

However, the expression in Eq. (48) is very challenging to evaluate. To
proceed, we generalize the Jeffreys distance in a way similar to Ref. 46:

$\displaystyle D^{(n)}(\rho,\sigma)$
$\displaystyle\equiv\frac{1}{n-1}\Big{(}\log\mathrm{tr}(\rho^{n})+\log\mathrm{tr}(\sigma^{n})$
$\displaystyle\qquad-\log\mathrm{tr}(\rho\sigma^{n-1})-\log\mathrm{tr}(\sigma\rho^{n-1})\Big{)}$
(50)

This $n$-th Jeffreys distance is symmetric and non-negative 333However, it
does not monotonically decrease under quantum channels anymore.. Furthermore,
in the limit $n\rightarrow 1^{+}$, $D^{(n)}\rightarrow D_{\textrm{J}}$. At
$n=2$, we find that the expression behaves like an overlap between Choi states
of two density matrices in the doubled Hilbert space description:

$\displaystyle
D^{(2)}(\rho,\sigma)\equiv-\log(\frac{\langle\langle\rho\|\sigma\rangle\rangle}{\sqrt{\langle\langle\rho\|\rho\rangle\rangle\cdot\langle\langle\sigma\|\sigma\rangle\rangle}}),$
(51)

With this definition, we evaluate the derivatives of the “distance” between
$\rho^{D}$ and its perturbed version $\rho^{D}_{\delta}$ with respect to
$\delta$ (See Appendix. B). This quantity exhibits interesting behaviors when
$\delta\rightarrow 0$ and $L\rightarrow\infty$. The two different orders of
limit lead to different results which have natural interpretations from a
standard condensed-matter perspective and from a quantum information
perspective respectively.

In the limit where the thermodynamical limit $L\rightarrow\infty$ is taken
first and $\delta\rightarrow 0^{+}$ afterward, the conventional choice of
limit for a condensed-matter system experiencing SSB, the second-order
derivative $\partial^{2}_{\delta}D^{(2)}$ produces a quantity proportional to
the susceptibility:

$\displaystyle\lim_{\delta\rightarrow
0^{+}}\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\partial^{2}_{\delta}D^{(2)}=\chi,$
(52)

which diverges at the SSB phase transition. Based on the nature of this SSB
transition and the interpretation of $\delta$ as the symmetry-breaking field,
we conclude that the decohered density matrix undergoes a qualitative change
(with respect to this distance) as a function of $p$. Beyond critical
$p_{c}^{(2)}$, the density matrix changes significantly under weak
measurements in the $Z$ basis.

However, when we consider taking the limit $\delta\rightarrow 0$ before the
thermodynamical limit, the second-order derivative produces a quantity that
aligns with the information-theoretic intuition:

$\displaystyle\lim_{L\rightarrow\infty}\lim_{\delta\rightarrow
0}\frac{1}{L^{4}}\partial^{2}_{\delta}D^{(2)}=\sum_{v,v^{\prime}}\frac{\mathrm{tr}\\{\rho^{D}Z_{v}Z_{v^{\prime}}\rho^{D}Z_{v}Z_{v^{\prime}}\\}}{L^{4}\mathrm{tr}\\{(\rho^{D})^{2}\\}}=M,$
(53)

where $M$ can be interpreted as the expectation value of the squared order
parameter, which vanishes in the paramagnetic phase but acquires a finite
value in the SSB phase. Since this SSB transition belongs to the 2d Ising
universality class, the standard phenomenology of the Ising model can be
directly translated into the language of our study. In an Ising ferromagnet,
the magnetic order of the ground state of the system is very sensitive to an
infinitesimal external Zeeman field. This well-known feature corresponds to
the sensitivity of the density matrix to the decoherence in the $Z_{v}$ basis
(as in Eq. (III.5)), in the phase where the
$\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ symmetry is spontaneously broken
down to the diagonal $\mathbb{Z}_{2}$. This is quantified by the value of
$\partial^{2}_{\delta}D^{(2)}\big{|}_{\delta=0}$ (where $\delta$ is set to 0
before $L$ grows large) scaling as $L^{4}$ in the SSB phase, which aligns with
the Fisher information of the GHZ state in the context of quantum metrology
[48].

It is straightforward to argue that the second derivatives of the distance
$D^{(n>2)}$ in both limits would also exhibit similar behavior across a
certain critical value (that depends on $n$), particularly because
$\mathrm{tr}((\rho^{D})^{n})$ is mathematically equivalent to the partition
function of the coupled $n$-copies of Ising model which is expected to undergo
a transition from a paramagnetic to a ferromagnetic phase potentially at a
different critical strength $p_{c}^{(n)}\geq p_{c}^{(2)}$. Accordingly, we
expect the behavior of $\partial^{2}_{\delta}D^{(n)}$ to extrapolate in the
limit $n\rightarrow 1$. Therefore, there should be two different phases
separated by a critical point at strength $p=p_{c}^{(1)}$. This is indeed the
case, as we will see by using a dual description of the model.

###  III.6 Duality, Intrinsic transition, and Decodability

Under the Kramers-Wannier (KW) duality, the $\mathbb{Z}_{2}$ paramagnet under
symmetric decoherence maps to the toric code under dephasing noise. The toric
code under dephasing noise, in turn, is well known to exhibit an information
transition at critical noise strength, beyond which the quantum information
(logical qubits) stored in the toric code is not decodable [31]. This hints at
an intimate connection between the criticality discussed above and a well-
known information transition.

First, we illustrate how the doubled Hilbert space formalism provides us some
insights into the transition in the decohered toric code. Under the KW duality
(See Appendix. A.5), the Hamiltonian in Eq. (III.4) maps to the two copies of
toric code coupled by local anyon tunneling terms, as schematically shown in
Fig. 5. At critical strength of the tunneling, the anyon condenses and the
topological order reduces into a single toric code. The critical behavior is
associated with a Higgs transition signified by the development of the anyon
condensation amplitude
$\sim\langle\langle\rho^{D}\|\hat{\gamma}_{u}\hat{\gamma}_{l}\|\rho^{D}\rangle\rangle$,
where $\hat{\gamma}_{u/l}=\prod_{e\in\gamma^{\perp}}Z_{e,u/l}$ creates a pair
of $e$-anyon at the end of the string $\gamma^{\perp}$. In turn, this implies
that $\hat{\rho}^{D}$ and $\hat{\gamma}\hat{\rho}^{D}\hat{\gamma}$ have an
appreciable overlap and they become less and less distinguishable for
$p>p_{c}^{(2)}$ in the doubled Hilbert space.

In this coupled toric code in the doubled Hilbert space, the existence of the
criticality can be directly seen by calculating the norm of the wave function,
which is equivalent to the purity of the decohered toric code. The said
quantity is given by the partition function of the 2d Ising model at
$\beta=\tanh^{-1}(1-2p)^{2}$ as shown in Appendix. A.3, which has a transition
at $\beta=\ln(1+\sqrt{2})/2$. As expected from the KW duality, this transition
from an ordered to disordered phase upon increasing $p$ coincides with the
transition of the original Choi state Eq. (46) from a disordered to an ordered
phase, giving rise to the same critical point $p_{c}^{(2)}=0.178$.

As a next step, we calculate the von Neumann entropy of the decohered toric
code state, a quantity highly nonlinear in the density matrix that enters the
expression in Eq. (48). Following the calculations in Appendix. A.2, we show
that the von Neumann entropy of the decohered density matrix is proportional
to the free energy of the random bond Ising model along the Nishimori line
[49, 50] at $\beta=\tanh^{-1}(1-2p)$, which is known to be critical at
$p=0.1094$ [51]. Therefore, the transition behavior indeed extends down to the
$n\rightarrow 1$ limit for a toric code under dephasing noise, and by the KW
duality, for the original paramagnetic spin model under symmetric decoherence.
As remarked, the transition behavior in the limit $n\rightarrow 1$ corresponds
to the singularity of the Fisher information, a well-known information-
theoretic quantity. We remark that this series of transitions for different
$n$ is _intrinsic_ to the decohered density matrix (and the distance in use),
as these transitions are associated with spontaneous symmetry breaking.
Interestingly, the $n\rightarrow 1$ limit of this intrinsic transition
coincides with the decodability transition point demonstrated in Ref. 31,
which is based on a certain decoding procedure. Such a connection may imply
that the intrinsic transition point provides an upper bound for some
information retrieval protocols to be successful in this setting.

##  IV Non-local operators

As we mentioned previously, without any postselection, the correlation
function for local operators linear with the decohered density matrix should
have the same scaling as the undecohered correlation function. But in this
section, we will show that nonlocal quantities can still have qualitatively
different behaviors even if we only consider expectation values linear with
the density matrix.

###  IV.1 $1d$ Quantum Rotor

To illustrate the behavior of nonlocal operators under decoherence, let us
start with the $1d$ systems with a description in terms of the quantum rotor,
such as the spin-1/2 chain. In terms of the Abelian bosonization, the Néel and
valence bond solid (VBS) order parameters of a spin-1/2 chain are represented
as

$\displaystyle\left(N^{x},\ N^{y},\ N^{z},\ V\right)\sim\left(\sin\theta,\
\cos\theta,\ \sin\phi,\ \cos\phi\right).$ (54)

Under decoherence or weak measurement of (for example) local operator
$e^{\mathrm{i}\hat{\phi}}$, the density matrix becomes

$\displaystyle\hat{\rho}^{D}={\cal E}[\hat{\rho}_{0}],\ \ \ {\cal
E}=\prod_{x}{\cal E}_{x},$ (55) $\displaystyle{\cal E}_{x}[\hat{\rho}_{0}]$
$\displaystyle\sim$ $\displaystyle(1-p)\hat{\rho}_{0}+\frac{p}{2}\
e^{\mathrm{i}\hat{\phi}(x)}\hat{\rho}_{0}e^{-\mathrm{i}\hat{\phi}(x)}$ (57)
$\displaystyle+$ $\displaystyle\frac{p}{2}\
e^{-\mathrm{i}\hat{\phi}(x)}\hat{\rho}_{0}e^{\mathrm{i}\hat{\phi}(x)}.$ (59)

$\hat{\rho}_{0}$ is the undecohered density matrix of the spin-1/2 chain, and
$\hat{\rho}^{D}={\cal E}[\hat{\rho}_{0}]$ still keeps
$\mathrm{tr}[\hat{\rho}^{D}]=1$.

We can first evaluate the correlation function of local operator $O(r)$:

$\displaystyle
C^{D}(r)=\mathrm{tr}\\{\hat{\rho}^{D}\hat{O}(r)\hat{O}(0)\\}=\mathrm{tr}\\{\hat{\rho}_{0}{\cal
E}[\hat{O}(r)\hat{O}(0)]\\},$ (60) (61) $\displaystyle{\cal
E}_{x}[\hat{O}(r)\hat{O}(0)]\sim(1-p)\hat{O}(r)\hat{O}(0)+\frac{p}{2}\
e^{\mathrm{i}\hat{\phi}(x)}\hat{O}(r)\hat{O}(0)e^{-\mathrm{i}\hat{\phi}(x)}+\frac{p}{2}\
e^{-\mathrm{i}\hat{\phi}(x)}\hat{O}(r)\hat{O}(0)e^{\mathrm{i}\hat{\phi}(x)}$
(62)

For local bosonic order parameters at position $r$, such as the Néel and VBS,
their nontrivial commutation with $e^{\mathrm{i}\hat{\phi}(x)}$ would be
exponentially suppressed by the separation $|r-x|$ at long distance, and thus
for $|r-x|\gg 1$, $\hat{O}(r)$ and $e^{\mathrm{i}\hat{\phi}(x)}$ should
commute. When $\hat{O}\sim\cos(\phi)$ or $\sin(\phi)$, since
$e^{i\hat{\phi}(x)}$ always commute with $\hat{O}(0)$ and $\hat{O}(r)$, ${\cal
E}_{x}[\hat{O}(r)\hat{O}(0)]=\hat{O}(r)\hat{O}(0)$, and the decoherence does
not affect the correlator. When $\hat{O}=N^{x}$ or $N^{y}$, the decoherence
channel does not commute with $\hat{O}(r)$ only in the vicinity of $r$.
However, ($i$) the field $(N^{x},N^{y})$ is already the primary field with the
lowest scaling dimension that forms an irreducible representation of the
$\mathrm{U}(1)$ rotation symmetry generated by $\sum_{x}S^{z}_{x}$ of the spin
chain ($\theta\mapsto\theta+\alpha$), and ($ii$) the decoherence channel
explicitly preserves this $\mathrm{U}(1)$ symmetry,
$e^{\mathrm{i}\hat{\phi}(r)}(N^{x}_{r},N^{y}_{r})e^{-\mathrm{i}\hat{\phi}(r)}$
belong to the same representation as $(N^{x}_{r},N^{y}_{r})$ and would not
generate any field with lower scaling dimension. Therefore, under decoherence
the correlation functions of local operators $(N^{x},N^{y},N^{z},V)$ acquire
only a constant amount of multiplicative factor aside from potential
subleading power-law contributions, implying that they should still have the
same power-law with scaling dimensions of the undecohered spin-1/2 chain
444The conclusion here should generally hold for decoherence channel and
correlation functions of local operators, i.e. as long as (i) the local
operator is the primary field with the lowest scaling dimension of a certain
representation of the symmetry, and (ii) the decoherence channel explicitly
preserves this symmetry (with a “doubled” symmetry condition) and commute with
the local operator except for the close vicinity of the operator, then the
decoherence should not change the long distance scaling of the correlation
function. .

However, for nonlocal operators, the situation can be very different. A
particular family of nonlocal operators is the disorder operators [53, 54, 55,
56, 57, 58, 59, 60] 555It was also called by other names, such as patch
operator [54], and order diagnosis operator [55, 57], etc., which have
attracted great interests recently. These operators have been used as an
auxiliary diagnosis for the states of matter, especially for critical states
of matter. For a $1d$ quantum rotor, if we view $\hat{\phi}$ as the phase
angle of a local boson creation operator, then $(\nabla_{x}\hat{\theta})/2\pi$
is the boson density $\hat{n}_{\phi}$. The following operator is called a
disorder operator

$\displaystyle\tilde{O}_{r}$ $\displaystyle=$
$\displaystyle\exp\left(\mathrm{i}\alpha\int_{0}^{r}dx\
\hat{n}_{\phi}(x)\right)$ (63) $\displaystyle=$
$\displaystyle\exp\left(\mathrm{i}\frac{\alpha}{2\pi}(\hat{\theta}(r)-\hat{\theta}(0))\right).$
(65)

If $\hat{\phi}$ carries a full U(1) symmetry, then $\alpha\in\mathbb{R}$; if
$\hat{\phi}$ only carries a $\mathbb{Z}_{N}$ symmetry, then $\alpha=\frac{2\pi
k}{N}$ with $k\in\\{1,...,N\\}$, as $\hat{n}_{\phi}$ is defined modulo $N$.

For example, for the spin-1/2 chain, if we view $\hat{\phi}$ as a local
operator, then $e^{\mathrm{i}\hat{\theta}/2}$ is a disorder operator of
$\hat{\phi}$, and $e^{\mathrm{i}\hat{\theta}/2}$ plays two roles
simultaneously: it first creates a fractionalized spin-1/2 excitation (i.e. a
spinon), it also creates a domain wall of the VBS order parameter, i.e.
$e^{\mathrm{i}\hat{\theta}(r)/2}$ shifts
$\hat{\phi}(x)\rightarrow\hat{\phi}(x)+\pi$ for $x<r$. Indeed, it is well-
known that a spin-1/2 is localized at the domain wall between two VBS orders.
We can evaluate the correlation function of $e^{\mathrm{i}\hat{\theta}/2}$ for
the decohered density matrix:

$\displaystyle C^{D}(r)$ $\displaystyle=$
$\displaystyle\mathrm{tr}\\{\hat{\rho}^{D}e^{\mathrm{i}\hat{\theta}(r)/2}\
e^{-\mathrm{i}\hat{\theta}(0)/2}\\}$ (66) $\displaystyle\sim$ $\displaystyle
e^{-r/\xi}\mathrm{tr}\\{\hat{\rho}_{0}\ e^{\mathrm{i}\hat{\theta}(r)/2}\
e^{-\mathrm{i}\hat{\theta}(0)/2}\\},$ (68)

where $\mathrm{tr}\\{\hat{\rho}_{0}e^{\mathrm{i}\hat{\theta}(r)/2}\
e^{-\mathrm{i}\hat{\theta}(0)/2}\\}$ is the correlation function of
$e^{\mathrm{i}\hat{\theta}/2}$ for undecohered spin-1/2 chain, and the
“correlation length” $\xi$ is $\xi\sim-1/\ln(1-2p)$ for small $p$.

Hence our calculation for the $1d$ quantum rotor system implies that, although
the disorder operator has a power-law correlation with the absence of
decoherence, it can be rendered short-ranged under decoherence. As we will
show in the next subsection, similar behavior of the disorder operator happens
in higher dimensions as well. The spin-1/2 chain is also an example of spin
liquid with fractionalized spinon excitations since the spinon correlation
function decays as a power-law in the undecohered spin-1/2 chain. But under
decoherence or weak measurement on the VBS order parameter the spin chain
loses its fractionalization, as the spinon operator decays exponentially. This
can be intuitively understood as the fact that, if the VBS operator is
“measured”, in each measurement outcome the system is pinned to a certain
particular VBS pattern, which leads to confinement in this measurement
outcome. The confinement persists even if we average over all measurement
outcomes.

###  IV.2 $(2+1)d$ quantum critical points with a U(1) or $\mathbb{Z}_{N}$
symmetry

Now let us consider a $(2+1)d$ QCP or CFT with a global U(1) symmetry, or
$\mathbb{Z}_{N}$ symmetry that can be embedded into a U(1) that emerges in the
infrared. This U(1) symmetry is dual to a noncompact U(1) gauge field [62, 63,
64]. We always turn on decoherence on the scalar boson creation operator which
carries the U(1) charge, or equivalently the monopole operator of the dual
U(1) gauge field. The decohered density matrix takes the same form as Eq. 59:

$\displaystyle\hat{\rho}^{D}={\cal E}[\hat{\rho}_{0}],\ \ \ {\cal
E}=\prod_{\bm{x}}{\cal E}_{\bm{x}},$ (69) $\displaystyle{\cal
E}_{\bm{x}}[\hat{\rho}_{0}]$ $\displaystyle\sim$
$\displaystyle(1-p)\hat{\rho}_{0}+\frac{p}{2}\
e^{\mathrm{i}\hat{\phi}({\bm{x}})}\hat{\rho}_{0}e^{-\mathrm{i}\hat{\phi}({\bm{x}})}$
(71) $\displaystyle+$ $\displaystyle\frac{p}{2}\
e^{-\mathrm{i}\hat{\phi}({\bm{x}})}\hat{\rho}_{0}e^{\mathrm{i}\hat{\phi}({\bm{x}})}.$
(73)

Here $e^{\mathrm{i}\hat{\phi}({\bm{x}})}$ is the monopole operator, which
creates a scalar boson, or a $2\pi$ gauge flux at location ${\bm{x}}$.

We evaluate the expectation value of the following quantity defined for a
closed loop ${\cal C}=\partial{\cal A}$:

$\displaystyle\tilde{O}_{\cal C}=\exp\bigg(\sum_{{\bm{x}}\in{\cal
A},\partial{\cal A}={\cal
C}}\frac{\mathrm{i}2\pi}{N}\hat{n}_{\phi}({\bm{x}})\bigg{missing}).$ (74)

In the undecohered density matrix, and in the dual formalism, this quantity
reduces to the evaluation of the Wilson loop:

$\displaystyle\langle\tilde{O}_{\cal C}\rangle$ $\displaystyle\sim$
$\displaystyle\langle\exp\bigg(\frac{\mathrm{i}}{N}\oint_{\cal
C}d{\bm{x}}\cdot\hat{{\bm{a}}}({\bm{x}})\bigg{missing})\rangle,$ (75)

and as was shown previously, it should obey a perimeter law, with a universal
logarithmic contribution from the sharp corners of the loop ${\cal C}$ [57,
58, 65]. The coefficient of the universal logarithmic contribution arising
from the corner is proportional to the universal conductivity of the scalar
boson current at the $(2+1)d$ CFT in the AC limit $\omega/T\rightarrow\infty$.

However, under decoherence, the operator $\tilde{O}_{\cal C}$ will shift
$\hat{\phi}({\bm{x}})$ by angle $2\pi/N$ for ${\bm{x}}\in{\cal A}$. Hence we
expect the decoherence to change the behavior of $\langle\tilde{O}_{\cal
C}\rangle$ significantly:

$\displaystyle\langle\tilde{O}_{\cal
C}\rangle\sim\left((1-p)+p\cos\left(\frac{2\pi}{N}\right)\right)^{\cal A},$
(76)

namely $\langle\tilde{O}_{\cal C}\rangle$ should now decay with an area law.
Just like the $1d$ example discussed in the previous subsection, an area law
decay of the Wilson loop is a sign of confinement. It means that the vortex of
the $\mathrm{U}(1)$ boson, which is also the gauge charge of the dual gauge
field $\hat{{\bm{a}}}$, should be confined under decoherence of the scalar
boson creation operator. Here we would like to remark that, one of the tools
for diagnosing fractionalization and deconfinement is the dynamic structure
factor, where the fractionalization would lead to a continuum [66, 67, 68].
Computing real-time dynamics is beyond the current set-up of our current
manuscript as it requires the formalism that involves the Lindbladian. Here we
use the behavior of the Wilson loop as the sign of confinement/deconfinement.

##  V summary and future directions

In this study, we examined the effects of decoherence and weak measurement on
quantum critical points in $(2+1)d$ space-time. We found that this problem is
mathematically equivalent to the boundary or defect criticality of $(2+1)d$
conformal field theories, which have been extensively researched in recent
years. Our results indicate that when a QCP is exposed to decoherence or weak
measurement, observers may observe peculiar behaviors, including the
extraordinary-log correlation recently discovered in the context of boundary
criticality. Additionally, as the strength of decoherence or weak measurement
increases, the system can experience an information-theoretic transition that
is captured by quantities nonlinear in the decohered density matrix. This
transition is linked to spontaneous symmetry breaking when we consider
quantities to the $n$-th power of the density matrix; in particular using the
“doubled formalism”, we provided a specific example where the transition
belongs to the $2d$ Ising universality class for $n=2$. By KW duality, this
transition is connected to the _error threshold_ transition in the decohered
toric code, which is another type of information-theoretic transition in
quantum systems.

There are many related directions that are very much worth exploring in the
future. We list two such directions as follows:

(1) Unconventional quantum criticality under decoherence: It is known that in
the world of quantum many-body systems, there are two types of quantum
critical points: conventional and unconventional. Conventional QCPs correspond
to quantum phase transitions between a disordered state that can be
adiabatically connected to a direct product state, and a state that
spontaneously breaks certain symmetry. This type of QCPs has classical analogs
such as the Wilson-Fisher fixed points discussed in this work. Well-understood
examples of such conventional QCP can be found in Ref. [69], including the QCP
of the transverse field quantum Ising model, which is between the Ising
ordered phase that spontaneously breaks the Ising symmetry, and a symmetric
disordered phase that is adiabatically connected to a direct product state.
For a two-dimensional quantum Ising model, the QCP is described by the $3D$
Ising Wilson-Fisher fixed point. Another well-known example of such
conventional QCP is the quantum phase transition between the superfluid phase
and the Mott insulator phase in the Bose Hubbard model at integer filling
[70], which belongs to the 3D XY Wilson-Fisher universality class if the model
is built on a two-dimensional lattice.

On the other hand, unconventional QCPs are those that do not have a simple
classical analog and may involve transitions between two ordered phases with
different symmetries, or between an ordered phase and a topological order. The
most well-known example of unconventional QCPs is the deconfined QCP [71, 72],
which has many desirable phenomena such as deconfinement and a duality web as
was summarized in Ref. 73. There is another large class of unconventional QCPs
which involve quantum disordered states that cannot be adiabatically connected
to a direct product state, such as the topological orders. One early example
of such quantum phase transitions can be found in Ref. [74]. It is reasonable
to expect that the unconventional QCPs under decoherence can also be mapped to
certain boundary criticality problems, and it is going to be an unusual
boundary criticality with unconventional QCP in the bulk. As we have already
seen in the current work, decoherence may be at odds with deconfinement, as
deconfinement is often signified by nonlocal operators such as the disordered
operators or the Wilson loops, whose behavior can be strongly affected by
decoherence. It would be interesting to study the fate of unconventional QCPs
under decoherence in general in the future.

(2) The Strange Correlator: The notion of a strange correlator was originally
proposed as a tool to diagnose SPT states using their bulk wave functions
[17], rather than edge states. The strange correlator is defined as the
following quantity

$\displaystyle
C^{S}({\bm{r}})=\frac{\langle\Omega|\hat{O}({\bm{0}})\hat{O}({\bm{r}})|\Psi\rangle}{\langle\Omega|\Psi\rangle},$
(77)

where $|\Psi\rangle$ is the wave function that awaits diagnosis, and
$|\Omega\rangle$ is the trivial direct product disordered state with the same
symmetry $G$ and Hilbert space as $|\Psi\rangle$. $\hat{O}$ is an order
parameter that carries a nontrivial representation of $G$. The arguments given
in Ref. 17 suggest that, although the ordinary correlation functions in both
$|\Psi\rangle$ and $|\Omega\rangle$ must be short-ranged, this strange
correlator Eq. 77 must have either long-ranged or power-law correlation, for
$1d$ and $2d$ states. In the past decade, the strange correlator has been used
as a tool for both conceptual understanding and numerical diagnosis for SPT
states and also topological states [75, 76, 77, 78, 79, 80, 81, 82, 83, 84,
85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95].

One of the future directions worth pursuing is the strange correlator between
a quantum critical state $|\Omega\rangle$, and an SPT wave function
$|\Psi\rangle$. Let us still focus on two-dimensional systems. Using the
formalism developed in this work, this problem may be mapped to the $2d$
interface between a quantum critical point on the temporal domain $\tau<0$,
and an SPT state on the other domain $\tau>0$ in the Euclidean space-time
path-integral. Under space-time rotation, the strange correlator is mapped to
the spatial interface between a quantum criticality and an SPT state. This
kind of interface has two types of boundary effects: the boundary states
arising from the bulk topology, and also the boundary criticality originating
from the bulk critical modes. This is a subject under very active research
lately, both theoretically and numerically [18, 19, 20, 21, 22, 24, 27]. In
particular, some novel interface criticality especially a $(1+1)d$ deconfined
quantum critical point was identified in the literature [24]. One potentially
highly interesting direction in the future is to analyze the strange
correlator (and its generalized form defined in Ref. 12) between quantum
criticality and SPT state, and explore the possible novel phenomena,
especially when either the bulk quantum critical state $|\Omega\rangle$, or
the SPT state $|\Psi\rangle$, or both are under decoherence.

###### Acknowledgements.

We thank Ehud Altman, Soonwon Choi, Matthew P. A. Fisher, Sam Garrett, Yi-
Zhuang You for inspiring discussions and previous collaborations. J.Y.L. is
supported by the Gordon and Betty Moore Foundation under the grant GBMF8690
and by the National Science Foundation under the grant PHY-1748958. C. X.
acknowledges the support from the Simons Foundation through the Simons
Investigator program. C.-M. J. is supported by a faculty startup grant at
Cornell University. _Note Added:_ While finishing up this work, we became
aware of an independent related work [96, 97], which should appear on arXiv on
the same day as our work.

##  Appendix A Toric code under decoherence

The goal of this section is to show that the entanglement entropy of the toric
code state under dephasing noise is given by the free energy of the random
bond Ising model along the Nishimori line. We will show that such a decohered
state is dual to the disordered product state under $\mathbb{Z}_{2}$ symmetric
decoherence channel in Eq. (44).

First, the toric code Hamiltonian is defined as

$\displaystyle H$ $\displaystyle=-\sum_{v}\prod_{e\ni
v}{Z}_{e}-\sum_{p}\prod_{e\in p}X_{e}$
$\displaystyle=-\sum_{v}A_{v}-\sum_{p}B_{p}$ (78)

The ground state is characterized by $A_{v}=B_{p}=1$. Furthermore, on the
torus, the ground state is 4-fold degenerate with two logical qubits. Logical
qubits reside on the space where the following effective Pauli operators act
on $\bm{C}^{X}_{i}\equiv\prod_{e\in C_{i}}X_{e}$ and
$\bm{C}^{Z}_{i}\equiv\prod_{e\in C^{\perp}_{i}}Z_{e}$ where $C_{i}$ is a cycle
along the $i$-th axis; while $C_{i}$ is along the bond, $C_{i}^{\perp}$
crosses the bond. Note that
$\\{\bm{C}^{X}_{1},\bm{C}^{Z}_{2}\\}=\\{\bm{C}^{Z}_{1},\bm{C}^{X}_{2}\\}=0$,
while $[\bm{C}^{Z}_{i},\bm{C}^{X}_{i}]=0$. As an example we choose to study
one of the four ground states denoted as $|\psi_{\textrm{tc}}\rangle$, whose
pure state density matrix is
$\rho_{\textrm{tc}}=|\psi_{\textrm{tc}}\rangle\langle\psi_{\textrm{tc}}|$. The
ground state is the eigenstate of the qubits
$\bm{C}^{X}_{i}|\psi_{\textrm{tc}}\rangle=a_{i}|\psi_{\textrm{tc}}\rangle$,
with $a_{1}=a_{2}=1$.

###  A.1 Decomposition

We consider the toric code ground state decohered under the following channel:

$\displaystyle{\cal E}_{e}:\rho$ $\displaystyle\rightarrow(1-p)\rho+pZ_{e}\rho
Z_{e},\quad{\cal E}=\prod_{e}{\cal E}_{e}$ (79)

In order to understand the structure of ${\cal E}[\rho_{\textrm{tc}}]$, first
we evaluate the matrix elements of the decohered toric code density matrix. To
do the job, consider
$\rho_{{\bm{s}},{\bm{s}}^{\prime}}\equiv|\Omega_{{\bm{s}}^{\prime}}\rangle\langle\Omega_{\bm{s}}|$
where $|\Omega_{{\bm{s}}}\rangle$ is a generic product state characterized by
${\bm{s}}=\\{s_{e}\\}$, $s_{e}=\pm 1$:

$\displaystyle|\Omega_{\bm{s}}\rangle\equiv\prod_{e}Z^{(1-s_{e})/2}|+\rangle^{\otimes{2N_{v}}},$
(80)

where $N_{v}=L^{2}$ is the number of vertices. Then

$\displaystyle\langle\Omega_{\bm{s}}|{\cal
E}[\rho_{\textrm{tc}}]|\Omega_{{\bm{s}}^{\prime}}\rangle$
$\displaystyle=\mathrm{tr}(\rho_{{\bm{s}},{\bm{s}}^{\prime}}{\cal
E}[\rho_{\textrm{tc}}])=\mathrm{tr}({\cal
E}[\rho_{{\bm{s}},{\bm{s}}^{\prime}}]\rho_{\textrm{tc}})$ (81)

where ${\cal E}[\rho_{{\bm{s}},{\bm{s}}^{\prime}}]$ is given as

$\displaystyle\rho_{{\bm{s}},{\bm{s}}^{\prime}}$
$\displaystyle=\frac{1}{2^{{2N_{v}}}}\prod_{e}(1+s_{e}X_{e})Z_{e}^{(1-s_{e}s^{\prime}_{e})/2}$
$\displaystyle{\cal E}[\rho_{{\bm{s}},{\bm{s}}^{\prime}}]$
$\displaystyle=\frac{1}{2^{{2N_{v}}}}\prod_{e}(1+s_{e}(1-2p)X_{e})Z_{e}^{(1-s_{e}s^{\prime}_{e})/2}$
(82)

For $\mathrm{tr}({\cal
E}[\rho_{{\bm{s}},{\bm{s}}^{\prime}}]\rho_{\textrm{tc}})$ not to vanish,
$\partial({\bm{s}}\cdot{\bm{s}}^{\prime})=0$ so that product of $Z_{e}$ does
not create anyons. In such a case, the product of $Z$-strings always commutes
with a loop of $X$-strings along the bond. In fact, we can show that

$\displaystyle\langle\psi_{\textrm{tc}}|\prod_{e\in
l}X_{e}\prod_{e}Z_{e}^{(1-s_{e}s_{e}^{\prime})/2}|\psi_{\textrm{tc}}\rangle=F(l,{\bm{s}}\cdot{\bm{s}}^{\prime})\delta_{\partial
l,0}\delta_{\partial({\bm{s}}\cdot{\bm{s}}^{\prime}),0}$ (83)

where ${\bm{s}}\cdot{\bm{s}}^{\prime}$ defines a non-trivial link
configuration along the _dual link_ whenever it takes a negative value. For
$\langle\bm{C}_{1,2}^{X,Y,Z}\rangle=c^{x,y,z}_{1,2}$ we have

$\displaystyle
F(l,{\bm{s}}\cdot{\bm{s}}^{\prime})=\langle\psi_{\textrm{tc}}|\bm{C}_{h(l)}^{X}\bm{C}_{h({\bm{s}}\cdot{\bm{s}}^{\prime})}^{Z}|\psi_{\textrm{tc}}\rangle$
(84)

where $h(l)\in\pi_{1}(\mathbb{T}^{2})$ is the element of the homotopy group of
the torus. For the simplest case where $\bm{C}_{1,2}^{X}=1$, we remark that
$F(l,{\bm{s}}\cdot{\bm{s}}^{\prime})$ is non-zero if and only
if$h({\bm{s}},{\bm{s}}^{\prime})$ is trivial. Then, the overlap in Eq. (81) is
given as

$\displaystyle\mathrm{tr}({\cal
E}[\rho_{{\bm{s}},{\bm{s}}^{\prime}}]\rho_{\textrm{tc}})$
$\displaystyle=\frac{1}{2^{{2N_{v}}}}\sum_{l}(1-2p)^{\absolutevalue{l}}\prod_{e\in
l}s_{e}$ $\displaystyle\times\langle\psi_{\textrm{tc}}|\prod_{e\in
l}X_{e}\prod_{e}Z_{e}^{(1-s_{e}s_{e}^{\prime})/2}|\psi_{\textrm{tc}}\rangle$
$\displaystyle=\frac{\delta_{h({\bm{s}}\cdot{\bm{s}}^{\prime}),\bm{1}}}{2^{N_{v}}(2\cosh\beta)^{{2N_{v}}}}Z_{\textrm{RBIM}}[{\bm{s}},\beta]$
(85)

where the summation is over all possible edge configuration $l$,
$\tanh\beta=(1-2p)$, and

$\displaystyle
Z_{\textrm{RBIM}}[{\bm{s}},\beta]\equiv\sum_{{\bm{\sigma}}}\prod_{e=(v,v^{\prime})}e^{\beta
s_{e}\sigma_{v}\sigma_{v^{\prime}}},$ (86)

which turns out to be the partition function of an Ising model with random
signs, specified by $\\{s_{e}\\}$, in the near-neighbor spin-spin interaction.
At $p=0$, $\beta^{-1}=0$, i.e., zero temperature limit, It has an interesting
consequence: the matrix element $\langle\Omega_{\bm{s}}|{\cal
E}[\rho_{\textrm{tc}}]|\Omega_{{\bm{s}}^{\prime}}\rangle$ vanishes unless
${\bm{s}}$ and ${\bm{s}}^{\prime}$ belong to the same equivalence class, i.e.,
$\partial({\bm{s}}\cdot{\bm{s}}^{\prime})=0$. Furthermore, if
${\bm{s}}\sim{\bm{s}}^{\prime}$, then
$Z_{\textrm{RBIM}}[{\bm{s}},\beta]=Z_{\textrm{RBIM}}[{\bm{s}}^{\prime},\beta]$.
Accordingly, ${\cal E}[\rho_{\bm{s}}]$ is block-diagonal. Therefore, ${\cal
E}[\rho_{\textrm{tc}}]$ decomposes as the following:

$\displaystyle{\cal E}[\rho_{\textrm{tc}}]$
$\displaystyle=\sum_{{\bm{s}},{\bm{s}}^{\prime}}\rho_{{\bm{s}},{\bm{s}}^{\prime}}\mathrm{tr}(\rho_{{\bm{s}},{\bm{s}}^{\prime}}{\cal
E}[\rho_{\textrm{tc}}])$
$\displaystyle=\frac{1}{2^{N_{v}}(2\cosh\beta)^{{2N_{v}}}}\sum_{\bm{m}}Z_{\textrm{RBIM}}[{\bm{s}}_{\bm{m}},\beta]\,\rho_{{\bm{m}}}.$
(87)

where the summation is taken over the equivalence class of ${\bm{s}}$, denoted
by ${\bm{m}}$; the equivalence class is defined as
$\partial({\bm{s}}\cdot{\bm{s}}^{\prime})=0$. ${\bm{s}}_{\bm{m}}$ is the
representative of the equivalence class ${\bm{m}}$, and $\rho_{\bm{m}}$ is
defined as

$\displaystyle\rho_{\bm{m}}\equiv\sum_{{\bm{s}},{\bm{s}}^{\prime}\sim{\bm{s}}_{\bm{m}}}|\Omega_{\bm{s}}\rangle\langle\Omega_{{\bm{s}}^{\prime}}|.$
(88)

Therefore, the decohered density matrix has the following block-diagonal
structure (in $X$ basis)

$\displaystyle{\cal
E}[\rho_{\textrm{tc}}]=\begin{bmatrix}\,\framebox{$\bm{B}_{1}$}&0&0&\,\,\cdots\,\\\
0&\framebox{$\bm{B}_{2}$}&0&\,\,\cdots\\\
0&0&\framebox{$\bm{B}_{3}$}&\,\,\cdots\\\
\vdots&\vdots&\vdots&\,\,\ddots\end{bmatrix}$ (89)

where each block is $2^{N_{v}-1}$ by $2^{N_{v}-1}$ dimensional matrix labeled
by the equivalence class ${\bm{m}}$ and its entries are all equal, i.e.,

$\displaystyle\bm{B}_{i}\propto\underbrace{\begin{bmatrix}1&1&1&\,\cdots\\\
1&1&1&\,\cdots\\\ 1&1&1&\,\cdots\\\
\raisebox{3.0pt}{\scalebox{0.75}{$\vdots$}}&\raisebox{3.0pt}{\scalebox{0.75}{$\vdots$}}&\raisebox{3.0pt}{\scalebox{0.75}{$\vdots$}}&\,\raisebox{3.0pt}{\scalebox{0.75}{$\ddots$}}\end{bmatrix}}_{\displaystyle
2^{N_{v}-1}}\left.\vphantom{\begin{bmatrix}1&1&1&\,\cdots\\\ 1&1&1&\,\cdots\\\
1&1&1&\,\cdots\\\
\raisebox{3.0pt}{\scalebox{0.75}{$\vdots$}}&\raisebox{3.0pt}{\scalebox{0.75}{$\vdots$}}&\raisebox{3.0pt}{\scalebox{0.75}{$\vdots$}}&\,\raisebox{3.0pt}{\scalebox{0.75}{$\ddots$}}\end{bmatrix}}\right\\}2^{N_{v}-1}=2^{N_{v}-1}|\phi_{\bm{m}}\rangle\langle\phi_{\bm{m}}|$
(90)

where
$|\phi_{\bm{m}}\rangle=\frac{1}{\sqrt{2^{N_{v}-1}}}\sum_{{\bm{s}}\sim{\bm{s}}_{\bm{m}}}|\Omega_{\bm{s}}\rangle$.
There are total $2^{N_{v}+1}$ equivalence classes originated from $(N_{v}-1)$
independent stabilizers $B_{p}$ and two logical operators $\bm{C}^{X}_{i}$.

###  A.2 Entanglement Entropy

After reorganizing terms, we get

$\displaystyle\rho_{\textrm{tc}}^{D}$
$\displaystyle=\sum_{\bm{m}}p_{\bm{m}}|\phi_{\bm{m}}\rangle\langle\phi_{\bm{m}}|,\quad
p_{\bm{m}}=\frac{Z_{\textrm{RBIM}}[{\bm{s}}_{\bm{m}},\beta]}{2\cdot(2\cosh\beta)^{{2N_{v}}}}.$
(91)

Note that the entanglement entropy has a very interesting structure:

$\displaystyle S$
$\displaystyle=-\mathrm{tr}(\rho_{\textrm{tc}}^{D}\ln\rho_{\textrm{tc}}^{D})$
$\displaystyle=-\sum_{\bm{m}}p_{\bm{m}}\log p_{\bm{m}}$
$\displaystyle\propto-\sum_{\bm{m}}Z[{\bm{s}}_{\bm{m}},\beta]\log
Z[{\bm{s}}_{\bm{m}},\beta]$ (92)

which is nothing but a disorder averaged random bond Ising model’s free energy
along the Nishimori line, whose transition point is located at $p_{c}=0.1094$
[51]. Therefore, there is an _intrinsic_ phase transition of the entanglement
entropy of the decohered toric code state at $p_{c}=0.1094$, which coincides
with the decodability transition point obtained in Ref. 31.

###  A.3 Purity

Interestingly, the purity of the decohered density matrix maps to the
partition function of the Ising model: (Here
$C\equiv({2^{2}\cdot(2\cosh\beta)^{4N_{v}}})^{-1}$):

$\displaystyle\mathrm{tr}((\rho_{\textrm{tc}}^{D})^{2})=C\sum_{{\bm{m}}}Z^{2}_{\textrm{RBIM}}[{\bm{s}}_{\bm{m}},\beta]=C\sum_{\bm{s}}\frac{Z^{2}_{\textrm{RBIM}}[{\bm{s}},\beta]}{2^{N_{v}-1}}$
$\displaystyle=\frac{C(\cosh\beta)^{4N_{v}}}{2^{N_{v}-1}}\sum_{\bm{s}}\sum_{{\bm{\sigma}}}\sum_{{\bm{\sigma}}^{\prime}}\sum_{L_{1},L_{2}}(1-2p)^{|L_{1}|+|L_{2}|}$
$\displaystyle\qquad\times\prod_{e\in L_{1}}s_{e}\prod_{e^{\prime}\in
L_{2}}s_{e^{\prime}}\prod_{i\in\partial L_{1}}\sigma_{i}\prod_{j\in\partial
L_{2}}\sigma^{\prime}_{j}$ (93)

where $L_{i}$ is the link configuration defined on the square lattice. This
expression can be further simplified by the following:

$\displaystyle\mathrm{tr}((\rho_{\textrm{tc}}^{D})^{2})$
$\displaystyle=\frac{1}{2^{4N_{v}+N_{v}+1}}\sum_{{\bm{\sigma}}}\sum_{{\bm{\sigma}}^{\prime}}\sum_{L_{1},L_{2}}(1-2p)^{|L_{1}|+|L_{2}|}$
$\displaystyle\qquad\times 2^{{2N_{v}}}\delta_{L_{1},L_{2}}\prod_{i\in\partial
L_{1}}\sigma_{i}\prod_{j\in\partial L_{2}}\sigma^{\prime}_{j}$
$\displaystyle=\frac{1}{2^{4N_{v}+N_{v}+1}}\sum_{L_{1},L_{2}}(1-2p)^{|L_{1}|+|L_{2}|}\cdot
2^{{2N_{v}}}\delta_{L_{1},L_{2}}$ $\displaystyle\qquad\times
2^{N_{v}}\delta_{\partial L_{1},0}\cdot 2^{N_{v}}\delta_{\partial L_{2},0}$
$\displaystyle=\frac{1}{2^{N_{v}+1}}\sum_{L}(1-2p)^{2\absolutevalue{L}}\delta_{\partial
L,0}$
$\displaystyle=\frac{Z_{\textrm{FIM}}[\beta^{\prime}]}{2(2\cosh\beta^{\prime})^{{2N_{v}}}},\quad\tanh\beta^{\prime}=(1-2p)^{2}$
(94)

where
$Z_{\textrm{FIM}}[\beta^{\prime}]=Z_{\textrm{RBIM}}[\bm{1},\beta^{\prime}]$ is
the partition function of the ferromagnetic Ising model. In the last equality,
we used that

$\displaystyle\sum_{\partial\gamma=0}(\tanh\beta)^{|\gamma|}\prod_{e\in\gamma}s_{e}=\frac{Z_{\textrm{RBIM}}[s,\beta]}{2^{N_{v}}(\cosh\beta)^{{2N_{v}}}}.$
(95)

Since the ferromagnetic 2d Ising model has a transition at
$\beta=\ln(1+\sqrt{2})/2=0.441$, correlation functions in the doubled Hilbert
space (in the next section) would exhibit a critical behavior at
$p^{(2)}_{c}=0.178$.

###  A.4 Choi Isomorphism

One may study the decohered toric code state in the doubled Hilbert space
under Choi isomorphism. The decohered density matrix maps into the following
Choi state:

$\displaystyle\|{\cal
E}[\rho_{\textrm{tc}}]\rangle\rangle=\sum_{i}|i\rangle\otimes({\cal
E}[\rho_{\textrm{tc}}]|i\rangle)=\sum_{\bm{m}}p_{\bm{m}}|\phi_{\bm{m}}\rangle|\phi_{\bm{m}}\rangle$
(96)

where we used Eq. (91). The dephasing channel maps to the following Choi
operator in the doubled Hilbert space:

$\displaystyle\hat{{\cal E}}$ $\displaystyle=\prod_{e}(1-2p)^{1/2}e^{\tau
Z_{e,u}Z_{e,l}},\quad\tanh\tau=\frac{p}{1-p}$ (97)

which can be considered as an imaginary time evolution by an Ising
Hamiltonian. Note that $\cosh\tau=(1-p)/\sqrt{1-2p}$ and
$\sinh\tau=p/\sqrt{1-2p}$. Then, we see that

$\displaystyle\hat{{\cal E}}X_{e,u}\hat{{\cal E}}^{-1}=X_{e,u}e^{-2\tau
Z_{e,u}Z_{e,l}}$ $\displaystyle\Rightarrow\quad\hat{{\cal E}}B_{p}\hat{{\cal
E}}^{-1}=B_{p}\prod_{e\in p}e^{-2\tau Z_{e,u}Z_{e,l}}$ (98)

Now, consider the following parent Hamiltonian:

$\displaystyle H_{\textrm{parent}}$
$\displaystyle=\frac{1}{2}\sum_{v}(1-A_{v})^{\dagger}(1-A_{v})$
$\displaystyle+\frac{1}{2}\sum_{p}(1-B_{p})^{\dagger}(1-B_{p})$ (99)

for some $\alpha,\beta>0$. Following the procedure in Ref. 12, we can show
that the Choi state $\|{\cal E}[\rho_{\textrm{tc}}]\rangle\rangle=\hat{{\cal
E}}\|\rho_{\textrm{tc}}\rangle\rangle$ is the ground state of the following
Hamiltonian:

$\displaystyle\hat{{\cal H}}^{D}$ $\displaystyle=\hat{{\cal
H}}_{u}^{D}+\hat{{\cal H}}_{l}^{D}+\hat{{\cal H}}_{\textrm{int}}^{D}$
$\displaystyle\hat{{\cal H}}_{u}^{D}$
$\displaystyle=-\sum_{v}A_{v,u}-\sum_{p}\cosh(2\tau\sum_{e\in
p}Z_{e,u}Z_{e,l})B_{p,u}$ $\displaystyle\hat{{\cal H}}_{\textrm{int}}^{D}$
$\displaystyle=\sum_{p}\prod_{e\in p}e^{-4\tau Z_{e,u}Z_{e,l}}$ (100)

However, using that $\cosh(2\tau\sum_{e\in p}Z_{e,u}Z_{e,l})\geq 0$ and it
commutes with $A_{v}$ and $B_{p}$, we can simplify it as

$\displaystyle\hat{{\cal H}}^{D}$ $\displaystyle=\hat{{\cal
H}}_{u}^{D}+\hat{{\cal H}}_{l}^{D}+\hat{{\cal H}}_{\textrm{int}}^{D}$
$\displaystyle\hat{{\cal H}}_{u}^{D}$
$\displaystyle=-\sum_{v}A_{v,u}-\sum_{p}B_{p,u}$ $\displaystyle\hat{{\cal
H}}_{\textrm{int}}^{D}$ $\displaystyle=\sum_{p}\prod_{e\in p}(\cosh 2\tau-
Z_{e,u}Z_{e,l}\sinh 2\tau)$ (101)

In this doubled system, the coupling $\hat{{\cal H}}_{\textrm{int}}^{D}$
breaks two microscopic _magnetic_ one-form symmetries into their diagonal
subgroup, and we expect the phase transition from a doubled toric code order
to a single toric code order. The transition should be captured by the
condensation of $e$-anyons, which is diagnosed by non-vanishing expectation
values of

$\displaystyle\langle(\prod_{e\in\gamma^{\perp}}Z_{e,u})(\prod_{e\in\gamma^{\perp}}Z_{e,l})\rangle\sim\textrm{const},$
(102)

where $\gamma^{\perp}$ is the open string defined along the dual lattice.

###  A.5 Kramers-Wannier Duality

Under Kramers-Wannier duality, the toric code maps to the trivially disordered
state, $|\Omega_{0}\rangle=|+\rangle^{\otimes N}$ on the vertices of the dual
lattice. The mapping is explicitly given as the following:

$\displaystyle\prod_{e\ni v}X_{e}$ $\displaystyle\leftrightarrow X_{v}$
$\displaystyle Z_{e\perp(v,v^{\prime})}$ $\displaystyle\leftrightarrow
Z_{v}Z_{v^{\prime}}$ (103)

Note that in this dual model, the system has two 0-form $\mathbb{Z}_{2}$
symmetries even under decoherence. Also, $v$ that labels the vertices in the
dual lattice labels the plaquette in the original lattice. Since
$\prod_{v}X_{v}$ is dual to $\prod_{p}B_{p}=1$ in the toric code, the mapped
states must be symmetric under the 0-form $\mathbb{Z}_{2}$ symmetry. The above
relation makes it clear that the dephasing channel in Eq. (79) maps to the
$\mathbb{Z}_{2}$ symmetric decoherence channel in Eq. (44).

Now, applying the Kramers-Wannier duality on the doubled Hamiltonian in Eq.
(A.4), we can obtain the Hamiltonian for the dual model in the doubled Hilbert
space as the following:

$\displaystyle\hat{{\cal H}}^{D}$ $\displaystyle=\hat{{\cal
H}}_{u}^{D}+\hat{{\cal H}}_{l}^{D}+\hat{{\cal H}}_{\textrm{int}}^{D}$
$\displaystyle\hat{{\cal H}}_{u}^{D}$ $\displaystyle=-2\cosh
2\tau\sum_{v}X_{v}$ $\displaystyle\hat{{\cal H}}_{\textrm{int}}^{D}$
$\displaystyle=2\sum_{v}\prod_{v^{\prime}\in v}(\cosh 2\tau-
Z_{v,u}Z_{v^{\prime},u}Z_{v,l}Z_{v^{\prime},l}\sinh 2\tau)$ (104)

In this model, The transition should be captured by the development of an
order parameter that breaks off-diagonal
$\mathbb{Z}_{2}^{u}\times\mathbb{Z}_{2}^{l}$ symmetry, which is diagnosed by
non-vanishing expectation values of

$\displaystyle\langle(Z_{v,u}Z_{v,l})(Z_{v^{\prime},u}Z_{v^{\prime},l})\rangle\sim\textrm{const}$
(105)

for any well separated $v$ and $v^{\prime}$.

###  A.6 Purity calculation

As stated in the main text, the (unnormalized) groundstate of the above
Hamiltonian in the doubled Hilbert space is given as

$\displaystyle\hat{{\cal
E}}\|\rho_{0}\rangle\rangle=(1-p)^{2N_{v}}\sum_{\bm{l}}(\tanh\tau)^{\absolutevalue{{\bm{l}}}}|\partial{\bm{l}}\rangle\otimes|\partial{\bm{l}}\rangle.$
(106)

The norm of this wavefunction is given as

$\displaystyle\mathrm{tr}\\{(\rho^{D})^{2}\\}$
$\displaystyle=\langle\langle\rho^{D}\|\rho^{D}\rangle\rangle\propto\sum_{{\bm{l}}_{1},{\bm{l}}_{2}}\delta_{\partial{\bm{l}}_{1},\partial{\bm{l}}_{2}}(\tanh\tau)^{|{\bm{l}}_{1}|+|{\bm{l}}_{2}|}$
$\displaystyle\propto\sum_{\\{s_{v}\\}}\Big{[}\prod_{e}\big{(}1+\tanh\tau\prod_{v\in\partial
e}s_{v}\big{)}\Big{]}^{2}$
$\displaystyle\propto\sum_{\\{s_{v}\\}}\prod_{e}\big{(}1+\tanh
2\tau\prod_{v\in\partial e}s_{v}\big{)}$ $\displaystyle\propto
Z_{\textrm{FIM}}[2\tau]$ (107)

where we used the following identity:

$\displaystyle\delta_{\partial{\bm{l}}_{1},\partial{\bm{l}}_{2}}$
$\displaystyle=\frac{1}{2^{N_{v}}}\sum_{s_{v}\in\\{\pm
1\\}}\prod_{v\in\partial{\bm{l}}_{1}}s_{v}\prod_{v^{\prime}\in\partial{\bm{l}}_{2}}s_{v^{\prime}}.$
(108)

Note that at $p=p_{c}^{(2)}$, $2\tau$ here agrees with $\beta^{\prime}$ from
Eq. (A.3), which establishes the self duality in the 2d Ising model.

##  Appendix B Derivatives of the Distance

In this section, we calculate the second derivative of the distance defined in
Eq. (51) for the decohered density matrix $\rho^{D}={\cal
E}[|\Omega_{0}\rangle\langle\Omega_{0}|]$ from Eq. (44). The perturbation is
defined by the following channel

$\displaystyle{\cal M}_{\delta,v}:\rho\rightarrow(1-\delta)\rho+\delta
Z_{v}\rho Z_{v}$ $\displaystyle\rho_{\delta}\equiv{\cal
M}_{\delta}[\rho],\quad{\cal M}_{\delta}\equiv\prod_{v}{\cal M}_{\delta,v}.$
(109)

By defining $h$ such that $\tanh h=\delta/(1-\delta)$
($e^{h}=1/\sqrt{1-2\delta}$), the channel can be mapped to the following
operator under Choi isomorphism:

$\displaystyle\hat{\cal
M}_{\delta}\equiv\prod_{v}(1-2\delta)^{1/2}e^{hZ_{v,u}Z_{v,l}}$ (110)

Note that $\partial h/\partial\delta=1/(1-2\delta)$. Then, we can show that

$\displaystyle\partial_{\delta}D^{(2)}$
$\displaystyle=\frac{\langle\langle\rho_{\delta}^{D}\|\partial_{\delta}\rho^{D}_{\delta}\rangle\rangle}{\langle\langle\rho_{\delta}^{D}\|\rho^{D}_{\delta}\rangle\rangle}-\frac{\langle\langle\rho^{D}\|\partial_{\delta}\rho^{D}_{\delta}\rangle\rangle}{\langle\langle\rho^{D}\|\rho^{D}_{\delta}\rangle\rangle}$
$\displaystyle\partial_{\delta}^{2}D^{(2)}$
$\displaystyle=\frac{\langle\langle\partial_{\delta}\rho_{\delta}^{D}\|\partial_{\delta}\rho^{D}_{\delta}\rangle\rangle}{\langle\langle\rho_{\delta}^{D}\|\rho^{D}_{\delta}\rangle\rangle}-\Bigg{(}\frac{\langle\langle\rho^{D}\|\partial_{\delta}\rho^{D}_{\delta}\rangle\rangle}{\langle\langle\rho^{D}\|\rho^{D}_{\delta}\rangle\rangle}\Bigg{)}^{2}$
(111)

Let $O\equiv\sum_{v}Z_{v,u}Z_{v,l}$. Then, at $\delta\rightarrow 0$, we
evaluate that

$\displaystyle\partial_{\delta}{\cal M}_{\delta}\big{|}_{\delta\rightarrow 0}$
$\displaystyle=(O-L^{2})$ (112)

Furthermore, exactly at $\delta=0$, $\langle\langle
O\rangle\rangle\big{|}_{\delta=0}=0$ due to the symmetric nature of the
initial decohered density matrix under symmetric decoherence channel. Then,
plugging Eq. (112) into the Eq. (B), it is straightforward to show that the
first derivative of the distance vanishes as expected, and the second
derivative, depending on the order of limit, would be given as Eq. (52) or Eq.
(53).

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