# Symmetry protected topological phases under decoherence

Jong Yeon Lee  Kavli Institute for Theoretical Physics, University of
California, Santa Barbara, CA 93106  Yi-Zhuang You  Department of Physics,
University of California San Diego, La Jolla, CA 92093, USA  Cenke Xu
Department of Physics, University of California, Santa Barbara, CA 93106

###### Abstract

We study a class of symmetry protected topological (SPT) phases under various
types of decoherence, which can drive a pure SPT state into a mixed state. We
demonstrate that the system can still retain the nontrivial topological
information from the SPT ground state even under decoherence, which can arise
from noise or weak measurement. The main quantity that we investigate is the
“strange correlator” proposed previously as a diagnosis for the SPT ground
states, and in this work, we generalize the notion of the strange correlator
to mixed-state density matrices. Using both exact calculations of the
stabilizer Hamiltonians and field theory evaluations, we demonstrate that
under decoherence the nontrivial features of the SPT state can persist in the
two types of strange correlators: type-I and type-II. We show that the
nontrivial type-I strange correlator corresponds to the presence of the SPT
information that can be efficiently identified from experiments, as well as
the persistence of anomalous boundary physics. The nontrivial type-II strange
correlator corresponds to the presence of the original SPT information in the
density matrix, which in principle can be identified to distinguish decohered
mixed states of an SPT and trivial states. Therefore, our work provides a
unified framework to understand decohered SPT phases from the information-
theoretic viewpoint.

##  I Introduction

With rapid progress in various quantum simulator platforms, the study of
quantum many-body states in noisy intermediate-scale quantum (NISQ) platforms
has emerged as an active area of research, attracting growing interest from
both the condensed matter and quantum information communities. In particular,
various exotic quantum states have been realized in these platforms [1, 2, 3,
4, 5, 6, 7], demonstrating the possibility of realizing and manipulating
quantum matter in near-term quantum devices. Among them, symmetry-protected
topological (SPT) [8, 9, 10] phases are of great importance since the SPT
states provide a class of non-trivially entangled quantum systems that can be
realized in the NISQ platforms, and they are also relevant to measurement-
based quantum computation and state preparation [11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. However, despite the fact that
quantum decoherence of various types inevitably occurs in nature, most
theoretical studies have been focusing on the pure SPT states, except for a
few [28, 29, 30]. Therefore, it is timely to ask what defining features of
this highly interesting class of topological quantum phases can be identified
under the presence of decoherence.

It is commonly accepted that SPT states usually exhibit _trivial_ bulk
features, in the sense that the properties such as the correlation between
local order parameters are not distinguishable from those of trivial quantum
disordered states. Since the defining symmetries of the SPT states are
preserved, there are no bulk observables such as macroscopic magnetization. As
a result, characteristic features of SPT states are often understood in terms
of the nontrivial boundary physics, or nontrivial quantum numbers carried by
the symmetry defects. However, it is not so easy to directly extract this
information from a bulk ground state wave function of the SPT phase. In
particular, when the defining symmetry of the SPT state involves spatial
symmetries, an arbitrary open boundary may break the spatial symmetry and
render the boundary state trivial, unless the boundary is carefully designed
to preserve all the spatial symmetries. In order to overcome these
complexities, a notion _strange correlator_ was proposed [31], to diagnose
nontrivial SPT properties using just the bulk ground state wave function,
without reference to either boundaries or defects. With an analogy to the
Wick-rotated correlator in the imaginary space-time at the interface against a
trivial disordered state, the strange correlator defined purely based on the
bulk wave function must be “nontrivial”, $i.e.$ they must either saturate to a
constant or decay as a power-law for $1d$ or $2d$ SPT states. As we will
demonstrate in this work, the strange correlator can also be viewed as a
general “order parameter” for the $1d$ and $2d$ SPT phases, analogous to the
well-known string order parameter of the Haldane phase [32, 33]. Hence the
behavior of the strange correlator can also be regarded as the defining
feature of an SPT state. The strange correlator has been adopted as a useful
tool in both conceptual understanding and numerical studies of SPT phases, for
both bosonic and fermionic systems [34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55].

Figure 1:  Schematic representations of: (a) The density matrix of a SPT state
under decoherence as an imaginary-time path integral. The two boundaries at
the temporal direction are decoupled for a pure state. However, decoherence
acts like a coupling between two boundaries, collapsing the doubled symmetries
down to its diagonal subgroup. (b) Under Wick rotation
$(x,\tau)\rightarrow(\tilde{x},\tilde{\tau})$, the two boundaries along the
temporal direction $\tau$ would become two opposite boundaries along the
spatial direction $\tilde{x}$; also, decoherence acts as a perturbation
connecting two spatial boundaries. (c) The type-I strange correlator where
blue squares corresponding to charged operators (order parameters) are present
only at the $\tau=-\infty$ boundary. The grey (white) sheet corresponds to the
path integral formulation of the SPT (trivial state). (d) The type-II strange
correlator, where charged operators present at both $\tau=\pm\infty$
boundaries.

In any experimental implementation and transmission of an entangled quantum
state, the state experiences certain amount of decoherence that may erase a
characteristic signature defined for a pure state, where the key information
of the original quantum state needs be _decoded_ by the receiving end of the
process. In this work, we exploit the idea of strange correlators to
illustrate that SPT states under decoherence can still retain nontrivial
topological information that is in principle decodable. Nature gives quantum
systems a natural mechanism of decoherence: thermalization. At thermal
equilibrium, all degrees of freedom of the system experience decoherence whose
strength depends on the temperature, and such a “massive” thermal decoherence
drives SPT phases with on-site symmetries to be trivial [56]. Instead, we will
consider a “selective” decoherence to certain degrees of freedom of the
system, and we will demonstrate that this will expose an extremely rich
possibility of physics under decoherence. With decoherence, the quantum state
of interest becomes an _ensemble_ expressed by the density matrix. In the
density matrix formalism, two types of strange correlators naturally emerge,
which we denote as “type-I” and “type-II” strange correlators,  whose
graphical illustration is shown in Fig. 1(c,d). In the pure state limit, the
former one is essentially identical to the original definition of the strange
correlator [31], while the latter is the squared version of the original
strange correlator. From both exact lattice model and field-theoretic
calculations, we will show that the type-II strange correlator still retains
the information of an SPT state under certain selective decoherence, hence the
type-II strange correlator may serve as  a tool to decode the mixed state
originated from the SPT state under a noisy quantum channel, and also a
general method of classifying mixed state density matrices. We also found that
for some SPT states, the type-I strange correlator can retain SPT features
below critical decoherence strength, above which the type-I strange correlator
would become short-ranged and decay by an “area law”.

Taking further steps, we show that nontrivial behaviors of type-I strange
correlators can be probed from experiments. Since the strange correlator is
defined as the operator matrix element between two different states, it is
experimentally challenging to measure and has been used mostly as a conceptual
notion and a numerical tool to diagnose SPT phases in the past. However, we
show that for a broad class of SPT states with decorated defect constructions,
the _marginalized_ version of the type-I strange correlator can be
experimentally probed using measurements and additional classical
computational steps [24]. More interestingly, we reveal that this marginalized
type-I strange correlator provides a unifying framework to understand the non-
local order parameters of an SPT state and the feasibility of preparing a
long-range entangled quantum state by measuring the SPT state. Finally, the
type-II strange correlator can be in principle probed by fidelity estimation
using randomized measurements [57, 58, 59].

The rest of the paper is organized as the following. In Sec. II, we define the
type-I and type-II strange correlators. With exact calculations on stabilizer
Hamiltonians, and numerical computations for models away from the exactly
soluble limit, we illustrate that under selective decoherence, the type-I
strange correlator may be short-ranged, but the type-II strange correlator
stays nontrivial and retain the memory of an underlying SPT phase.
Furthermore, using the example of the 2d cluster state, we show that the
type-I strange correlator can stay nontrivial for a weak decoherence and
undergo a transition into a short-ranged phase at a critical decoherence
strength.

In Sec. III, we evaluate both the type-I and type-II strange correlators using
effective field theory descriptions for generic $1d$ and $2d$ SPT states. In
this formalism, the strange correlators of decohered SPT states at the $d$
spatial dimension reduce to the ordinary correlation functions in the space-
time of the boundary of the SPT state, and the two opposite boundaries are
coupled by interactions in the selected channel.

In Sec. IV, we map a decohered SPT mixed state into a pure state in the
doubled Hilbert space using the so-called Choi-Jamiołkowski isomorphism [60,
61]. We also argue that the type-II strange correlator can be viewed as a tool
to decode the key information of a quantum state transmitted through a noisy
channel.

In Sec. V, we establish the relation between the type-I strange correlators
and more general non-local order parameters, such as the string-order
parameter in the Haldane phase, showing that the type-I strange correlators
provide an upper-bound for non-local order parameters. We also propose a
method to experimentally probe the type-I strange correlator.

##  II Strange Correlators

In this section, we investigate the strange correlators of SPT states defined
by stabilizer Hamiltonians subject to decoherence. We remark that any
stabilizer SPT with zero correlation length has a corresponding decorated
domain wall (defect) construction [62], which facilitates the analytic
calculations.

###  II.1 Basic formalism

Let us first elaborate on the basic formalism used in the paper. Decoherence
is defined as the process that incurs the loss of quantum coherence, evolving
pure states into mixed states. Throughout this paper, we consider local
decoherence models described by the quantum channel ${\cal E}_{i}$ in the
Kraus representation:

${\cal E}_{i}[\rho]=\sum_{m}K^{\vphantom{\dagger}}_{m,i}\rho
K_{m,i}^{\dagger}$ (1)

where $K_{m,i}$ is a Kraus operator with local support in the neighborhood of
the $i$-th site, satisfying
$\sum_{m}K^{\vphantom{\dagger}}_{m,i}K_{m,i}^{\dagger}=1$. The global
decoherence model would be defined by the composition of local decoherence
models, ${\cal E}=\circ_{i}{\cal E}_{i}$.

Once we use the density matrix to describe a state that is invariant under
certain symmetry $G$, the density matrix enjoys a “doubled” symmetry
transformations, $i.e.$ the density matrix $\rho=|\Psi\rangle\langle\Psi|$ is
invariant under separate “left” and “right” multiplication of the symmetry
transformation, as $|\Psi\rangle$ is invariant under $G$: $\rho=g_{L}\rho
g^{\dagger}_{R}$, $g_{L},g_{R}\in G$. However, the notion of symmetry may be
weakened to the density matrix being invariant $\rho=g\rho g^{\dagger}$ (for
$g\in G$) under the adjoint action of the symmetry from both left and right,
which is in connection to the average symmetry discussed in [30, 63]. An
average symmetry $G$ was defined in Ref. 30 for a random ensemble
$\\{\ket{\Psi}\\}$ of states subject to a probability distribution
$P(\ket{\Psi})$ that is invariant under the symmetry transformation, i.e.
$P(g\ket{\Psi})=P(\ket{\Psi})$ for $g\in G$, even though
$g\ket{\Psi}\neq\ket{\Psi}$. This directly implies that the density matrix
$\rho=\sum_{\Psi}p(\ket{\Psi})\ket{\Psi}\bra{\Psi}$ is only invariant under
the adjoint action $\rho=g\rho g^{\dagger}$ of the average symmetry as we
introduced above. For some of the prototypical SPT states considered in this
work, the system is defined with two symmetries, $G_{A}$ and $G_{B}$, and the
ground state wave function of the system $|\Psi\rangle$ is symmetric under
$G_{A}$ and $G_{B}$, meaning its density matrix is invariant under separate
left and right transformation of $G_{A}$ and $G_{B}$. Throughout the work, we
often introduce _symmetric_ decoherence on degrees of freedom charged under
$G_{B}$, but not $G_{A}$, which implies that the mixed density matrix under
decoherence, denoted by $\rho^{D}$, is invariant under _simultaneous_ left and
right transformation of $G_{B}$: $\rho^{D}=g_{B}\rho^{D}g_{B}^{\dagger}$ for
$g_{B}\in G_{B}$, while $\rho^{D}$ must still remain invariant under separate
left and right transformation of $G_{A}$.

###  II.2 $1d$ Cluster State

Let $\ket{\Psi}$ be a nontrivial SPT state, and $\ket{\Omega}$ be a trivial
disordered state, then the following quantity

$C(r)\equiv\frac{\langle\Psi|O(0)O(r)|\Omega\rangle}{\langle\Psi|\Omega\rangle}$
(2)

is called the _strange correlator_ , and it is expected to either saturate to
a non-zero constant value or decay as power-law, for SPT states in $1d$ and
$2d$ [31]. $O$ is an operator that transforms nontrivially under symmetries
that define the SPT states. For noninteracting or weakly interaction fermionic
topological insulators (TI) and topological superconductors (TSC), the strange
correlators in higher dimensions should also decay with a power law.

It is helpful to write the strange correlator in a slightly different form:
$C(r)\equiv\langle\Psi|\hat{C}(r)|\Psi\rangle$, where

$\displaystyle\hat{C}(r)=\frac{1}{|\langle\Psi|\Omega\rangle|^{2}}\Big{[}O(0)O(r)|\Omega\rangle\langle\Omega|\Big{]}.$
(3)

Hence the strange correlator can be viewed as the expectation value of an
“order parameter” $\hat{C}(r)$ of $1d$ and $2d$ SPT phases, and the nontrivial
behavior of this order parameter, either long-ranged or power-law decaying in
its expectation value, can be viewed as a defining feature of an SPT wave
function $|\Psi\rangle$.

As an example, let us consider a $1d$ cluster state [11] with $2N$ sites,
defined by the stabilizer Hamiltonian with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry:

$H=-\sum_{i}Z_{i-1}X_{i}Z_{i+1},$ (4)

where the symmetry action is defined by the product of $X$ on even/odd
sublattices. The periodic boundary condition is assumed, such that the
Hamiltonian has a unique SPT ground state without degeneracies arising from
boundary modes.  We remark that under open boundaries, the system has
anomalous boundary zero modes (spin-1/2) at each end, whose degeneracy is
protected by symmetries; this is often considered to be a defining feature of
the SPT state.

To evaluate a strange correlator, we use the following product state for a
trivial disordered state: $\ket{\Omega}=\ket{+}^{\otimes 2N}$ for a $1d$
system with $2N$ sites, where $\ket{+}$ denotes the $X_{n}=+1$ eigenstate on
every site. For the strange correlator of $\mathbb{Z}_{2}^{\textrm{odd}}$
charged operators separated by the $2n$ lattice spacings, we get

$\displaystyle C_{\textrm{odd}}(2n)$
$\displaystyle=\frac{\langle\Psi|Z_{1}Z_{2n+1}|\Omega\rangle}{\langle\Psi|\Omega\rangle}$
$\displaystyle=\frac{\langle\Psi|\prod_{m=1}^{n}X_{2m}|\Omega\rangle}{\langle\Psi|\Omega\rangle}=1,$
(5)

where we have used $Z_{1}\prod_{m=1}^{n}X_{2m}Z_{2n+1}\ket{\Psi}=\ket{\Psi}$
for the SPT state $\ket{\Psi}$. This result is expected from the presence of a
spin-1/2 zero mode at the boundary in the Wick-rotated picture of the strange
correlator.

Similarly, the strange correlators of $\mathbb{Z}_{2}^{\textrm{even}}$ charged
operators $Z_{0}$ and $Z_{2n}$ take a unit value, where we identify
$Z_{0}\equiv Z_{2N}$. On the other hand, if we replace $|\Psi\rangle$ with a
trivial product state, strange correlators would vanish. Later in Sec. V we
will show that the strange correlator for the stabilizer Hamiltonian discussed
here is directly connected to the well-known string order parameter of the
Haldane phase [32, 33]; therefore, the strange correlator can be indeed viewed
as an “order parameter” defining the SPT phase.

###  II.3 Decoherence

The strange correlators probe nontrivial information of the SPT state. What
would happen to strange correlators if the SPT state is decohered through the
noise channel that destroys, say the $\mathbb{Z}_{2}^{\textrm{even}}$
symmetry? First of all, we remark that as we discuss decoherence, we should
use the density matrix formalism. In this case, the strange correlator
expression in Eq. (2) should generalize as the following:

$C^{\textrm{I}}(r)\equiv\frac{\tr(\rho_{\textrm{spt}}O(0)O(r)\rho_{0})}{\tr(\rho_{\textrm{spt}}\rho_{0})}$
(6)

where $\rho_{\textrm{spt}}$ and $\rho_{0}$ are density matrices of the
decohered SPT and trivial states, respectively, and the superscript “I” stands
for the “type-I” strange correlator. In the pure state limit, the above
expression reduces into Eq. (2).

Now, consider a noise channel defined as the composition of local noises:

$\displaystyle\mathcal{E}_{i}:\rho\rightarrow(1-p)\rho+pZ_{i}\rho
Z_{i},\quad{\cal E}\equiv{\cal E}_{2}\circ{\cal E}_{4}\circ\cdots{\cal
E}_{2N}$ $\displaystyle\Rightarrow{\cal
E}[\rho]=\sum_{\bm{\eta}}P(\bm{\eta})\,K^{\dagger}_{\bm{\eta}}\rho
K^{\vphantom{\dagger}}_{\bm{\eta}},\quad
K_{\bm{\eta}}\equiv\prod_{m}Z_{2m}^{\eta_{2m}}.$ (7)

where $p$ is the probability of having a dephasing noise locally,
${\bm{\eta}}\equiv(\eta_{2},\eta_{4},...,\eta_{2N})$ is a bit-string of
$\\{0,1\\}$ characterizing the $Z$ noise operator $K_{\bm{\eta}}$, and
$P(\bm{\eta})=\prod_{m=1}^{N}P(\eta_{2m})$ with $P(0)=1-p$ and $P(1)=p$ is the
probability of having the noise $\bm{\eta}$. This noise channel perturbs the
system in a way that locally breaks the $\mathbb{Z}_{2}^{\textrm{even}}$
symmetry along every quantum trajectory.

Under this channel, the pure SPT state becomes a mixed state ensemble, denoted
by $\rho^{D}_{\textrm{spt}}\equiv{\cal E}[\rho_{\textrm{spt}}]$. Now we
evaluate the type-I strange correlator of $\rho^{D}_{\textrm{spt}}$ against
the trivial SPT state density matrix $\rho_{0}=|\Omega\rangle\langle\Omega|$:

$\displaystyle\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}\rho^{\vphantom{\dagger}}_{0}\big{)}$
$\displaystyle=\sum_{\bm{\eta}}P(\bm{\eta})|{\langle\Psi|K_{\bm{\eta}}|\Omega\rangle}|^{2}=\frac{1}{2}|{\langle\Psi|\Omega\rangle}|^{2}.$
(8)

This is because odd number of $Z$ operators can change the
$\mathbb{Z}_{2}^{\textrm{even}}$ parity, whose expectation value vanishes
between two states with the same parity. Such a case corresponds to
$\sum_{m}\eta_{2m}\equiv 1\mod 2$, which happens about half the time in the
limit $N\rightarrow\infty$. Interestingly, this implies that we can use the
trivial product state $\ket{\Omega}$ with an arbitrary parity, since the
decoherence would flip the parity of the SPT state half the time, i.e., ${\cal
E}[\rho_{\textrm{spt}}]=\frac{1}{2}\rho_{\textrm{spt}}^{\textrm{e}}+\frac{1}{2}\rho_{\textrm{spt}}^{\textrm{o}}$
in the limit $N\rightarrow\infty$ where
$\tr\rho_{\textrm{spt}}^{\textrm{e,o}}=1$. For an even parity error
$K_{\bm{\eta}}$ such as $Z_{2a}Z_{2b}$, it satisfies
$K_{\bm{\eta}}\ket{\Psi}=\prod_{m=a}^{b-1}X_{2m+1}\ket{\Psi}$ and then the
action of the product of $X$ on $\langle\Omega|$ should square to one.

Similarly, the numerator of the strange correlator for the
$\mathbb{Z}_{2}^{\textrm{odd}}$ charged operator is

$\displaystyle\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}Z^{\vphantom{\dagger}}_{1}Z^{\vphantom{\dagger}}_{2n+1}\rho^{\vphantom{\dagger}}_{0}\big{)}$
$\displaystyle\quad=\sum_{\bm{\eta}}P(\bm{\eta}){\langle\Psi|K_{\bm{\eta}}Z_{1}Z_{2n+1}|\Omega\rangle}{\langle\Omega|K^{\dagger}_{\bm{\eta}}|\Psi\rangle}$
$\displaystyle\quad=\sum_{\bm{\eta}}P(\bm{\eta}){\langle\Psi|\prod_{m=1}^{n}X_{2m}K_{\bm{\eta}}|\Omega\rangle}{\langle\Omega|K^{\dagger}_{\bm{\eta}}|\Psi\rangle}$
$\displaystyle\quad=\frac{1}{2}|{\langle\Psi|K_{\bm{\eta}}|\Omega\rangle}|^{2}\Big{[}\prod_{m=1}^{n}\sum_{\eta_{2m}}(-1)^{\eta_{2m}}P(\eta_{2m})\,\Big{]}$
$\displaystyle\quad=\frac{1}{2}|{\langle\Psi|\Omega\rangle}|^{2}e^{-n/\xi},\,\,\,\,\,\,\xi=1/\ln(1/(1-2p)).$
(9)

However, for the strange correlator of $\mathbb{Z}_{2}^{\textrm{even}}$
charged operators, its numerator is

$\tr(\rho^{D}_{\textrm{spt}}Z_{0}Z_{2n}\rho_{0})=\tr(\rho^{D}_{\textrm{spt}}\rho_{0})=\frac{1}{2}|{\langle\Psi|\Omega\rangle}|^{2},$
(10)

because the operator $Z_{0}Z_{2n}$ can commute through $K_{\bm{\eta}}$ to act
on the state $\bra{\Psi}$ and become $\prod_{m=1}^{n}X_{2m-1}$ to commute back
and hit $\ket{\Omega}$. Following Eq. (6), we obtain

$C^{\textrm{I}}_{\textrm{even}}(2n)=1,\qquad
C^{\textrm{I}}_{\textrm{odd}}(2n)=e^{-n/\xi}.$ (11)

for the SPT state decohered under $\mathbb{Z}_{2}^{\textrm{even}}$ noise
channel. Therefore, the odd-sited strange correlator becomes exponentially
decaying, and its length scale depends on the strength of the decoherence. For
$p\ll 1$, we see that $\xi$ is very large, and the strange correlator would
look the same as a pure state SPT phase for small system size. However, in the
thermodynamic limit, the odd-sited strange correlator is always exponentially
decaying to zero while the even-sited strange correlator behaves the same as
the pure state.

A recent work pointed out that the notion of SPT order can still be defined
even if the symmetry only holds in the average sense [30]. Under decoherence,
although the $\mathbb{Z}_{2}^{\textrm{even}}$ symmetry is broken for every
quantum trajectory, the ensemble of all trajectories still preserves the
symmetry on average, thus an “average SPT” order should still be expected. Our
result Eq. (11) appears to suggest that this average SPT behavior cannot be
detected. However, one can define a new type of odd-sited correlator as
follows:

$C^{\textrm{II}}(r)\equiv\frac{\tr(\rho^{D}_{\textrm{spt}}O(0)O(r)\rho^{\vphantom{\dagger}}_{0}O(r)^{\dagger}O(0)^{\dagger})}{\tr(\rho^{D}_{\textrm{spt}}\rho^{\vphantom{\dagger}}_{0})}$
(12)

which is called the “type-II” strange correlator. Then, following the same
calculation as above, one can show that for $O=Z$,

$C^{\textrm{II}}_{\textrm{even}}(2n)=C^{\textrm{II}}_{\textrm{odd}}(2n)=1.$
(13)

which is nontrivial for both even and odd sites. This is because the noise
only flips the sign of the operator overlap between $|\Omega\rangle$ and
$K_{\bm{\eta}}|\Psi\rangle$, which squares to one in the type-II strange
correlator evaluation. Hence the type-II strange correlator is analogous to
the Edwards-Anderson correlator of the type-I strange correlator.

Figure 2:  The DMRG numerics is performed at the system size $L=100$ for
$H=-\sum_{n}(Z_{n-1}X_{n}Z_{n+1}+hX_{n})$ under periodic boundary condition.
(a) Odd-sited strange correlators at the distance $L/2$ as functions of the
transverse field $h$. (b) Odd-sited strange correlators as functions of the
distance at $h=1/2$. Red (blue) curves represent type-I (II) strange
correlators. Solid (dashed) lines are obtained at the decoherence strength
$p=0$ ($p=0.1$). Here, the decohered density matrix is obtained for the
ensemble of $4\times 10^{3}$ different disorder realizations on even-sites.

###  II.4 Away from the fixed point

So far, we have discussed the strange correlators for the $1d$
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT state in the stabilizer limit with a
zero correlation length ($\xi=0$). Even away from the stabilizer limit, the
aforementioned properties of the type-I and type-II strange correlators
generally hold: while the type-I strange correlator would decay exponentially
with distance under decoherence, the type-II strange correlator would remain
long-ranged under decoherence. This behavior will be more systematically
discussed in the later section using field theory.

To demonstrate the behavior, consider introducing a transverse field
$-h\sum_{n}X_{n}$ to the Hamiltonian in Eq. (4), which drives the ground state
away from the soluble limit. In Fig. 2, we numerically obtained both types of
$\mathbb{Z}_{2}^{\textrm{odd}}$-charged strange correlators
$C^{\textrm{I}}_{\textrm{odd}}$ and $C^{\textrm{II}}_{\textrm{odd}}$ computed
against $|+\rangle^{\otimes L}$ as functions of $r$ at $h=0.5$, with and
without decoherence on the even sites as in Eq. (II.3). The plot illustrates
two important points: $(i)$ away from the fixed point, the magnitudes of the
strange correlators decrease, which completely vanish at the known
topological-trivial transition point $h=1$; and $(ii)$ under decoherence,
while $C^{\textrm{I}}$ decays exponentially with distance, $C^{\textrm{II}}$
stays robust. Therefore, the type-II strange correlator serves as a probe for
the SPT physics under decoherence. As a side remark, we found that the
magnitude of the type-I strange correlator in each quantum trajectory is very
close to each other; the main difference among the type-I strange correlators
in different quantum trajectories is their signs.

Although not shown in the figure, we remark that for $h>0$, both
$C^{\textrm{I}}_{\textrm{even}}$ and $C^{\textrm{II}}_{\textrm{even}}$ may
increase under the decoherence on even sites. This is because while
$C^{\textrm{I,II}}_{\textrm{even}}$ probe correlations between exactly
localized charged operators, away from the fixed point,
$\mathbb{Z}_{2}$-charged operator become diffused across the correlation
length $\xi$. As decoherence effectively diffuses a localized charged operator
$Z$ over multiple sites, decoherence can help enhance the strange correlator.

###  II.5 Generic Noise Model

One can consider more generic decoherence channels as defined in Eq. (1). For
example, one may consider the following depolarization channel on site $i$

$\mathcal{E}_{i}:\rho\rightarrow(1-p)\rho+\frac{p}{3}(X_{i}\rho
X_{i}+Y_{i}\rho Y_{i}+Z_{i}\rho Z_{i}).$ (14)

When we turn on this noise on all the even sites, it is straightforward to see
that the behavior of the type-I and type-II strange correlators would remain
the same as was discussed in the previous sections.

Instead, one may also consider a noise model coherent over multiple sites,
i.e., Kraus operators $K_{i}$s in Eq. (1) are extended over multiple sites. In
this case, the global decoherence channel acts as a stochastic random short-
depth local unitary transformation. While doing so, we can still require the
noise channel to respect the doubled symmetry of
$\mathbb{Z}_{2}^{\textrm{odd}}$, and the adjoint action of
$\mathbb{Z}_{2}^{\textrm{even}}$. For each quantum trajectory, such a noise
action can be decomposed into strictly a local noise in Eq. (II.3) and short-
depth local unitary transformations. Accordingly, a generic complicated noise
model can change the magnitude of type-II strange correlator as a generic
unitary transformation away from the fixed point decreases the magnitude of
the type-II strange correlator in Fig. 2(b).

###  II.6 2d Cluster State

A 2d generalization of the cluster state [12, 13] in the Lieb lattice is an
SPT state [64] with the mixed anomaly between 0-form $\mathbb{Z}_{2}^{(0)}$
and 1-form $\mathbb{Z}_{2}^{(1)}$ symmetries defined on the square lattice,
where qubits reside on both vertices and edges. For a $L\times L$ square
lattice, there are $N_{v}=L^{2}$ vertex qubits and $N_{e}=2L^{2}$ edge qubits.
Its stabilizer Hamiltonian is defined as

$H=-\sum_{v}\quantity(X_{v}\prod_{e\in\mathrm{d}v}\bm{Z}_{e})-\sum_{e}\quantity(\bm{X}_{e}\prod_{v\in\partial
e}Z_{v}),$ (15)

where $\partial$ stands for the boundary operator on the lattice and
$\mathrm{d}=\star\partial\star$ is the coboundary operator (with $\star$ being
the Hodge dual). Bold symbols $\bm{Z}$ and $\bm{X}$ act on edges, and unbold
symbols $Z$ and $X$ act on vertices. Here, all terms in the Hamiltonian
commute with one another, and the groundstate satisfies that each term be 1.
This implies that $B_{p}\equiv\prod_{e\in\partial p}\bm{X}_{e}=1$ for any
plaquette $p$. We denote the symmetry group by
$G_{A}\equiv\mathbb{Z}_{2}^{(0)}$ and $G_{B}\equiv\mathbb{Z}_{2}^{(1)}$, where
the 0-form symmetry charge (generator) $g\equiv\prod_{v}X_{v}\in G_{A}$ and
the 1-form symmetry charge $h_{\gamma}\equiv\prod_{e\in\gamma}\bm{X}_{e}\in
G_{B}$ for any closed loop $\gamma$ along the bonds. Again, the ground state
of Eq. (15) has the decorated domain wall (defect) structure: the defect of
the 1-form symmetry measured by $\prod_{e\ni v}\bm{Z}_{e}$ is bound to the
charge of the 0-form symmetry measured by $X_{v}$; also the defect (domain
wall) of the 0-form symmetry measured by $Z_{v}Z_{v^{\prime}}$ is bound to the
charge of the 1-form symmetry measured by $X_{e}$.  As shown in [64], the
mixed anomaly between $\mathbb{Z}_{2}$ 0-form and $\mathbb{Z}_{2}$ 1-form
symmetries in the SPT state enforces the boundary to be nontrivial under open
boundary condition; when vertex qubits are exposed, the boundary should
spontaneously break the 0-form symmetry, forming a $\mathbb{Z}_{2}$
ferromagnet.

Now, we are ready to calculate the strange correlators for this state. There
are two operators we can inspect: $Z_{v}Z_{v^{\prime}}$ which is the
correlation of $G_{A}$-charged operators, and
$\prod_{e\in\gamma^{\star}}\bm{Z}_{e}$ which is the Wilson loop of
$G_{B}$-charged operators where $\gamma^{\star}$ is a closed loop on the dual
lattice.

First, we evaluate strange correlators for the stabilizer state. Let
$\rho_{0}=|\Omega\rangle\langle\Omega|$, where
$|\Omega\rangle=|+\rangle^{\otimes(N_{e}+N_{v})}$. Then,

$\displaystyle
C_{A}^{\textrm{I}}\equiv\frac{\tr(\rho_{\textrm{spt}}Z_{v}Z_{v^{\prime}}\rho_{0})}{\tr(\rho_{\textrm{spt}}\rho_{0})}$
$\displaystyle=\frac{\tr(\rho_{\textrm{spt}}\prod_{e\in
l}\bm{X}_{e}\rho_{0})}{\tr(\rho_{\textrm{spt}}\rho_{0})}=1$ (16)

where $l$ is an open string connecting two vertices $v$ and $v^{\prime}$.
Since $B_{p}=1$ for both $\ket{\Psi}$ and $\ket{\Omega}$, the RHS is
independent of the choice of $l$. Similarly, for any closed loop
$\gamma=\partial{\cal A}$, where ${\cal A}$ is the region enclosed by the loop
$\gamma$,

$\displaystyle C_{B}^{\textrm{I}}$
$\displaystyle\equiv\frac{\textrm{tr}(\rho_{\textrm{spt}}\prod_{e\in\partial{\cal
A}}\bm{Z}_{e}\rho_{0})}{\tr(\rho_{\textrm{spt}}\rho_{0})}=\frac{\tr(\rho_{\textrm{spt}}\prod_{v\in{\cal
A}}X_{v}\rho_{0})}{\tr(\rho_{\textrm{spt}}\rho_{0})}=1$ (17)

Accordingly, the type-II strange correlators would simply be
$C_{A}^{\textrm{II}}=C_{B}^{\textrm{II}}=1$.

Interestingly, under decoherence, $C_{A}$ and $C_{B}$ exhibit a qualitative
difference. For example, consider the decoherence of
$G_{B}=\mathbb{Z}_{2}^{\textrm{(1)}}$ symmetry by the noise channel in Eq.
(II.3) for edge qubits, i.e., $\rho_{\textrm{spt}}^{D}={\cal
E}_{B}[\rho_{\textrm{spt}}]$. It is convenient to define the projection
operator ${\cal P}$ for vertex and edge qubits as the following:

$\displaystyle{\cal P}_{0}^{v}$
$\displaystyle=\prod_{v}\frac{1+X_{v}}{2},\qquad{\cal
P}_{0}^{e}=\prod_{e}\frac{1+\bm{X}_{e}}{2}.$ (18)

Then, the trivial state density matrix is given by the product of two
projectors, $\rho_{0}={\cal P}_{0}^{v}\otimes{\cal P}_{0}^{e}$. Using this,

$\displaystyle\tr({\cal E}_{B}[\rho_{\textrm{spt}}]\rho_{0})$
$\displaystyle=\tr(\rho_{\textrm{spt}}{\cal
E}_{B}[\rho_{0}])=\langle\Psi|{\cal P}^{v}_{0}\otimes{\cal P}^{e}_{\cal
E}|\Psi\rangle,$ $\displaystyle{\cal P}^{e}_{\cal E}\equiv{\cal E}[{\cal
P}^{e}_{0}]$
$\displaystyle=\prod_{e}\frac{1}{2}(1+(1-p)\bm{X}_{e}+p\bm{Z}_{e}\bm{X}_{e}\bm{Z}_{e})$
(19)

where we use self-adjointness of the decoherence channel in the first line.
Note that ${\cal P}_{\cal E}^{e}=\prod_{e}\frac{1}{2}(1+(1-2p)\bm{X}_{e})$
since $ZXZ=-X$. By expanding the above expressions and using the Lemma in
Appendix. A, we obtain that

$\displaystyle\textrm{tr}\big{(}\rho_{\textrm{spt}}^{D}\rho^{\vphantom{\dagger}}_{0}\big{)}$
$\displaystyle=\frac{2}{2^{N_{e}}\cdot
2^{N_{v}}}\sum_{\gamma}(1-2p)^{|\gamma|}$ (20)

where the summation over $\gamma$ is taken for all closed loops. Similarly,

$\displaystyle\tr({\cal
E}_{B}[\rho_{\textrm{spt}}]Z_{v}Z_{v^{\prime}}\rho_{0})=\tr(\rho_{\textrm{spt}}Z_{v}Z_{v^{\prime}}{\cal
E}_{B}[\rho_{0}])$ $\displaystyle=\langle\Psi|\prod_{e\in l}\bm{X}_{e}{\cal
P}^{v}_{0}\otimes{\cal P}^{e}_{\cal E}|\Psi\rangle=\frac{2}{2^{N_{e}}\cdot
2^{N_{v}}}\sum_{\gamma^{\prime}}(1-2p)^{|\gamma^{\prime}|}$ (21)

where the summation over $\gamma^{\prime}$ is taken over all possible loops
added by the open string $l$ (they are $\mathbb{Z}_{2}$-valued). The channel
${\cal E}_{B}$ could be moved to $\rho_{0}$ since it acts trivially on the
vertices. Therefore, we see that the strange correlator

$\displaystyle
C_{A}^{\textrm{I}}=\frac{\textrm{tr}(\rho^{D}_{\textrm{spt}}Z_{v}Z_{v^{\prime}}\rho_{0})}{\textrm{tr}(\rho^{D}_{\textrm{spt}}\rho_{0})}=\expectationvalue{Z_{v}Z_{v^{\prime}}}_{\beta}^{\textrm{2dIsing}},$
(22)

which is the correlation function of classical 2d ferromagnetic Ising model at
the inverse temperature $\beta=\tanh^{-1}(1-2p)$. In other words, the strange
correlator exhibit transition behavior from long-ranged to short-ranged
depending on the decoherence strength $p$.  To compute the type-II strange
correlator, we calculate its numerator as the following:

$\displaystyle\tr({\cal
E}_{B}[\rho_{\textrm{spt}}]Z_{v}Z_{v^{\prime}}\rho_{0}Z_{v}Z_{v^{\prime}})=\tr(\rho_{\textrm{spt}}Z_{v}Z_{v^{\prime}}{\cal
E}_{B}[\rho_{0}]Z_{v}Z_{v^{\prime}})$ $\displaystyle=\langle\Psi|\prod_{e\in
l}\bm{X}_{e}{\cal P}^{v}_{0}\otimes{\cal P}^{e}_{\cal E}\prod_{e\in
l}\bm{X}_{e}|\Psi\rangle=\textrm{tr}\big{(}\rho_{\textrm{spt}}^{D}\rho_{0}\big{)}.$
(23)

Therefore, $C^{\textrm{II}}_{A}=1$. Note that $C_{B}^{\textrm{I,II}}$ are
unaffected by the noise ${\cal E}_{B}$, taking a unit value.

It is instructive to understand how the conventional non-local order parameter
for the 2d SPT state behaves. In the 2d SPT pure state, the following string
order parameter takes a finite value in the limit where its length diverges
[64]:

$\lim_{|l|\rightarrow\infty}\Big{\langle}Z_{v}\Big{[}\prod_{e\in
l}\bm{X}_{e}\Big{]}Z_{v^{\prime}}\Big{\rangle}\neq 0,$ (24)

where $l$ is an open string along the bonds and $v,v^{\prime}$ are two
vertices where the string ends. This can be easily understood if we imagine
applying a symmetric quantum circuit $U$ that moves the state away from the
fixed point wave function. Since $U$ commutes with a string of $\bm{X}$, it
only acts nontrivially on $Z_{v,v^{\prime}}$. Accordingly, for any order
parameter of the above form, it will be corrected by a constant amount as long
as one is within the same phase. However, this order parameter is actually
short-ranged for the decohered mixed state. Using the above formalism, it is
straightforward to show that

$\displaystyle\textrm{tr}\Big{(}{\cal
E}_{B}[\rho_{\textrm{spt}}]Z_{v}\Big{[}\prod_{e\in
l}\bm{X}_{e}\Big{]}Z_{v^{\prime}}\Big{)}$
$\displaystyle\quad=(1-2p)^{\absolutevalue{l}}\textrm{tr}\Big{(}\rho_{\textrm{spt}}Z_{v}\Big{[}\prod_{e\in
l}\bm{X}_{e}\Big{]}Z_{v^{\prime}}\Big{)}=e^{-l/\xi}$ (25)

where $\xi=1/\ln(1/(1-2p))$. Therefore, it implies that the conventional non-
local order parameter would fail to detect nontrivial structure of the
underlying SPT state if there is decoherence, while the type-I or type-II
strange correlator can still detect in this case. This issue will be discussed
more in the Sec. V.

On the other hand, if we consider the decoherence of
$G_{A}=\mathbb{Z}_{2}^{\textrm{(0)}}$ symmetry by the noise channel in Eq.
(II.3) for vertex qubits, we can evaluate that

$\displaystyle C_{B}^{\textrm{I}}(\partial{\cal A})$
$\displaystyle=(1-2p)^{|{\cal A}|}$ (26)

This indicates that the type-I strange correlator probing the 1-form symmetry
decays in an area-law manner, i.e., short-ranged. This result is consistent
with the earlier discussion about $1d$ $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
SPT state, as upon compactification into a cylindrical geometry which is
effectively 1d, the 1-form symmetry becomes 0-form, and we have shown that the
corresponding type-I strange correlator is short-ranged under the decoherence
of the other symmetry.

##  III Effective Field Theory Evaluation

Many SPT phases classified and constructed through the group cohomology
formalism in Ref. 65, 66 can be described by a nonlinear sigma model (NLSM)
effective field theory [67, 68]. Various different SPT phases share similar
physics captured by a NLSM with a topological $\Theta$-term. For example, $1d$
bosonic SPT phases can often be viewed as the descendants of the previously
well-known Haldane phase of a spin-1 chain, through reducing the SO(3) spin
symmetry down to its subgroups, as long as the subgroup still has a nontrivial
projective representation carried by the edge states. Hence all these $1d$
bosonic SPT states can be described as a O(3) NLSM in the $(1+1)d$ space-time
with a $\Theta-$term at $\Theta=2\pi$ [69]. The $\Theta-$term of the NLSM
reduces to the physical boundary as a WZW term in the $(0+1)d$ space-time
[70], which gives us two-fold degenerate boundary states carrying a projective
representation of the underlying symmetry group.

The effective field theory description also makes the physical interpretation
of the construction (such as decorated domain wall, or more generally
decorated defects construction) of these SPT states transparent. Hence an
effective field theory evaluation of the strange correlator of the SPT states
under decoherence would be more universal, and is applicable to situations
with continuous symmetries, as well as higher dimensions.

###  III.1 $1d$ SPT states

As was observed in Ref. 71, 31, the wave function of many bosonic SPT states
in $1d$ can be inferred from the bulk topological $\Theta-$term of the NLSM,
and written as

$\displaystyle\quad|\Psi\rangle\sim\int D[{\bm{n}}(x)]\exp\left(-{\cal
S}[{\bm{n}}(x)]\right)|{\bm{n}}(x)\rangle$ $\displaystyle\quad{\cal S}=\int
dx\ \frac{1}{g}(\nabla_{x}{\bm{n}}(x))^{2}+\mathrm{WZW}[{\bm{n}}(x)]$
$\displaystyle\mathrm{WZW}[{\bm{n}}(x)]=\int_{0}^{1}du\int dx\
\frac{2\pi\mathrm{i}}{4\pi}\epsilon_{abc}\tilde{n}^{a}\partial_{u}\tilde{n}^{b}\partial_{x}\tilde{n}^{c}$
(27)

where ${\bm{n}}$ is a unit three component vector with length $|{\bm{n}}|=1$,
and $\tilde{{\bm{n}}}(x,u)$is an extension of ${\bm{n}}(x)$ into the space
$(x,u)$ where $\tilde{{\bm{n}}}(x,0)={\bm{n}}(x)$,
$\tilde{{\bm{n}}}(x,1)=\hat{z}$. This wave function can describe SPT phases in
$1d$ with symmetries $\mathrm{SO}(3)$, $\mathrm{U}(1)\rtimes Z_{2}$,
$Z_{2}\times Z_{2}$, $\mathrm{U}(1)\times Z_{2}^{T}$, $Z_{2}^{T}$, etc. The
Wess-Zumino-Witten term in Eq. 27 can be viewed as the termination of the bulk
topological $\Theta-$term at a temporal boundary [71], which is the “space-
time dual” of the WZW term at the physical real space boundary [70] of the SPT
state.

To make connection with the computation based on the stabilizer Hamiltonian of
the SPT states presented in the previous section, let us assume the symmetry
of the system is $Z_{2}^{A}\times Z_{2}^{B}$, which act on the vector
${\bm{n}}$ as

$\displaystyle Z_{2}^{A}:(n_{x},n_{y},n_{z})\rightarrow(-n_{x},-n_{y},n_{z});$
(28) (29) $\displaystyle
Z_{2}^{B}:(n_{x},n_{y},n_{z})\rightarrow(n_{x},-n_{y},-n_{z}).$ (30)

The density matrix of the pure SPT state $|\Psi\rangle\langle\Psi|$ is given
as the following in the basis of $|{\bm{n}}(x)\rangle$:

$\displaystyle\rho_{\textrm{spt}}\sim\int{\cal
D}\\{{\bm{n}}(x),{\bm{n}}^{\prime}(x)\\}\ e^{-{\cal S}[{\bm{n}}]-{\cal
S}^{\ast}[{\bm{n}}^{\prime}]}|{\bm{n}}(x)\rangle\langle{\bm{n}}^{\prime}(x)|.$
(31)

As a pure state density matrix, all the symmetries of the system have
manifestly “doubled” as we explained in the introduction: this pure state
density matrix is invariant under separate “left” and “right” symmetry
transformations, which act on ${\bm{n}}(x)$ and ${\bm{n}}^{\prime}(x)$
respectively. We would like to consider a mixed density matrix built based
upon the SPT wave function $|\Psi\rangle$. One way to achieve this is
increasing the “weight” of the diagonal elements of the density matrix to
decrease the purity (or increase the number of non-zero Schmidt eigenvalues).
This is equivalent to turning on some “interaction” ${\cal
S}^{\textrm{int}}[{\bm{n}}(x),{\bm{n}}^{\prime}(x)]$ between ${\bm{n}}(x)$ and
${\bm{n}}^{\prime}(x)$ in the density matrix, as diagrammatically represented
in Fig. 1(a):

$\displaystyle\rho^{D}_{\textrm{spt}}$ $\displaystyle\sim\int{\cal
D}\\{{\bm{n}},{\bm{n}}^{\prime}\\}e^{-{\cal S}[{\bm{n}}]-{\cal
S}^{\ast}[{\bm{n}}^{\prime}]-{\cal
S}^{\textrm{int}}[{\bm{n}},{\bm{n}}^{\prime}]}|{\bm{n}}(x)\rangle\langle{\bm{n}}^{\prime}(x)|.$
(32)

Since ${\cal S}^{\textrm{int}}[{\bm{n}}(x),{\bm{n}}^{\prime}(x)]$ should favor
specific combinations of ${\bm{n}}(x)$ and ${\bm{n}}^{\prime}(x))$, the system
would be no longer invariant under all left and right symmetry
transformations; still, it must remain invariant under the simultaneous left
and right transformations on ${\bm{n}}(x)$ and ${\bm{n}}^{\prime}(x)$ as long
as ${\cal S}^{\textrm{int}}$ is invariant under the simultaneous actions.

To proceed, we define the trivially disordered state $|\Omega\rangle$ as an
equal weight superposition of all the configurations of ${\bm{n}}(x)$:

$\displaystyle|\Omega\rangle\sim\int D{\bm{n}}(x)\,|{\bm{n}}(x)\rangle.$ (33)

Then, the type-I (6) and type-II (12) strange correlators are expressed in
terms of vectors ${\bm{n}}$ as follows:

$\displaystyle C^{\mathrm{I}}_{ab}(x)$
$\displaystyle=\frac{\mathrm{tr}\left(n^{a}(x)n^{b}(0)\rho^{D}_{\textrm{spt}}\rho_{0}\right)}{\mathrm{tr}\left(\rho^{D}_{\textrm{spt}}\rho_{0}\right)}.$
$\displaystyle C^{\mathrm{II}}_{ab}(x)$
$\displaystyle=\frac{\mathrm{tr}\left(n^{a}(x)n^{b}(0)\rho^{D}_{\mathrm{spt}}n^{a}(x)n^{b}(0)\rho_{0}\right)}{\mathrm{tr}\left(\rho^{D}_{\textrm{spt}}\rho_{0}\right)}.$
(34)

Formally, after the space-time rotation ($x\rightarrow\tau$)  illustrated in
Fig. 1(b), the original type-I strange correlator is mapped to the spin-spin
correlation along the temporal direction of two interacting spin-1/2 degrees
of freedom (the $0d$ boundary of a $1d$ SPT state), one from $\bm{n}$ and the
other from $\bm{n}^{\prime}$:

$C^{\mathrm{I}}_{ab}(x)\sim\langle n_{a}(x)n_{b}(0)\rangle\sim\langle
S_{a}(\tau)S_{b}(0)\rangle,$ (35)

as in the path integral formalism, an isolated spin-1/2 is represented by a
$(0+1)d$ NLSM with a WZW term at level-1. The evaluation of the type-II
strange correlator will be mapped formally to an evaluation of the “doubled”
temporal spin-spin correlation of the two interacting spins, ${\bm{S}}$ and
${\bm{S}}^{\prime}$:

$\displaystyle C^{\mathrm{II}}_{ab}(x)\sim\langle S_{a}(\tau)S_{b}(0)\
S^{\prime}_{a}(\tau)S^{\prime}_{b}(0)\rangle.$ (36)

As an example, let us introduce the following “interaction” in the density
matrix

$\displaystyle\mathcal{S}^{\textrm{int}}_{1d,1}[{\bm{n}}(x),{\bm{n}}^{\prime}(x)]\sim\int
dx\ u\left({\bm{n}}(x)\cdot{\bm{n}}^{\prime}(x)\right).$ (37)

Physically this interaction corresponds to introducing decoherence of all
degrees of freedom of the system, as now the density matrix in Eq. 32 is no
longer invariant under any separate left or right $Z_{2}^{A}$ or $Z_{2}^{B}$
transformation, though it is still invariant under simultaneous left and right
transformations; in other words, the decoherence introduced by
$S^{\textrm{int}}_{1d,1}$ should be analogous to introducing temperature to
the density matrix, which thermalizes all degrees of freedom. A similar idea
of generating mixed density matrices has been explored for states constructed
with loop degrees of freedom [72]. Most naturally, when the density matrix is
driven into a mixed state under decoherence, the interaction
$\mathcal{S}^{\textrm{int}}_{1d,1}$ should favor the “diagonal” configurations
${\bm{n}}(x)\sim{\bm{n}}^{\prime}(x)$, $i.e.$ we need $u<0$ in Eq. 37. When
${\bm{n}}(x)\sim{\bm{n}}^{\prime}(x)$, the two WZW terms, $i.e.$
$\mathrm{WZW}[{\bm{n}}(x)]$ and $\mathrm{WZW}[{\bm{n}}^{\prime}(x)]$ tend to
cancel each other. Without the WZW term, both the type-I and type-II strange
correlators would be short-range, for all components $a$, $b$. This is
consistent with the picture under space-time rotation, as the interaction term
$S^{\textrm{int}}_{1d,1}$ would translate into the following spin-spin
interaction of the zero-dimensional system

$\displaystyle H^{\textrm{int}}_{0d,1}\sim
J{\bm{S}}\cdot{\bm{S}}^{\prime}+\cdots.$ (38)

The ellipsis includes terms that reduce the SO(3) spin symmetry down to
$Z_{2}^{A}\times Z_{2}^{B}$. When $J>0$, the ground state is a spin singlet;
when $J<0$, as long as there are terms that reduce the symmetry to
$Z_{2}^{A}\times Z_{2}^{B}$, the ground state is in general nondegenerate,
hence any spin correlation along the temporal direction would still be short
ranged.

As was understood before, multiple copies of fermionic topological insulators
(TI) and topological superconductors (TSC) of fermions may be mapped to
bosonic SPT states under interaction [73, 74, 75, 76]. For example, four
copies of Kitaev’s chains of Majorana fermions, or two copies of the spinless
Su–Schrieffer–Heeger (SSH) models of complex fermions [77], can be mapped to
the Haldane phase with different defining symmetries. Our evaluation above
also implies that, under general decoherences on fermion bilinear operators
that respect the symmetry of the system, the classification of the fermionic
TIs and TSCs would collapse, analogous to the collapse of the classification
of TIs and TSSs under short-range interactions [78, 79]). Similar behavior of
TIs and TSCs under decoherence in higher dimensions is also expected, and it
was noticed that the averaged symmetry caused by disorder could also lead to
the collapse of classifications of TIs [30]. We will defer a more complete
discussion of decohered TIs and TSCs to future work.

Now let’s consider another type of interaction:

$\displaystyle\mathcal{S}^{\textrm{int}}_{1d,2}[{\bm{n}}(x),{\bm{n}}^{\prime}(x)]\sim\int
dx\ u\left(n_{z}(x)n^{\prime}_{z}(x)\right).$ (39)

Here only the $z$ components of the two ${\bm{n}}$ vectors are coupled, which
corresponds to introducing decoherence on $Z_{2}^{B}$ but not $Z_{2}^{A}$, as
now the density matrix is only invariant under one simultaneous $Z_{2}^{B}$
transformation on $n_{z}(x)$ and $n_{z}^{\prime}(x)$, but it is still
invariant under two separate $Z_{2}^{A}$ symmetry transformations on
${\bm{n}}$ and ${\bm{n}}^{\prime}$. Then after space-time rotation the strange
correlator calculation is mapped to a temporal spin-spin correlation (either
Eq. 35 or Eq. 36) of the following two interacting spins:

$\displaystyle H^{\textrm{int}}_{0d,2}\sim JS_{z}S^{\prime}_{z}.$ (40)

Then the ground state of the system would be a doublet, rather than a singlet.
For example, when $J<0$, the ground states of the two-spin system are

$\displaystyle|\mathrm{GS}_{1}\rangle=|S_{z}=+1/2,\
S^{\prime}_{z}=+1/2\rangle,$ (41) (42)
$\displaystyle|\mathrm{GS}_{2}\rangle=|S_{z}=-1/2,\
S^{\prime}_{z}=-1/2\rangle.$ (43)

The type-I strange correlator $C^{\mathrm{I}}_{xx}(x)$ defined in Eq. 6
evaluated as Eq. 35 would be short ranged, as a single $S_{x}$ operator does
not connect these two states within the doublet; however, the type-II strange
correlator $C^{\mathrm{II}}_{xx}(x)$ is still long-ranged, as the operator
$S_{x}S^{\prime}_{x}$ can connect the two states within the doublet. Both
$C^{\mathrm{I}}_{zz}$ and $C^{\mathrm{II}}_{zz}$ are long-ranged, as both the
ground states are eigenstates of $S_{z}$ and $S^{\prime}_{z}$.

It was concluded before that the edge state of the SPT state with
$\mathbb{Z}_{2}\times\mathbb{Z}^{\text{avg}}_{2}$ symmetry is trivial (though
the SPT state was still considered nontrivial) [30], where
$\mathbb{Z}^{\text{avg}}_{2}$ is the average symmetry that only exists after
disorder average. Our calculation indicates that the boundary state, which is
directly related to the type-I strange correlator, may not be the best
diagnosis for systems under decoherence; instead, the type-II strange
correlator serves as another method of characterizing mixed state SPT phases.
The type-II strange correlator is to some extent analogous to the Edwards-
Anderson correlator of spin glass systems. Indeed, as remarked in the
numerical lattice model calculation, for each quantum trajectory the type-I
strange correlator fluctuates in its sign while magnitude stays robust,
signalling the presence of nontrivial correlation. Although the type-I strange
correlator vanishes over averaging, the squared order parameter, i.e., the
type-II strange correlator, still captures the nontrivial information from the
SPT state.

###  III.2 $2d$ SPT states

Now let us consider the wave function of a class of $2d$ SPT states:

$\displaystyle|\Psi\rangle$ $\displaystyle\sim\int
D[{\bm{n}}(\mathbf{x})]\exp\left(-\mathcal{S}[{\bm{n}}(\mathbf{x})]\right)|{\bm{n}}(\mathbf{x})\rangle$
$\displaystyle\mathcal{S}$ $\displaystyle=\int d^{2}x\
\frac{1}{g}(\partial{\bm{n}}(\mathbf{x}))^{2}+\mathrm{WZW}[{\bm{n}}(\mathbf{x})]$
$\displaystyle\mathrm{WZW}[{\bm{n}}(\mathbf{x})]$
$\displaystyle=\int_{0}^{1}du\int d^{2}x\
\frac{2\pi\mathrm{i}}{\Omega_{3}}\epsilon_{abcd}\tilde{n}^{a}\partial_{u}\tilde{n}^{b}\partial_{x}\tilde{n}^{c}\partial_{y}\tilde{n}^{c}.$
(44)

The pure density matrix is analogous to Eq. 31. This formalism can describe
SPT phases with symmetries $\mathrm{O}(4)$,
$\mathrm{U}(1)\times\mathrm{U}(1)$, $\mathrm{SO}(3)\rtimes Z_{2}$,
$\mathrm{SO}(3)\times Z_{2}^{T}$, $Z_{2}\times Z_{2}$, or even just one
$\mathrm{U}(1)$ or $Z_{2}$ symmetry, etc.

Let us start with the case with the maximum $\mathrm{O}(4)$ symmetry. We can
introduce some weak decoherence on ${\bm{n}}$, and consider the mixed density
matrix Eq. 32. We start with an interaction term analogous to Eq. 37:

$\displaystyle{\cal S}^{\textrm{int}}_{2d,1}=\int d^{2}x\
u\left({\bm{n}}(\mathbf{x})\cdot{\bm{n}}^{\prime}(\mathbf{x})\right).$ (45)

This interaction respects the simultaneous left and right $\mathrm{O}(4)$
symmetry (which act on ${\bm{n}}(\mathbf{x})$ and
${\bm{n}}^{\prime}(\mathbf{x})$ simultaneously). Like the $1d$ case, the
decoherence is supposed to drive the system into a mixed state with enhanced
weight on the diagonal configurations, hence $u$ is most naturally negative.
But regardless of the sign of $u$, this interaction drives the computation of
both type-I and type-II strange correlators into a field theory calculation of
correlation functions of a $(2+0)d$ or $(1+1)d$ nonlinear Sigma model (NLSM)
without any WZW term. Both type-I and type-II strange correlators should be
short-ranged, as a NLSM in $(2+0)d$ without any topological term will flow to
the disordered phase.

Now let us consider a $2d$ bosonic symmetry protected topological state with
symmetry $\mathrm{U}(1)^{A}\times\mathrm{U}(1)^{B}$. This is a SPT phase
considered in Ref. 80. The two $\mathrm{U}(1)$ symmetries act on the four-
component vector ${\bm{n}}$ as the following:

$\displaystyle\mathrm{U}(1)^{A}$ $\displaystyle:$
$\displaystyle(n_{1}+\mathrm{i}n_{2})\rightarrow
e^{\mathrm{i}\theta}(n_{1}+\mathrm{i}n_{2}),$ (46)
$\displaystyle\mathrm{U}(1)^{B}$ $\displaystyle:$
$\displaystyle(n_{3}+\mathrm{i}n_{4})\rightarrow
e^{\mathrm{i}\phi}(n_{3}+\mathrm{i}n_{4}).$ (48)

Let us turn on the following interaction, which corresponds to introducing
decoherence on $\mathrm{U}(1)^{B}$ charges:

$\displaystyle{\cal
S}^{\textrm{int}}_{2d,2}[{\bm{n}}(\mathbf{x}),{\bm{n}}^{\prime}(\mathbf{x})]\sim\int
d^{2}x\ \sum_{a=3}^{4}u(n_{a}(\mathbf{x})n^{\prime}_{a}(\mathbf{x})).$ (49)

This interaction can be conveniently analyzed through Abelian bosonization of
the $(1+1)d$ NLSM. Under Abelian bosonization, the NLSM of ${\bm{n}}(\bf{x})$
with the WZW term corresponds to the following $(1+1)d$ Lagrangian, and its
dual:

$\displaystyle{\cal
L}=\frac{1}{2K}\left((\partial_{\tau}\theta)^{2}+v^{2}(\partial_{x}\theta)^{2}\right),$
(50) (51) $\displaystyle{\cal
L}_{d}=\frac{K}{2}\left((\partial_{\tau}\phi)^{2}+v^{2}(\partial_{x}\phi)^{2}\right).$
(52)

The four-component vector ${\bm{n}}$ has the following schematic
representation in the Abelian bosonized formalism:

$\displaystyle{\bm{n}}\sim\left(\cos\theta,\sin\theta,\cos(2\pi\phi),\sin(2\pi\phi)\right).$
(53)

The computation of the strange correlator, especially type-II, requires making
two copies of the system of Eq. 50, and now the interaction term ${\cal
S}^{\textrm{int}}_{2d,2}$ becomes the following coupling in the Abelian
bosonization language:

$\displaystyle H^{\textrm{int}}_{1d,2}\sim\int dx\
\alpha\cos(2\pi\phi-2\pi\phi^{\prime}).$ (54)

We evaluate the following type-I and type-II strange correlators for the
operator $\Phi\sim n_{1}-\mathrm{i}n_{2}\sim e^{-\mathrm{i}\theta}$ charged
under $\mathrm{U}(1)^{A}$:

$\displaystyle C^{\mathrm{I}}(\mathbf{x})$ $\displaystyle=$
$\displaystyle\frac{\mathrm{tr}\left(\Phi(0)\Phi^{\ast}(\mathbf{x})\rho^{D}_{\mathrm{spt}}\rho^{\vphantom{\dagger}}_{0}\right)}{\mathrm{tr}\left(\rho^{D}_{\mathrm{spt}}\rho^{\vphantom{\dagger}}_{0}\right)}\sim\langle
e^{-\mathrm{i}\theta(0,0)}\ e^{+\mathrm{i}\theta(x,\tau)}\rangle,$ (55)
$\displaystyle C^{\mathrm{II}}(\mathbf{x})$ $\displaystyle=$
$\displaystyle\frac{\mathrm{tr}\left(\Phi(0)\Phi^{\ast}(\mathbf{x})\rho^{D}_{\mathrm{spt}}\Phi(0)\Phi^{\ast}(\mathbf{x})\rho_{0}\right)}{\mathrm{tr}\left(\rho^{D}_{\mathrm{spt}}\rho^{\vphantom{\dagger}}_{0}\right)}$
(58) $\displaystyle\sim$ $\displaystyle\langle
e^{-\mathrm{i}(\theta(0,0)+\theta^{\prime}(0,0))}\
e^{+\mathrm{i}(\theta(x,\tau)+\theta^{\prime}(x,\tau))}\rangle,$ (61)

where the expectation values are taken with respect to the two copies of
Luttinger liquids coupled by Eq. (54).

As the scaling dimension of $e^{i2\pi\phi}$ is given by $\pi/K$, the coupling
$H^{\textrm{int}}_{1d,2}$ is relevant (irrelevant) when $K\,{>}\,K_{c}$
($K\,{<}\,K_{c}$) with $K_{c}=\pi$. When $K\,{>}\,K_{c}$, $i.e.$
$H^{\textrm{int}}_{1d,2}$ is relevant, it will gap out the channel
$\theta_{-}=(\theta-\theta^{\prime})/\sqrt{2}$,
$\phi_{-}=(\phi-\phi^{\prime})/\sqrt{2}$, and the type-I strange correlator
becomes short-ranged. Hence by tuning $K$, there is a phase transition
signified by the behavior of the type-I strange correlator. Since the
Luttinger parameter $K$ can receive renormalization from $\alpha$ in Eq. 54,
this phase transition of the type-I strange correlator can also be driven by
the strength of interaction. But the
$\theta_{+}=(\theta+\theta^{\prime})/\sqrt{2}$ and
$\phi_{+}=(\phi+\phi^{\prime})/\sqrt{2}$ channel remains a gapless CFT,
regardless of the fate of $H^{\textrm{int}}_{1d,2}$ under renormalization
group. Since the type-II strange correlator in Eq. (61) only involves the
symmetric linear combinations, it always decays with a power law. Here once
again the type-II strange correlator captures the “memory” of the pure state
SPT wave function.

###  III.3 Bosonic Integer Quantum Hall state

We would also like to discuss the $2d$ bosonic integer quantum Hall state
discussed in Ref. 80. The bIQH state can be obtained from the same NLSM
description Eq. 44, and reducing the
$\mathrm{U}(1)^{A}\times\mathrm{U}(1)^{B}$ symmetry in Eq. 48 to one diagonal
$\mathrm{U}(1)$ symmetry. The edge state of the bIQH state is nonchiral, whose
left-moving modes carry the $\mathrm{U}(1)$ charge, while the right-moving
modes are neutral. The Hall conductivity of the bIQH state must be an even
integer due to the bosonic nature of the underlying particles.

If we only look at the degrees of freedom that carry the $\mathrm{U}(1)$
charge, the density matrix of the bIQH state can be mapped to a pair of chiral
bosons in $(1+1)d$ space-time under Wick rotation:

$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle\int
d^{2}x\Big{[}+\frac{1}{2\pi}\partial_{\tau}\varphi\partial_{x}\varphi+v^{2}K(\partial_{x}\varphi)^{2}\Big{]},$
(62) $\displaystyle\mathcal{S}^{\prime}$ $\displaystyle=$ $\displaystyle\int
d^{2}x\Big{[}-\frac{1}{2\pi}\partial_{\tau}\varphi^{\prime}\partial_{x}\varphi^{\prime}+v^{2}K(\partial_{x}\varphi^{\prime})^{2}\Big{]}.$
(64)

Again the $\mathrm{U}(1)$ symmetry is doubled, carried by $\varphi$ and
$\varphi^{\prime}$ respectively. If we turn on decoherence (or noise) on the
phase angle of charges, this effect is mapped to the following interaction
between $\varphi$ and $\varphi^{\prime}$:

$\displaystyle H^{\textrm{int}}_{1d,3}=\int dx\
\alpha\cos(\varphi-\varphi^{\prime}).$ (65)

Again, depending on the parameter $K$, $H^{\textrm{int}}_{1d,3}$ may become
relevant, and render the strange correlators short-ranged.

Here we note that a similar analysis also applies to the fermionic integer
quantum Hall states, whose boundary states also become chiral bosons after
bosonization. A full analysis of fermionic topological insulators under
decoherence will be presented in another work.

###  III.4 Boundary states under decoherence

As we have seen in this section, the density matrix of an SPT state
corresponds to a system evolving in the imaginary time, and terminating on two
opposite boundaries $\tau=\pm\infty$ at the temporal direction, see Fig. 1(a).
After Wick rotation, these two temporal boundaries become the left and right
boundaries of the system in real space, and calculations about the decoherence
on the density matrix is formally mapped to a system whose two boundaries are
coupled in the selected channel as in Fig. 1(b).

One may also ask about the fate of the physical boundary state of the system
under decoherence. This can be answered by first deriving the pure state
density matrix of the boundary theory only, then turning on decoherence. In
this process, we have ignored the bulk. Let us still take the $2d$ SPT state
with $\mathrm{U}(1)^{A}\times\mathrm{U}(1)^{B}$ symmetry as an example. The
boundary theory is captured by the $(1+1)d$ Lagrangians Eq. 50, and the
schematic density matrix of the ground state of Eq. 50 reads

$\displaystyle\rho_{b}\sim\int{\cal
D}\\{{\bm{\phi}}(x),{\bm{\phi}}^{\prime}(x)\\}e^{-{\cal S}[\phi]-{\cal
S}[\phi^{\prime}]}|\phi(x)\rangle\langle\phi^{\prime}(x^{\prime})|,$ (66) (67)
$\displaystyle{\cal S}[\phi]\sim\sum_{q}K|q||\phi_{q}|^{2}.$ (68)

The problem is now formally analogous to the transmission of Luttinger liquid
through a zero dimension defect [81].

The decoherence on $\mathrm{U}(1)^{B}$ is still modeled with a coupling
between $\phi$ and $\phi^{\prime}$:
$\sim\alpha\cos(2\pi\phi-2\pi\phi^{\prime})$, and there is again a critical
$K_{c}$ above which the coupling $\alpha$ becomes relevant, and the
correlation function at the physical boundary is rendered short-ranged by the
decoherence. Hence the type-I strange correlator indeed reflects what could
happen at the physical boundary under decoherence, though the exact transition
point of the boundary depends on the details of the boundary Hamiltonian.

In this section, the effective field theories used to compute the strange
correlators were unitary theories after Wick rotation. This is not necessarily
true for all wave functions within the SPT phase. But in most cases, there
exists a submanifold in the parameter space in which the SPT phase admits a
description in terms of a Lorentz invariant field theory, then since the
strange correlator is mapped to the ordinary correlation functions at the
physical boundary of the system under Wick rotation, the strange correlators
can indeed be evaluated with a unitary field theory.

##  IV Doubled System

In the previous section, we have developed a field-theoretic understanding of
two types of strange correlators and their behaviors under decoherence. In
doing so, the idea of doubled systems has been crucial. In the following, we
make this idea more concrete using the Choi-Jamiołkowski isomorphism [60, 61].

###  IV.1 Choi Representation of Decohered SPT States

Let $\rho_{\textrm{spt}}$ be the density matrix of the pure SPT state with
$G\,{=}\,G_{A}\,{\times}\,G_{B}$ symmetry. By using the Choi-Jamiołkowski
isomorphism, we can represent an operator as a state in the following way

$\|\rho\rangle\rangle\equiv\frac{1}{\sqrt{\textrm{Dim}[\rho]}}\sum_{i}|i\rangle\otimes\rho|i\rangle$
(69)

Accordingly, the pure state density matrix can be represented as two copies of
independent SPT states:

$\|\rho_{\textrm{spt}}\rangle\rangle=|\Psi^{*}\rangle_{u}\otimes|\Psi\rangle_{l}=|\Psi^{*}_{u},\Psi_{l}\rangle.$
(70)

where $u$ and $l$ denote the upper and lower copies respectively. The Choi
state is defined on the doubled Hilbert space as ${\cal H}_{ul}\equiv{\cal
H}_{u}\otimes{\cal H}_{l}$. Here, the star superscript is to denote that its
amplitude is complex-conjugated relative to the original wavefunction. In this
doubled Hilbert space, the symmetry group is also doubled as
$(G^{u}_{A}\,{\times}\,G^{u}_{B})\,{\times}\,(G^{l}_{A}\,{\times}\,G^{l}_{B})$,
and the Choi state is the SPT state under the doubled symmetry group. See
Appendix. B for more details. Similarly, the Choi state of a trivial
disordered state density matrix is given as

$\|\rho_{0}\rangle\rangle=|\Omega^{*}\rangle_{u}\otimes|\Omega\rangle_{l}\equiv|\tilde{\Omega}\rangle$
(71)

which is a trivial disordered state in the doubled Hilbert space.

Under the Choi isomorphism, the decoherence quantum channel ${\cal E}$ in Eq.
(1) maps into an operator $\hat{{\cal E}}$ acting in the doubled Hilbert space
as the following:

$\displaystyle\hat{{\cal E}}=\sum_{m}K^{*}_{m,u}\otimes K_{m,l}$ (72)

where $K_{m}$ is a Kraus operator. For example, the aforementioned local $Z$
noise channel under the Choi mapping would be expressed as

$\hat{{\cal
E}}=\prod_{i}\big{[}(1-p)\mathbb{I}_{i,u}\otimes\mathbb{I}_{i,l}+pZ_{i,u}\otimes
Z_{i,l}\big{]}.$ (73)

Then, the decohered SPT state can be represented as

$|\tilde{\Psi}\rangle\equiv\|{\cal
E}[\rho_{\text{spt}}]\rangle\rangle=\hat{{\cal E}}|\Psi_{u},\Psi_{l}\rangle$
(74)

Although $\hat{{\cal E}}$ is not a unitary map that preserves the norm of the
state, it is nevertheless a positive semi-definite map for the Choi states.
Therefore, $|\tilde{\Psi}\rangle$ is a valid state in the doubled Hilbert
space whose norm is given by the purity of the decohered density matrix,
$\textrm{tr}(\rho^{D}_{\textrm{spt}}\rho^{D}_{\textrm{spt}})>0$.

The Choi representation of the mixed state density matrix
$|\tilde{\Psi}\rangle$ can be shown to be the ground state of a certain parent
Hamiltonian in the doubled Hilbert space, which is perturbed from the doubled
SPT Hamiltonian. Starting from the pure state limit, we consider a pure SPT
state $\ket{\Psi}$ as the ground state of a parent Hamiltonian
$H_{\text{spt}}$ in the standard Hilbert space. With some constant energy
shift, it is always possible to make $H_{\text{spt}}$ a positive semidefinite
Hermitian operator, such that the ground state has exactly zero energy, i.e.
$H_{\text{spt}}\ket{\Psi}=0$. Correspondingly, in the doubled Hilbert space,
the Choi representation of the SPT density matrix
${\|\rho_{\text{spt}}\rangle\rangle}=\ket{\Psi^{*}}_{u}\otimes\ket{\Psi}_{l}$
should be the ground state of the following double Hamiltonian

$\hat{{\cal
H}}_{\text{spt}}=H_{\text{spt},u}^{*}\otimes\mathbb{I}_{l}+\mathbb{I}_{u}\otimes
H_{\text{spt},l},$ (75)

such that $\hat{{\cal H}}_{\text{spt}}{\|\rho_{\text{spt}}\rangle\rangle}=0$.
The double Hamiltonian $\hat{{\cal H}}_{\text{spt}}$ will inherit the positive
semidefinite property of the single Hamiltonian $H_{\text{spt}}$, such that it
admits a Cholesky decomposition as

$\hat{{\cal H}}_{\text{spt}}=\hat{{\cal A}}^{\dagger}\hat{{\cal A}},$ (76)

with $\hat{{\cal A}}$ being some generic (possibly non-Hermitian) operator in
the doubled Hilbert space. The fact that ${\|\rho_{\text{spt}}\rangle\rangle}$
is a zero-energy eigenstate of $\hat{{\cal H}}_{\text{spt}}$ implies

${\langle\langle\rho_{\text{spt}}\|}\hat{{\cal
H}}_{\text{spt}}{\|\rho_{\text{spt}}\rangle\rangle}={\langle\langle\rho_{\text{spt}}\|}\hat{{\cal
A}}^{\dagger}\hat{{\cal A}}{\|\rho_{\text{spt}}\rangle\rangle}=0,$ (77)

therefore $\hat{{\cal A}}{\|\rho_{\text{spt}}\rangle\rangle}=0$ must be a zero
vector, i.e. the ground state ${\|\rho_{\text{spt}}\rangle\rangle}$ should be
annihilated by the Cholesky operator $\hat{{\cal A}}$.

In our setup, the decohered SPT state $\rho_{\text{spt}}^{D}={\cal
E}[\rho_{\text{spt}}]$ is always given by sending the pure SPT state
$\rho_{\text{spt}}=|\Psi\rangle\langle\Psi|$ through a decoherence channel
${\cal E}$. In the Choi representation, this can be written as
${\|\rho_{\text{spt}}^{D}\rangle\rangle}=\hat{{\cal
E}}{\|\rho_{\text{spt}}\rangle\rangle}$. We claim that the state
${\|\rho_{\text{spt}}^{D}\rangle\rangle}$ must be the ground state of the
following deformed double Hamiltonian

$\hat{{\cal H}}_{\text{spt}}^{D}=(\hat{{\cal E}}\hat{{\cal A}}\hat{{\cal
E}}^{-1})^{\dagger}(\hat{{\cal E}}\hat{{\cal A}}\hat{{\cal E}}^{-1}),$ (78)

which is still a Hermitian positive semidefinite operator in the doubled
Hilbert space. To see this, we apply $\hat{{\cal H}}_{\text{spt}}^{D}$ on the
state ${\|\rho_{\text{spt}}^{D}\rangle\rangle}$ and obtain $\hat{{\cal
H}}_{\text{spt}}^{D}{\|\rho_{\text{spt}}^{D}\rangle\rangle}=(\hat{{\cal
E}}\hat{{\cal A}}\hat{{\cal E}}^{-1})^{\dagger}\hat{{\cal E}}\hat{{\cal
A}}{\|\rho_{\text{spt}}\rangle\rangle}=0$, which proves our claim. In this
way, given the original SPT Hamiltonian $H_{\text{spt}}$ and the decoherence
channel ${\cal E}$, we can in principle construct the parent Hamiltonian
$\hat{{\cal H}}_{\text{spt}}^{D}$ that stabilizes the decohered SPT state
${\|\rho_{\text{spt}}^{D}\rangle\rangle}$ as its unique ground state.

However, one concern is that the Hamiltonian constructed in Eq. (78) may not
be a local Hamiltonian. Nevertheless, if the original SPT Hamiltonian
$H_{\text{spt}}$ is made of local commuting projectors (e.g. a stabilizer
Hamiltonian), it is possible to define a set of local Cholesky operators
$\hat{{\cal A}}_{i}$ (using local projectors) such that $\hat{{\cal
H}}_{\text{spt}}=\sum_{i}\hat{{\cal A}}_{i}^{\dagger}\hat{{\cal A}}_{i}$, then
the same construction leads to a local Hamiltonian $\hat{{\cal
H}}_{\text{spt}}^{D}=\sum_{i}(\hat{{\cal E}}\hat{{\cal A}}_{i}\hat{{\cal
E}}^{-1})^{\dagger}(\hat{{\cal E}}\hat{{\cal A}}_{i}\hat{{\cal E}}^{-1})$ [82,
83, 84].

We now apply this general construction principle to $1d$ and $2d$ cluster
state examples discussed previously.

Figure 3:  (a) 1d
$\mathbb{Z}_{2}^{\textrm{odd}}\times\mathbb{Z}_{2}^{\textrm{even}}$ symmetric
cluster state density matrix and (b) 2d
$\mathbb{Z}_{2}^{0}\times\mathbb{Z}_{2}^{1}$ symmetric cluster state density
matrix under the Choi Isomorphism. The decoherence of the symmetry $G_{B}$
maps to the coupling between $G_{B}^{u}$ and $G_{B}^{l}$ charges in two
layers. Accordingly, two respective symmetries become identified, and the SPT
classification collapses.

####  IV.1.1 1d cluster state

For the $1d$ cluster state, the SPT projector Hamiltonian can be written as

$H_{\text{spt}}=\sum_{i}\hat{A}_{i}^{2},\quad\hat{A}_{i}\equiv\frac{1-Z_{i-1}X_{i}Z_{i+1}}{2}.$
(79)

Consider the decoherence model in Eq. (73), which has the following convenient
form

$\displaystyle\hat{{\cal E}}$ $\displaystyle=\prod_{m}\frac{e^{\tau
Z_{2m,u}Z_{2m,l}}}{\cosh\tau},\quad\tanh\tau=\frac{p}{1-p}$ (80)

Then, we see that

$\displaystyle\hat{{\cal E}}\hat{{\cal A}}_{2n}\hat{{\cal
E}}^{-1}=\frac{1}{2}\big{(}1-e^{2\tau
Z_{2n,u}Z_{2n,l}}Z_{2n-1}X_{2n}Z_{2n+1}\big{)}$ (81)

while $\hat{{\cal E}}\hat{{\cal A}}_{2n-1}\hat{{\cal E}}^{-1}\,{=}\,\hat{{\cal
A}}_{2n-1}$. As depicted in Fig. 3(a), the parent Hamiltonian $\hat{{\cal
H}}_{\text{spt}}^{D}$ for the decohered SPT state takes the form of

$\hat{{\cal H}}_{\text{spt}}^{D}=\hat{{\cal H}}_{\text{spt},u}^{D}+\hat{{\cal
H}}_{\text{spt},l}^{D}+\hat{{\cal H}}_{\textrm{int}}^{D},$ (82)

where the upper layer Hamiltonian reads

$\displaystyle\hat{{\cal H}}_{\text{spt},u}^{D}$
$\displaystyle=\sum_{m}\frac{\cosh^{2}2\tau-\cosh 2\tau
Z_{2m-1}X_{2m}Z_{2m+1}}{2}$
$\displaystyle\,+\sum_{m}\frac{1-Z_{2m}X_{2m+1}Z_{2m+2}}{2},$ (83)

and the lower layer Hamiltonian $\hat{{\cal H}}_{\text{spt},l}^{D}$ is
essentially the same as $\hat{{\cal H}}_{\text{spt},u}^{D}$ with all the label
$u$ replaced by the label $l$, together with the interlayer coupling

$\hat{{\cal H}}_{\textrm{int}}^{D}=-\frac{\sinh
4\tau}{2}\sum_{m}Z_{2m,u}Z_{2m,l}.$ (84)

The coupling vanishes in the pure state limit when the decoherence strength
$p\to 0$ ($\tau\to 0$), and diverges in the strong decoherence limit as $p\to
1/2$ (which corresponds to measuring $Z_{2m}$ operators projectively and
forgetting the measurement outcomes).

This parent Hamiltonian $\hat{{\cal H}}_{\text{spt}}^{D}$ in Eq. (82)
explicitly shows that the decohered SPT state
${\|\rho_{\text{spt}}^{D}\rangle\rangle}$ can be interpreted as two identical
layers of $\mathbb{Z}_{2}^{\textrm{odd}}\times\mathbb{Z}_{2}^{\textrm{even}}$
SPT states coupled together by the interlayer $\mathbb{Z}_{2}^{\textrm{even}}$
charge tunneling (i.e. the ferromagnetic interlayer $ZZ$ coupling on every
even site). The resulting state is a nontrivial SPT state protected by the
$\mathbb{Z}_{2}^{{\textrm{odd}},u}\times\mathbb{Z}_{2}^{{\textrm{odd}},l}\times\mathbb{Z}_{2}^{{\textrm{even}}}$
symmetry in the doubled Hilbert space. This lattice model analysis is in full
agreement with the field theory understanding in Sec. III.1.

####  IV.1.2 2d cluster state

For the $2d$ cluster state under the local $Z$ noise on edge qubits described
by Eq. (80), the Choi state of the decohered density matrix can be shown to be
the groundstate of the Hamiltonian in the form Eq. (82), where

$\displaystyle\hat{{\cal H}}_{\text{spt},u}^{D}$ $\displaystyle=-\cosh
2\tau\sum_{e}\bm{X}_{e}\prod_{v\in\partial
e}Z_{v}-\sum_{v}X_{v}\prod_{e\in\differential v}\bm{Z}_{e}$
$\displaystyle\hat{{\cal H}}_{\textrm{int}}^{D}$ $\displaystyle=-\sinh
4\tau\sum_{e}\bm{Z}_{e,u}\bm{Z}_{e,l}.$ (85)

In this doubled system, the coupling $\hat{{\cal H}}_{\textrm{int}}^{D}$
breaks two one-form symmetries into their diagonal subgroup, and the remaining
global symmetry would be
$\mathbb{Z}_{2}^{(0),u}\times\mathbb{Z}_{2}^{(0),l}\times\mathbb{Z}_{2}^{(1)}$
as illustrated in Fig. 3(b).

However, a 1-form symmetry can often be robust under perturbation; even if it
does not exist at a microscopic level, it can appear as an emergent symmetry
[85, 86]. The robustness of an SPT state under the local decoherence of
higher-form symmetries can be thought of as a consequence of extensively many
conserved quantities, which is in contrast with the single globally conserved
charge of a 0-form symmetry. Accordingly, some decoherence in local charge
configuration cannot destroy the macroscopic stability of emergent higher-form
symmetries, and the corresponding anomaly acquires noise resilience.

Therefore for $p\ll 1$, two emergent one-form symmetries may become robust,
providing a bulk anomaly in-flow such that if one imagines a fictitious
boundary of the doubled system, the boundary would exhibit spontaneous
symmetry breaking of individual $\mathbb{Z}_{2}^{(0),u}$ and
$\mathbb{Z}_{2}^{(0),l}$ symmetries. This, in turn, implies that the type-I
strange correlator for $G_{A}^{u}\equiv\mathbb{Z}_{2}^{(0),u}$ charged
operators must be nontrivial in the doubled (pure) state for small $p$. In
other words,

$\displaystyle C^{\textrm{I}}_{A_{u}}(v,v^{\prime})$
$\displaystyle\equiv\frac{\langle\tilde{\Omega}|\,Z_{v}Z_{v^{\prime}}\otimes\mathbb{I}\,|\tilde{\Psi}\rangle}{\langle\tilde{\Omega}|\tilde{\Psi}\rangle}$
(86)

would be nontrivial. Since this expression is nothing but the type-I strange
correlator for the original decohered mixed state, the nontrivial type-I
strange correlator under decoherence in this model can be understood in terms
of the stability of the emergent one-form symmetries in the doubled system.

####  IV.1.3 Back to the original mixed state

So far, we have examined the cases where the effect of decoherence can be
considered as perturbing the parent Hamiltonian for the Choi state. In the
doubled Hilbert space, decoherence generally amounts to a non-unitary state
evolution that only respects the diagonal part of the doubled symmetry, and
breaks the respective symmetry of each layer. Thus, one might expect that any
doubled SPT order protected by the doubled symmetry group would immediately
collapse under decoherence. Interestingly, we have shown that this is not the
case in several examples, including 2d cluster state (lattice model) and 2d
SPT phases (field theory).

In the Choi state language, the type-I and type-II strange correlators of the
decohered SPT mixed state would be expressed as the type-I strange correlators
for different symmetries:

$\displaystyle C^{\textrm{I}}_{A}$
$\displaystyle=\frac{\langle\tilde{\Omega}|\,(O^{A}_{i}O^{A}_{j})_{u}\otimes\mathbb{I}_{l}\,|\tilde{\Psi}\rangle}{\langle\tilde{\Omega}|\tilde{\Psi}\rangle}=C^{d,\textrm{I}}_{A_{u}}$
(87) $\displaystyle C^{\textrm{II}}_{A}$
$\displaystyle=\frac{\langle\tilde{\Omega}|\,(O^{A}_{i}O^{A}_{j})_{u}\otimes(O^{A}_{i}O^{A}_{j})_{l}\,|\tilde{\Psi}\rangle}{\langle\tilde{\Omega}|\tilde{\Psi}\rangle}=C^{d,\textrm{I}}_{A_{u}\cdot
A_{l}},$ (88)

where the superscript $d$ represents that the strange correlator is for the
pure doubled state, and the subscript $A_{u}\cdot A_{l}$ means that the
strange correlator is for the operator charged under both $G_{A}^{u}$ and
$G_{A}^{l}$. Interestingly, the original type-II strange correlator of the
mixed state $\rho_{\textrm{spt}}^{D}$ for $G_{A}$ becomes the type-I strange
correlator of the Choi state $\|\rho_{\textrm{spt}}^{D}\rangle\rangle$ for the
operator charged under both $G_{A}^{u}$ and $G_{A}^{l}$. In the Choi state,
this strange correlator naturally probes the anomaly inflow of the
$G_{A}^{u}\times G_{A}^{l}\times G_{B}$ SPT order. Since the decoherence acts
like a shallow quantum circuit acting on the doubled SPT state that respects
the symmetry, as long as the decoherence is not too strong, the
$G_{A}^{u}\times G_{A}^{l}\times G_{B}$ SPT order should be robust. Therefore,
this Choi isomorphism naturally provides a mechanism for the stability of the
type-II strange correlator under selective decoherence.

Figure 4:  Schematic diagram of how information is encoded, decohered, and
decoded. After the encoding step, a label $x$ is encoded in quantum
correlations of a prepared state $|\psi(x)\rangle$. The state undergoes
decoherence, becoming a mixed state $\rho^{D}(x)$. Finally, a receiver
measures this mixed state to learn about the label $x$ (assuming the state is
repeated prepared). After enough measurements, the receiver may generate an
output $\hat{x}$ which is the estimator of the label $x$. The decoding is
successful if $\hat{x}=x$.

###  IV.2 Implications of the strange correlators

As remarked, a nontrivial and robust type-I strange correlator can be
interpreted as the presence of nontrivial boundary physics. From the
conventional understanding, the boundary of a $1d$ or $2d$ topological phase
is equipped with an associated anomaly that makes the boundary nontrivial,
i.e., the boundary must either spontaneously break the symmetry or be power-
law correlated. Compared with the type-I strange correlator, a nontrivial
type-II strange correlator has a more subtle implication. The type-II strange
correlator can be interpreted as an information-theoretic quantity that allows
one to identify what the underlying state is for a given mixed-state density
matrix. As illustrated in Fig. 4, one may imagine a procedure where the
_label_ information $x$ is encoded as the quantum circuit that can repeatedly
generate a certain quantum state. For example, given a label
$x\,{=}\,\textrm{SPT}$, a corresponding quantum circuit that prepares a
certain SPT state can be constructed, whose output quantum state is sent to an
observer. While doing so, decoherence occurs for various reasons, such as
imperfection in the state preparation or transmission noise from environments,
converting the pure quantum state into a mixed state density matrix. Finally,
the observer repeatedly measures the mixed state to learn about the input
label $x$. This last step is formally denoted as _decoding_.

Whether one can accurately identify the label or not crucially depends on the
decoding strategy. For example, if there is no decoherence, decoding would be
straightforward; one can simply measure the type-I strange correlator, which
as we argued can be viewed as a non-local SPT order parameter for a given pure
state density matrix, and its value would immediately allow one to tell the
decoded label $\hat{x}$ (See Sec. V for more precise protocol). Such a
decoding strategy would also be efficient in the sense that the required
number of measurements would at most scale polynomially with the system size.
Although the type-I strange correlator may become short-ranged with strong
enough decoherence, at weak decoherence the type-I strange correlator may
remain nontrivial. Hence for weak decoherence, the type-I strange correlator
can still tell us whether the underlying state was a trivial disordered state
or a nontrivial SPT state.

But with strong decoherence, we have shown that there are examples where the
type-I strange correlator becomes short-ranged, while the type-II strange
correlator remains nontrivial. Indeed, the robustness of the type-II strange
correlator implies that there must be a method to distinguish between the
decohered mixed states originating from an SPT state and those originating
from a trivial disordered state. Without referring to any decoding strategy,
the fact that the decohered trivial state and decohered SPT state are
distinguished by the type-II strange correlator implies the existence of a
fundamental information-theoretic distinction between these two labels.
Although the corresponding decoding strategy may not be efficient, the type-II
strange correlator behavior provides a fundamental distinction between two
classes of density matrices. Finally, when both the type-I and type-II strange
correlators are trivial, one cannot tell whether the underlying state is an
SPT or trivial state based on any feature that involves the mixed state
density matrix. Therefore, behaviors of the type-I and type-II strange
correlators allow us to distinguish three different regimes of the mixed state
density matrix under decoherence as illustrated in Fig. 5.

Figure 5:  We draw schematic plots of the type-I and type-II strange
correlators as a function of decoherence strength, which distinguish three
different information-theoretic phases. For $p<p_{c,1}$, $C^{\textrm{I}}$ is
nontrivial, implying that the boundary physics of the original system should
be nontrivial. Furthermore, the SPT label information may be efficiently
decoded (see Sec. V). For $p_{c,1}<p<p_{c,2}$, $C^{\textrm{II}}$ is
nontrivial, quantifying the information that the original state under
decoherence was an SPT order. However, the quantity is very difficult to
extract from the bulk measurements. For $p>p_{c,2}$, the SPT label information
cannot be identified by the type-II strange correlator. In the local
decoherence models, we considered in the lattice calculations, $p_{c,2}$ does
not exist since the type-II strange correlator is always finite. However, for
a generic decoherence model, we would expect the transition into the “trivial”
information phase to exist.

Here we comment on the efficiency of measuring the type-II strange correlator.
Note that any observable in the doubled Hilbert space that involves operators
in both upper and lower Hilbert space would involve two density matrices,
which would make the quantity challenging to measure in general. This is
because the numerator and denominator of the type-II strange correlator should
be evaluated separately, and the denominator which is the purity of the
density matrix $\rho^{D}$ is exponentially small in general; it is
straightforward to show that

$\textrm{tr}(\rho^{D}_{\textrm{spt}}\rho^{D}_{\textrm{spt}})\sim\exp(-N)$ (89)

where $N$ is the number of sites for any $p>0$ in the decoherence model.
Likewise, the numerator decays exponentially. Therefore, although in principle
we can define a quantity to characterize an ordering in the doubled Hilbert
space, it only characterizes the presence of some nontrivial quantum
entanglements whose verification is exponentially hard. A similar situation
occurs in the famous black hole information paradox: the reconstruction of a
quantum state fallen into an _old_ blackhole based on the Hawking radiation is
theoretically possible, but it would require an exponential computational
complexity in general [87, 88, 89].

##  V Experimentally probing the strange correlator

So far, we have discussed how strange correlators can detect nontrivial SPT
features even under decoherence. Naively, strange correlators are just
theoretical tools since we cannot sandwich two different states in actual
experiments. However, in this section, we will show that with the help of
classical computations, one can experimentally probe nontrivial signatures of
the strange correlators through measurements, so-called computationally
assisted observables (CAO) [24, 90].

###  V.1 Strange correlator against a generic trivial state

To further proceed, we consider a strange correlator against a generic trivial
disordered state $\ket{\Omega_{\bm{s}}}$, which is defined as the product
state along the $X$-basis characterized by the bit-string
${\bm{s}}=\\{s_{1},s_{2},...,s_{N}\\}$:

$\ket{\Omega_{\bm{s}}}\equiv\bigotimes_{n=1}^{N}\big{[}Z^{(1-s_{n})/2}|+\rangle_{n}\big{]},\quad
s_{i}\in\\{-1,1\\}$ (90)

which is invariant under the global symmetry. Essentially, the label
${\bm{s}}$ specifies the symmetry charge configuration of the trivial state of
interest. Then, we define
$C^{\textrm{I}}_{\bm{s}}(r)=\tr(\rho_{\text{spt}}O(0)O(r)\rho_{0,{\bm{s}}})/\tr(\rho_{\text{spt}}\rho_{0,{\bm{s}}})$
as the type-I strange correlator computed against the ${\bm{s}}$-labeled
trivial state density matrix
$\rho_{0,{\bm{s}}}\equiv|\Omega_{\bm{s}}\rangle\langle\Omega_{\bm{s}}|$. We
emphasize that for the strange correlator to be well-defined, the overlap
between the SPT and trivial states $\tr(\rho_{\textrm{spt}}\rho_{0,{\bm{s}}})$
has to be non-zero. For that, it is necessary that they have components
carrying the same total charge under the global symmetry. In general, one may
expect that this generalized strange correlator $C^{\textrm{I}}_{\bm{s}}(r)$
still serves as a suitable probe for the SPT physics as the locally-charged
background (specified by ${\bm{s}}$) should not change the qualitative
behaviors.

In the following, we compute strange correlators of pure $G_{A}\times G_{B}$
SPT states against generic trivial states for two exemplary cases with or
without decoherence, where the decoherence model is given as Eq. (II.3):

$(i)$ _1d cluster
$\mathbb{Z}_{2}^{\textrm{odd}}\times\mathbb{Z}_{2}^{\textrm{even}}$ SPT
state_: For $|\Omega_{\bm{s}}\rangle$ to have a non-zero overlap with the 1d
cluster state, the trivial state should have a even parity on both symmetries,
i.e., $\prod_{n}s_{2n+1}=\prod_{n}s_{2n}=1$. Then,

$\displaystyle
C^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}}(2n)=\prod_{m=1}^{n}s_{2m},\quad
C^{\textrm{I}}_{{\textrm{even}},{\bm{s}}}(2n)=\prod_{m=1}^{n}s_{2m-1}.$ (91)

Under decoherence of $\mathbb{Z}_{2}^{\textrm{even}}$,
$\rho_{\textrm{spt}}^{D}$ would become an ensemble of configurations with
different total $\mathbb{Z}^{\textrm{even}}_{2}$ charges, and the constraint
$\prod_{n}s_{2n}=1$ does not have to hold anymore for a trivial state. In this
case, $C^{\textrm{I}}_{{\textrm{even}},{\bm{s}}}(2n)$ remains unchanged while
$C^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}}(2n)$ becomes short-ranged as in Eq.
(11). Even away from the stabilizer limit, the numerical results in Fig. 2 has
shown that the above sign structure persists, and the magnitude of
$C^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}}$ does not depend on the
$\mathbb{Z}_{2}^{\textrm{even}}$-charge configuration of the trivial state
${\bm{s}}_{\textrm{even}}=\\{s_{2},s_{4},...\\}$. However, the magnitude of
the $C^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}}$ can change depending on
${\bm{s}}_{\textrm{odd}}=\\{s_{1},s_{3},...\\}$.

$(ii)$ _2 $d$ cluster
$\mathbb{Z}^{\textrm{(0)}}_{2}\times\mathbb{Z}_{2}^{\textrm{(1)}}$ SPT state_:
For $|\Omega_{\bm{s}}\rangle$ to have a non-zero overlap with the 2d cluster
state, there are two necessary conditions: $(i)$ $\prod_{v}s_{v}=1$ (trivial
0-form global charge), and $(ii)$ $B_{p}\equiv\prod_{e\in p}s_{e}=1$ for all
plaquette $p$ (trivial 1-form charge configurations). Then, an ordinary
correlation $Z_{v}Z_{v^{\prime}}$ for $G_{A}=\mathbb{Z}^{\textrm{(0)}}_{2}$
and a Wilson loop operator $\prod_{e\in\partial{\cal A}}Z_{e}$ for
$G_{B}=\mathbb{Z}^{\textrm{(1)}}_{2}$ are given as follows

$\displaystyle C^{\textrm{I}}_{A,{\bm{s}}}(\partial l)=\prod_{e\in
l}s_{e},\quad C^{\textrm{I}}_{B,{\bm{s}}}(\partial{\cal A})=\prod_{v\in{\cal
A}}s_{v}.$ (92)

where $l$ is an open string connecting two vertices $v$ and $v^{\prime}$,
i.e., $\partial l=\\{v,v^{\prime}\\}$.

Under decoherence of $G_{A}$ (0-form), the above sign structure for
$C^{\textrm{I}}_{B}$ remains robust, while the magnitude can change depending
on the decoherence strength as in Eq. (26). In this case,
$C^{\textrm{I,II}}_{A}$ is unaffected. Under decoherence of $G_{B}$ (1-form),
however, $C^{\textrm{I}}_{A}$ gets modified in an interesting way. It is
straightforward to show that under decoherence,

$\displaystyle C^{\textrm{I}}_{A,{\bm{s}}}=\langle
Z_{v}Z_{v^{\prime}}\rangle_{\beta}^{\textrm{RBIM}({\bm{s}})}$ (93)

which is the correlation function of the random bond Ising model (RBIM) at the
inverse temperature $\beta=\tanh^{-1}(1-2p)$ for the bond configuration
${\bm{s}}$. As the correlation of the RBIM, the magnitude of
$C^{\textrm{I}}_{A,{\bm{s}}}$ only depends on the bond frustration, which
corresponds to the 1-form charge configuration $\\{B_{p}\\}$ of the trivial
state $|\Omega_{\bm{s}}\rangle$. Interestingly, unlike the 1d cluster state
case, this implies that the type-I strange correlator of the mixed state
$\rho^{D}_{\textrm{spt}}$ behaves differently depending on the trivial state
we computed against. For example, when $s_{e}=1$,
$C^{\textrm{I}}_{A,{\bm{s}}}$ behaves as the correlation of the ferromagnetic
Ising model. If we choose $s_{e}$ such that $B_{p}=+1$ for half the plaquettes
and $B_{p}=-1$ for the other half, $C^{\textrm{I}}_{A,{\bm{s}}}$ behaves as
the correlation of the maximally frustrated RBIM, decaying exponentially in
distance for any $p$. This raises the question of which strange correlator is
appropriate for one to use to diagnose the nontrivial behavior of the
underlying SPT physics. In fact, a proper way is to take a weighted average of
all type-I strange correlators, whose weight is determined by the overlap
between the trivial state from a given density matrix,
$P_{\bm{s}}=\langle\Omega_{\bm{s}}|\rho_{\textrm{spt}}^{D}|\Omega_{\bm{s}}\rangle$.
In this formalism, for small $p$ the weight for the latter scenario is
extremely small, and the _average_ behavior of the type-I strange correlator
would still be long-ranged. We will examine the implication of this in the
next section.

###  V.2 With post-selection

With measurements and post-selections, it is possible to experimentally
extract ($i$) the quantity very similar to the type-I strange correlator and
($ii$) the type-II strange correlator. The idea is that measurements and post-
selections allow one to project the SPT state into a certain trivial product
state to some degree, a step essential to compute the strange correlator. To
illustrate the idea, consider a $1d$ cluster state discussed in the previous
section. Assume that we want to understand the behavior of the type-I strange
correlators for the $\mathbb{Z}_{2}^{\textrm{odd}}$ symmetry. As the type-I
strange correlator is not a physical quantity that can be directly measured,
we instead define the _marginalized_ type-I strange correlator:

$\displaystyle\tilde{C}^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}^{\textrm{e}}}(2n)\equiv\frac{\textrm{tr}(\rho^{D}_{\textrm{spt}}Z_{1}Z_{2n+1}{\cal
P}_{{\bm{s}}^{\textrm{e}}})}{\textrm{tr}(\rho^{D}_{\textrm{spt}}{\cal
P}_{{\bm{s}}^{\textrm{e}}})}=\langle Z_{1}Z_{2n+1}\rangle_{{\cal
P}^{\textrm{e}}_{{\bm{s}}^{\textrm{e}}}\rho^{D}_{\textrm{spt}}}$ (94)

where ${\cal P}_{{\bm{s}}^{\textrm{e}}}$ is the projector onto a specific
$\mathbb{Z}_{2}^{\textrm{even}}$-charge configuration defined as

$\displaystyle{\cal
P}_{{\bm{s}}^{\textrm{e}}}\equiv\prod_{m=1}^{N}\frac{1+s_{2m}X_{2m}}{2}$ (95)

with the bit-string ${{\bm{s}}^{\textrm{e}}}=(s_{2},s_{4},...,s_{2N})$. We
remark that the form of the marginalized type-I strange correlator is very
similar to that of the type-I strange correlator. The crucial difference is
that instead of evaluating against a trivial state density matrix
$\rho_{0,{\bm{s}}}$, which is essentially the projector onto a specific
$\mathbb{Z}_{2}^{\textrm{odd}}\times\mathbb{Z}_{2}^{\textrm{even}}$ charge
configuration labeled by ${\bm{s}}$, here we evaluate against the projector
that only specifies the $\mathbb{Z}_{2}^{\textrm{even}}$ charge configuration.
Thus, in the computation of
$\tilde{C}^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}^{\mathbf{e}}}$, the
$\mathbb{Z}_{2}^{\textrm{odd}}$ charge configuration is _marginalized_.

One may immediately notice that the marginalized type-I strange correlator is
the correlation function of $Z$ operators for the _projected_ mixed state
${\cal P}_{{\bm{s}}^{\textrm{e}}}\rho^{D}_{\textrm{spt}}{\cal
P}_{{\bm{s}}^{\textrm{e}}}$. The projected mixed state can be obtained by
measuring all the even sites in the $X$-basis and _post-selecting_ the
resulting state with measurement outcomes equal to ${\bm{s}}^{\textrm{e}}$. As
the correlation function of the physical mixed state, the marginalized type-I
strange correlator is experimentally measurable. Also, as the name suggests,
it is related to the type-I strange correlators. To show this, we define
${\cal P}_{{\bm{s}}^{\textrm{o}}}$ as the projector onto a specific
$\mathbb{Z}_{2}^{\textrm{odd}}$ configuration labeled by
${\bm{s}}^{\textrm{o}}=(s_{1},s_{3},...,s_{2N-1})$. Together, ${\cal
P}_{{\bm{s}}^{\textrm{o}}}$ and ${\cal P}_{{\bm{s}}^{\textrm{e}}}$ project
onto a configuration where all symmetry charges are specified, i.e., a trivial
disordered state $|\Omega_{{\bm{s}}}\rangle$ with
${\bm{s}}=({\bm{s}}^{\textrm{o}},{\bm{s}}^{\textrm{e}})$. Therefore, by
inserting a completeness relation
$\mathbb{I}=\sum_{{\bm{s}}^{\textrm{o}}}{\cal P}_{{\bm{s}}^{\textrm{o}}}$ in
Eq. (94), we can show that

$\displaystyle\tilde{C}^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}^{\textrm{e}}}(2n)$
$\displaystyle=\sum_{{\bm{s}}^{\textrm{o}}}\frac{\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}{\cal
P}_{{\bm{s}}^{\textrm{e}}}{\cal
P}_{{\bm{s}}^{\textrm{o}}}\big{)}}{\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}{\cal
P}_{{\bm{s}}^{\textrm{e}}}\big{)}}\frac{\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}Z_{1}Z_{2n+1}{\cal
P}_{{\bm{s}}^{\textrm{e}}}{\cal
P}_{{\bm{s}}^{\textrm{o}}}\big{)}}{\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}{\cal
P}_{{\bm{s}}^{\textrm{e}}}{\cal P}_{{\bm{s}}^{\textrm{o}}}\big{)}}$
$\displaystyle=\sum_{{\bm{s}}^{\textrm{o}}}P({\bm{s}}^{\textrm{o}}|{\bm{s}}^{\textrm{e}})\,C^{\textrm{I}}_{{\textrm{odd}},({\bm{s}}^{\textrm{o}},{\bm{s}}^{\textrm{e}})}(2n),$
(96)

where $P({\bm{s}}^{\textrm{o}}|{\bm{s}}^{\textrm{e}})$ is the probability of
observing measurement outcome ${\bm{s}}^{\textrm{o}}$ on odd sites conditioned
on ${\bm{s}}^{\textrm{e}}$ on even sites for the SPT state $|\Psi\rangle$.
Therefore, as a weighted average of the type-I strange correlators, the
experimentally accessible quantity
$\tilde{C}^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}^{\textrm{e}}}(2n)$ would
quantify the overall non-trivialness of the type-I strange correlator. In the
stabilizer limit, as we showed in Eq. (91), the type-I strange correlator
$C^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}}$ is independent of the values of
${\bm{s}}$ on odd sites. Accordingly, the marginalized type-I strange
correlator $\tilde{C}^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}^{\textrm{e}}}$
would be equal to ${C}^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}}$ if ${\bm{s}}$
matches ${\bm{s}}^{\textrm{e}}$ on even sites.

The type-II strange correlator is a quantity that can be expressed as the
ratio of two fidelities:
$\textrm{tr}(\rho_{\textrm{spt}}^{D}\rho_{0,{\bm{s}}}^{\vphantom{\dagger}})$
and
$\textrm{tr}(\rho_{\textrm{spt}}^{D}{\rho}_{0,{\bm{s}}^{\prime}}^{\vphantom{\dagger}})$
where $\rho_{0,{\bm{s}}^{\prime}}\equiv
Z_{1}Z_{2n+1}\rho_{0,{\bm{s}}}Z_{2n+1}Z_{1}$ is also the pure state density
matrix where ${\bm{s}}^{\prime}$ differs from ${\bm{s}}$ in signs for two
sites:

$\displaystyle C^{\textrm{II}}_{{\textrm{odd}},{\bm{s}}}(2n)$
$\displaystyle=\frac{\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}Z_{1}Z_{2n+1}\rho^{\vphantom{\dagger}}_{0,{\bm{s}}}Z_{2n+1}Z_{1}\big{)}}{\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}\rho^{\vphantom{\dagger}}_{0,{\bm{s}}}\big{)}}$
$\displaystyle=\frac{\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}\rho^{\vphantom{\dagger}}_{0,{\bm{s}}^{\prime}}\big{)}}{\textrm{tr}\big{(}\rho^{D}_{\textrm{spt}}\rho^{\vphantom{\dagger}}_{0,{\bm{s}}}\big{)}}=\frac{P({\bm{s}}^{\prime})}{P({\bm{s}}\,)}$
(97)

Therefore, if we repeatedly measure the charge configuration ${\bm{s}}$ of the
mixed state and estimate the probability $P({\bm{s}})$ and
$P({\bm{s}}^{\prime})$ for the measurement outcome to be ${\bm{s}}$
(${\bm{s}}^{\prime}$), we can calculate
$C^{\textrm{II}}_{{\textrm{even}},{\bm{s}}}(2n)$. Without post-selection, this
is an extremely challenging task since there are exponentially many possible
outcomes ${\bm{s}}$. However, assuming we can post-select, one can compare the
relative frequency between ${\bm{s}}^{\prime}$ and ${\bm{s}}$ and estimate the
type-II strange correlator.

Another possibility to perform the fidelity estimation is to use classical
shadow tomography [91]. In this approach, the decohered SPT state
$\rho_{\text{spt}}^{D}$ will be repeatedly prepared and then measured in the
random Clifford basis. The measurement will collapse $\rho_{\text{spt}}^{D}$
to a random stabilizer state $\ket{\sigma}$ with probability
$\bra{\sigma}\rho_{\text{spt}}^{D}\ket{\sigma}$. These post-measurement states
$\ket{\sigma}$ will be recorded and post-processed on a classical computer. In
particular, the fidelity $\textrm{tr}(\rho_{\text{spt}}^{D}\rho_{0,{\bm{s}}})$
(both in the numerator and the denominator) in Eq. (V.2) can be estimated as
$\textrm{tr}(\rho_{\text{spt}}^{D}\rho_{0,{\bm{s}}})=\mathbb{E}_{\ket{\sigma}}(2^{2N}+1)\bra{\sigma}\rho_{0,{\bm{s}}}\ket{\sigma}-1$,
where $\bra{\sigma}\rho_{0,{\bm{s}}}\ket{\sigma}$ can be efficiently computed
given that $\ket{\sigma}$ is a stabilizer state and we have the freedom to
choose the trivial reference state $\rho_{0,{\bm{s}}}$ to be a stabilizer
state as well. Thus both the numerator and the denominator of Eq. (V.2) can be
obtained by fidelity estimation via classical shadow tomography. However, this
is not more efficient than the post-selection method. Because both fidelities
are exponentially small ($\sim 2^{-2N}$) with respect to the system size $2N$,
such that the number of measurements needed to accurately determine both the
numerator and the denominator is still exponentially large ($\sim 4^{2N}$) in
order to control the estimation standard deviation below the level of
$2^{-2N}$.

###  V.3 Without post-selection

Post-selecting specific measurement outcomes allow us to access a type-II
strange correlator as well as a weighted average of type-I strange
correlators. However, as the system size increases, the number of samples for
post-selection increases exponentially with the system size, and the whole
idea becomes experimentally impractical. Therefore, for this method to be
feasible, it is essential to avoid the post-selection issue.

####  V.3.1 Without decoherence

Before addressing this issue, it is important to understand that in the pure
SPT state, a nontrivial behavior of the type-I strange correlator in the SPT
phase is intimately tied to the presence of finite non-local SPT order
parameters, which can be experimentally measured to diagnose the underlying
SPT order. For example, the following _string order parameter_ in the 1d
$\mathbb{Z}^{\textrm{odd}}_{2}\times\mathbb{Z}^{\textrm{even}}_{2}$ SPT phase
allows one to distinguish an SPT and trivial phases [32, 33]:

$S_{2a+1,2b+1}=Z_{2a+1}Z_{2b+1}\prod_{m={a+1}}^{b}X_{2m},$ (98)

which takes a non-zero expectation value as $|a-b|\rightarrow\infty$ in the
SPT phase. One way to interpret this quantity is the following: in the SPT
phase, the application of the $\mathbb{Z}_{2}^{\textrm{even}}$ symmetry
transformation on a partial region is equivalent to threading a symmetry
defect (flux) at the boundaries of the region. For this SPT state, by
definition, such a $\mathbb{Z}_{2}^{\textrm{even}}$ symmetry defect should
induce a $\mathbb{Z}_{2}^{\textrm{odd}}$ charge, which can be canceled by the
application of the charge creation/annihilation operator. If we define $S$ as
such a partial symmetry transformation combined with the charge
creation/annihilation operators at the boundary, $S$ would behave non-
trivially, while for a trivial disordered state, the defect does not induce
any charge and $S$ would take a value exponentially decaying in its size.

For a given SPT state $\rho_{\textrm{spt}}$, the expectation value of the
string order parameter can be expressed in terms of strange correlators as the
following:

$\displaystyle\textrm{tr}\big{(}\rho_{\textrm{spt}}S_{1,2n+1}\big{)}$
$\displaystyle=\sum_{{\bm{s}}}P_{\bm{s}}\Big{[}\prod_{m=1}^{n}s_{2m}\Big{]}\,C^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}}(2n)$
$\displaystyle=\sum_{{\bm{s}}^{e}}P_{{\bm{s}}^{e}}\Big{[}\prod_{m=1}^{n}s_{2m}\Big{]}\,\tilde{C}^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}^{e}}(2n)$
(99)

where we used the completeness of the product states
$\sum_{{\bm{s}}}\rho_{0,{\bm{s}}}=\mathbb{I}$ for the derivation, and
$P_{\bm{s}}=\textrm{tr}(\rho_{\textrm{spt}}\rho_{0,{\bm{s}}})$ is the
probability to obtain the measurement outcome ${\bm{s}}$ by measuring
$\rho_{\textrm{spt}}$ in $X$-basis, satisfying $\sum_{\bm{s}}P_{\bm{s}}=1$.
The expression implies that the string order parameter is a weighted average
of _signed_ strange correlators. The implications of Eq. (V.3.1) can be
summarized as the following:

  * •

If the expectation value of the non-local parameter is non-trivial, the signed
average of the type-I strange correlators should be non-trivial.

  * •

If $(i)$ the type-I strange correlators are nontrivial and $(ii)$ their sign
structure cancels the prefactor $\prod s_{2m}$, the non-local parameter $S$
must be non-trivial. However, even if the condition $(i)$ holds, the condition
$(ii)$ may not hold under decoherence.

In the pure SPT state where the expectation value of the non-local order
parameter is nontrivial, indeed the strange correlators are nontrivial and
their signs are generally canceled by the term $\prod s_{2m}$. In other words,

$C^{\textrm{I}}_{{\textrm{odd}},{\bm{s}}}=\prod_{m=1}^{n}s^{\vphantom{\dagger}}_{2m}C^{\textrm{I}}_{{\textrm{odd}},\bar{{\bm{s}}}}\,\,\Rightarrow\,\,\expectationvalue{S_{1,2n+1}}=\tilde{C}^{\textrm{I}}_{{\textrm{odd}},\bar{s}^{e}}$
(100)

where $\bar{{\bm{s}}}$ is defined in such a way that its even part
$\bar{{\bm{s}}}^{e}=1$ and its odd part $\bar{{\bm{s}}}^{o}={\bm{s}}^{o}$.
Therefore, the string order parameter would be given as the marginalized
type-I strange correlator discussed in the previous section. Since the string
order parameter can be simply measured from the bulk wavefunction without any
post-selection, it implies that in the pure SPT state measuring string order
parameters allow one to learn about the type-I strange correlators.

To be more concrete, we outline the following experimental procedure to
measure non-local order parameters: for a given $\rho_{\textrm{spt}}^{D}$,
measure all even sites by $X$-basis and odd sites by $Z$-basis, where
${\bm{s}}^{\textrm{e}}=(s_{2},s_{4},...)$ and
${\bm{\sigma}}^{\textrm{o}}=(\sigma_{1},\sigma_{3},...)$ are measurement
outcome bit-strings. Now, we repeat this procedure repeatedly and obtain a set
of $M$ bit-strings $\\{{\bm{s}}^{\textrm{e},(i)}\\}_{i=1}^{M}$ and
$\\{{\bm{\sigma}}^{\textrm{o},(i)}\\}_{i=1}^{M}$. Then,

$\displaystyle\frac{1}{M}\sum_{i=1}^{M}\Big{[}\prod_{m=1}^{n}s^{(i)}_{2m}\Big{]}\,\sigma^{(i)}_{1}\sigma^{(i)}_{2n+1}\rightarrow\tilde{C}^{\textrm{I}}_{o,\bar{{\bm{s}}}}(2n)$
(101)

whose statistical uncertainty reduces _polynomially_ in the number of
measurement snapshots $M$. This is exactly equivalent to the measurement of
the marginalized type-I strange correlator except we multiply by the
additional factor $\prod s_{2m}$. In fact, the absence of post-selection is
exactly compensated by the presence of this additional factor. Therefore, even
without post-selection, we can identify a nontrivial behavior of the
marginalized type-I strange correlator with polynomial sample complexity.

To generalize this idea to higher dimension, consider an $G_{A}\times G_{B}$
SPT characterized by the mixed anomaly between $G_{A}$ and $G_{B}$ symmetries;
in other words, the domain wall (defect) of $G_{B}$ is decorated with
$G_{A}$-SPT states [62]. In this case, the non-local order parameter $S^{A}$
associated with $G_{A}$-symmetry on the area ${\cal A}$ is defined as the
following [64, 92]:

$\displaystyle S^{A}_{g}({\cal A})\equiv O_{g}(\partial{\cal A})\cdot
U_{g}(cA)$ (102)

where $g\in G_{B}$ is the element of the $G_{B}$ symmetry group,
$O_{g}(\partial{\cal A})$ is the operator that creates a $G_{A}$-SPT state
along the boundary $\partial{\cal A}$ decorating the $g$-domain wall, and
$U_{g}({\cal A})$ is the partial $G_{B}$ symmetry transformation on the region
${\cal A}$. By definition, this non-local order parameter would exhibit a
perimeter law in the SPT phase, i.e., it decays exponentially with its
perimeter $|\partial{\cal A}|$:

$\langle S_{g}^{A}({\cal A})\rangle\propto e^{-a|\partial{\cal A}|}$ (103)

When $G_{B}$ is a $n$-form symmetry, the region of support ${\cal A}$ is
$(d\,{-}\,n)$-dimensional. Accordingly, its boundary is
$(d\,{-}\,n\,{-}\,1)$-dimensional. Therefore, if $G_{B}$ is a
$(d\,{-}\,1)$-form symmetry, there always exist a non-local order parameter
for $G_{B}$ which takes a finite value in the SPT phase. On the other hand, if
$G_{B}$ is $(d\,{-}\,2)$-form symmetry, then it has support on a 2-dimensional
manifold, whose boundary is 1-dimensional. Accordingly, in the SPT phase, the
associated non-local order parameter would decay exponentially with its
boundary length; however, for the trivial state, this order parameter would
decay exponentially with its area, so one can distinguish an SPT state from a
trivial disordered state.

By inserting a completeness relation in the $G_{B}$-charge basis
$\mathbb{I}=\sum_{{\bm{s}}^{B}}{\cal P}_{{\bm{s}}^{B}}$, where ${\bm{s}}^{B}$
is the string specifying a $G_{B}$-charge configuration, the non-local order
parameter expectation value can be expressed as

$\displaystyle\textrm{tr}(\rho_{\textrm{spt}}^{D}S_{g})$
$\displaystyle=\sum_{{\bm{s}}^{B}}P_{{\bm{s}}^{B}}\,c(U_{g}|{\bm{s}}^{B})\,\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}(O_{g})$
(104)

where each term is defined as

$\displaystyle P_{{\bm{s}}^{B}}$ $\displaystyle\equiv\tr({\cal
P}_{{\bm{s}}^{B}}\rho_{\textrm{spt}}^{D})$ $\displaystyle
c(U_{g}|{\bm{s}}^{B})$
$\displaystyle\equiv\bra{\Omega_{{\bm{s}}^{B}}}U_{g}({\cal
A})\ket{\Omega_{{\bm{s}}^{B}}},$
$\displaystyle\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}(O_{g})$
$\displaystyle\equiv\sum_{{\bm{s}}^{A}}P({\bm{s}}^{A}|{\bm{s}}^{B})\,C^{\textrm{I}}_{A,({\bm{s}}^{A},{\bm{s}}^{B})}(O_{g}).$
(105)

$P_{{\bm{s}}^{B}}$ is the probability of obtaining a measurement outcome
${\bm{s}}^{B}$ on $\rho_{\textrm{spt}}^{D}$, and $c(U_{g}|{\bm{s}}^{B})$ is
the phase factor obtained by applying $U_{g}({\cal A})$ on the product state
labeled by ${\bm{s}}^{B}$, and
$\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}(O_{g})$ is the marginalized type-I
strange correlator. Therefore, for these SPT states the points made earlier
for the 1d $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT state immediately extend.

####  V.3.2 With decoherence

Despite the Eq. (104), nontrivial behaviors of the type-I strange correlator
under decoherence do not imply that the non-local order parameters are also
nontrivial under decoherence. As we have examined in Eq. (II.6), the non-local
order parameter immediately becomes short-ranged under decoherence while the
type-I strange correlator remains nontrivial for a weak decoherence strength.
The reason is that, $c(U|{\bm{s}}^{B})$ does not cancel the sign of
$\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}$, and the weighted average of the
marginalized strange correlators with unbalanced oscillating signs becomes
short-ranged.

However, this issue can be circumvented through some computational assistance
[24]. Theoretically, the question can be rephrased as finding a decoding
quantum channel ${\cal D}$ such that if we evaluate a non-local order
parameter of ${\cal D}[\rho_{\textrm{spt}}^{D}]$ instead of
$\rho_{\textrm{spt}}^{D}$, the quantity becomes non-trivial. Roughly speaking,
this can be achieved by multiplying an additional ${\bm{s}}^{B}$-dependent
factor $d(S_{g}|{\bm{s}}^{B})$ such that it cancels the oscillating sign in
the summation:

$\displaystyle\textrm{tr}({{\cal D}}[\rho_{\textrm{spt}}^{D}]S_{g})$
$\displaystyle=\sum_{{\bm{s}}^{B}}P_{{\bm{s}}^{B}}\,c(U_{g}|{\bm{s}}^{B})\,\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}(O_{g})\cdot
d(S_{g}|{\bm{s}}^{B})$ (106)

where $d(S_{g}|{\bm{s}}^{B})$ is defined in a way such that
$c\cdot\tilde{C}^{\textrm{I}}\cdot d\geq 0$ for as many ${\bm{s}}^{B}$ as
possible. In principle, if $\tilde{C}$ is nontrivial for each ${\bm{s}}^{B}$,
there must exist $d$ such that the above summation becomes non-trivial.

Experimentally, the procedure is as the following. First, we measure all
$G_{B}$-charges. Then, based on the measurement outcome ${\bm{s}}^{B}$, we can
predict or _decode_ the sign for the corresponding marginalized type-I strange
correlator, and apply a relevant transformation on the remaining qubits. This
step is captured by the decoding quantum channel ${\cal D}_{{\bm{s}}^{B}}$
acting on $G_{A}$-charges 111Although it sounds like we apply unitary
transformations, this procedure can be stochastic, implying that the decoding
step is captured by the quantum channel.. The combined action of measurement
and decoding can be described by the following quantum channel:

${\cal D}\big{[}\rho_{\textrm{spt}}^{D}\big{]}\equiv\sum_{{\bm{s}}^{B}}{\cal
D}_{{\bm{s}}^{B}}\big{[}{{\cal P}}_{{\bm{s}}^{B}}\rho_{\textrm{spt}}^{D}{{\cal
P}}_{{\bm{s}}^{B}}\big{]}.$ (107)

where ${\cal P}_{{\bm{s}}^{B}}$ is the projector onto the measurement outcomes
${\bm{s}}^{B}$ on the $G_{B}$-charges. In this representation,

$d(S_{g}|{\bm{s}}^{B})\equiv\frac{\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}({\cal
D}_{{\bm{s}}^{B}}[O_{g}])}{\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}(O_{g})}$
(108)

where we used the self-adjointness of the decoding channel ${\cal
D}_{{\bm{s}}^{B}}$ which only acts on $G_{A}$-charge associated with the
$O_{g}$ operator. Finally, we measure $O_{g}$ for this state projected into
${\bm{s}}^{B}$, which is equivalent to measuring the marginalized type-I
strange correlator as in Eq. (94). Let ${\bm{\sigma}}^{A}$ be the label for
the measurement outcome of $O_{g}$. One can repeat this procedure, and then
use the sequence of
$\\{({\bm{\sigma}}^{A,(i)},{\bm{s}}^{B,(i)})\,|\,i=1,2,...\\}$ to estimate the
_computationally assisted observable_.

This computationally assisted observable can exhibit nontrivial behavior in
the decohered SPT phase. Then, this would guarantee that the marginalized
type-I strange correlators must be non-trivial. It can be immediately seen
from the following inequality:

$\displaystyle\Big{|}\sum_{{\bm{s}}^{B}}P_{{\bm{s}}^{B}}\,c^{\cal
D}(U|{\bm{s}}^{B})\,\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}(O)\Big{|}\leq\sum_{{\bm{s}}^{B}}P_{{\bm{s}}^{B}}\Big{|}\tilde{C}^{\textrm{I}}_{A,{\bm{s}}^{B}}(O)\Big{|}.$
(109)

For example, if the LHS is decaying in power-law, the inequality implies that
the marginalized strange correlators cannot decay exponentially, i.e., non-
trivial. Therefore, this method would allow one to identify the SPT
information from the decohered mixed state using a polynomial number of
measurement snapshots.

On the contrary, if the type-I strange correlators are short-ranged, then no
matter how well we choose a phase factor $c^{\cal D}$, the LHS must be
trivial. Put it differently, there does not exist a decoding channel ${\cal
D}$ such that a computationally assisted observable takes a nontrivial value.
Therefore, the behavior of the type-I strange correlator imposes a fundamental
bound on topological information we can extract from a decohered mixed state
with polynomial sample complexity.

The Eq. (109) has another important consequence in the context of measurement-
assisted state preparations. It has been pointed out that by measuring SPT
(short-range entangled) states followed by an appropriate feedback, one can
realize long-range entangled quantum states, from GHZ states to non-abelian
topological orders, and even fractons [11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27]. In the measurement-assisted state preparations,
one prepares an SPT state with two symmetries $G_{A}$ and $G_{B}$, and charges
of one symmetry are measured, say $G_{B}$ [18]. Then, the resulting state has
a nontrivial correlation structure for $G_{A}$-charged operators; note that
this correlation function is exactly equivalent to our definition of the
marginalized type-I strange correlators in Eq. (V.2). Finally, one applies
appropriate feedback based on the measurement outcomes of $G_{B}$-charges so
that this nontrivial correlation of $G_{A}$-charged operators is transformed
into a way independent of the measurement outcomes. The last step corresponds
to finding a decoding quantum channel ${\cal D}$ in our language. If the last
step fails, the measurement-assisted state preparation fails as well. Since
our inequality imposes a fundamental bound on this last step, we get the
following theorem:

Theorem Under decoherence, an SPT state can be diagnosed efficiently in the
sample size only if the non-trivial marginalized type-I strange correlator
exists. Furthermore, a scheme to prepare a long-range entangled state by
measuring an SPT state can exist only if the corresponding marginalized type-I
strange correlator is nontrivial.

As we have examined for the 2d cluster state SPT, the type-I strange
correlator is generally nontrivial under weak decoherence of 1-form symmetry,
and its marginalized version can be shown to be nontrivial for small $p$.
Then, the theorem implies that there must exist a decoding protocol to capture
this quantity in a sample efficient manner, and also to facilitate the
measurement-assisted state preparation. Indeed, in Ref. [24], this was shown
explicitly where the decoding transformation ${\cal D}_{{\bm{s}}^{B}}$
corresponds to spin-flips on the vertex qubits based on the measurement
outcomes ${\bm{s}}^{B}$ on the edges (rotation in $Z$-axis followed by
measurement in $X$-basis is equivalent to having decoherence by a dephasing
channel). We remark that the theorem does not provide a decoding protocol, and
finding an optimal decoding transformation can be computationally challenging.
However, there exist several cases with sub-optimal yet computationally
efficient decoding solutions. Indeed, Ref. [24] also demonstrates a sample-
and computation-efficient protocol for the 2d SPT state where the 1-form
symmetry is decohered.

##  VI Conclusion and Outlook

In this work, we proposed several quantities that can be used to diagnose
nontrivial topological features of _mixed state_ density matrices, namely the
type-I and type-II strange correlators. Using these strange correlators, we
have shown that an SPT state under _selective_ decoherence can still persist.
We also introduced a “doubled Hilbert space” formalism based on the Choi-
Jamiołkowski isomorphism, in which the mixed state is mapped to a pure state,
and this doubled formalism gives us a unified picture of both type-I and type-
II strange correlators.  Under selective decoherence, the mapping provides a
natural explanation for the robustness of the type-II strange correlator as a
signature of the pure SPT state in the doubled Hilbert space under a shallow-
depth symmetric quantum circuit. Finally, we interpreted the type-I and type-
II strange correlators as indicators of the information-theoretic phases,
quantifying the presence of nontrivial topological information in the mixed-
state density matrix that may not be readily accessible in the experiments.

A natural yet speculative direction one may ask is the full implication of the
Choi-Jamiolski isomorphism in understanding mixed state properties. The
mapping allows one to understand nontrivial properties of the density matrix
as physical properties of the corresponding Choi state in the doubled Hilbert
space. Accordingly, one may classify density matrices based on the possible
quantum phases of Choi states constrained by Hermiticity, positive semi-
definiteness, and average symmetry. This may give an interesting way to
quantify the information content of the density matrices.

Many subjects remain unexplored along the direction of decohered quantum
states of matter. As we have mentioned, we will leave a more systematic study
of decohered fermionic TIs, and TSCs to the future. For each element of the
ten-fold way classification, there can be various types of decoherence
channels, which likely leads to very rich physics. As we have seen in the
paper, decoherence may be mapped to an interaction between the two boundaries
of the SPTs, hence decoherence may lead to various phenomena such as symmetric
mass generation [94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105], and
decoherence collapsed classification of TIs and TSCs [106, 107, 108, 109, 110,
111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123]. When the
classification of TIs and TSCs is collapsed due to interaction, it was shown
that the original topological transition between the topological insulator and
the trivial state may become an “unnecessary transition” [124, 125], namely
this transition is still a stable fixed point in the parameter space, but two
sides of the transition can still be connected adiabatically. We expect that
the unnecessary transition may also occur due to decoherence  in the doubled
Hilbert space.

Another big class of quantum states of matter is the gapless quantum states,
including quantum critical points. The effect of weak measurement on $(1+1)d$
conformal field theory (CFT) was studied in Ref. 90, and it was shown that the
effect of weak measurement can be mapped to the boundary operators of the CFT.
In the past few years, the boundary of $(2+1)d$ quantum critical points have
attracted a lot of attentions from both theoretical and numerical community
[126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 130, 139].
The problem becomes particularly interesting when the $(2+1)d$ bulk was a SPT
state driven to a critical point, hence two different boundary effects were at
play. Quantum critical points under decoherence will be another direction
worth exploration.

###### Acknowledgements.

We thank Ehud Altman, Soonwon Choi, Matthew P. A. Fisher, Sam Garrett, Joel
Moore, Xiao-Liang Qi, Shinsei Ryu, and Ryan Thorngren for inspiring
discussions. We would like to thank the KITP program “Quantum Many-Body
Dynamics and Noisy Intermediate-Scale Quantum Systems” where the collaboration
of this work was initiated. 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. Y.Z.Y. is supported by a startup fund at UCSD. C.
X. is supported by the NSF Grant No. DMR-1920434, and the Simons Investigator
program. _Note Added:_ While finishing up this work, we became aware of an
independent related work [140], which should appear on arXiv on the same day
as ours.

##  Appendix A Lemma

In this appendix, we demonstrate the following lemma that is used to
facilitate calculations in the main text:

Lemma Consider a $d$-dimensional cluster SPT state with a mixed anomaly
between two symmetries $G^{\vphantom{\dagger}}_{A}$ and
$G^{\vphantom{\dagger}}_{B}$, where $G_{A}\,{=}\,G^{(n)}_{1}$ and
$G_{B}\,{=}\,G^{(d-n-1)}_{2}$. Here, the superscripts represent that they are
$n$ and $(d\,{-}\,n\,{-}\,1)$ form symmetries. Also, $G_{A}$ and $G_{B}$ act
on different sublattices, say $L_{A}$ and $L_{B}$. Then, the expectation value
of an operator $O$ acting only on $L_{A}$ or $L_{B}$ does not vanish only if
$O$ contains a symmetry action on the corresponding sublattice.

_Proof_ : A $d$-dimensional cluster state with two symmetries $G_{1}$ and
$G_{2}$ is defined by the following Hamiltonian consisting of two types of
local terms:

$\displaystyle H$ $\displaystyle=-\sum_{a\in L_{1}}h_{1,a}-\sum_{b\in
L_{2}}h_{2,b},$ $\displaystyle h_{1,a}$ $\displaystyle={\cal
O}^{\textrm{d.w.}}_{1,a}\prod_{b\in\partial a}{\cal
O}^{\textrm{charge}}_{2,b}$ $\displaystyle h_{2,b}$ $\displaystyle={\cal
O}^{\textrm{d.w.}}_{2,b}\prod_{a\in\differential b}{\cal
O}^{\textrm{charge}}_{1,a}$ (110)

where ${\cal O}_{i}^{\textrm{charge}}$ (${\cal O}_{i}^{\textrm{d.w.}}$)
creates the charge (domain wall) of the symmetry $G_{i}$. As a cluster state
Hamiltonian, $[h_{i,a},h_{j,b}]=0$. In fact, the result is generic for any SPT
state with a stabilizer structure.

Now, consider an operator $O$ defined on the sublattice $L_{1}$. In general,
$O$ can be decomposed into terms that carry quantum numbers under the set of
stabilizers $\\{h_{1,a}\\}$ and $\\{h_{2,b}\\}$. However, note that any term
that carries a nontrivial quantum number would have a vanishing expectation
value since this stabilizer can be pulled out from the state and be absorbed
back freely for the groundstate. In other words, the only term $O_{A}$ that
can take a non-zero expectation value satisfies the following condition:

$\forall i,\,\,\,h_{1,i}O_{A}h_{1,i}=O_{A}\quad\textrm{ and }\quad\forall
j,\,\,\,h_{2,j}O_{A}h_{2,j}=O_{A}$ (111)

In fact, this already implies that $O_{A}$ is in fact a symmetry of the
system. Since $O_{A}$ is restricted to $L_{1}$, it means that $O\in G_{1}$. A
similar argument can be made for $O$ defined on the sublatice $L_{2}$. This
concludes the proof.

##  Appendix B Choi-Jamiołkowski isomorphism

For a given density matrix $\rho$ defined on the Hilbert space ${\cal H}$, the
Choi state is defined in the doubled Hilbert space ${\cal H}_{d}={\cal
H}_{u}\otimes{\cal H}_{l}$ as the following:

$\displaystyle\|\rho\rangle\rangle$
$\displaystyle\equiv\frac{1}{\sqrt{\textrm{Dim}[\rho]}}\sum_{i}|i\rangle_{u}\otimes\rho|i\rangle_{l}$
$\displaystyle=\frac{1}{\sqrt{\textrm{Dim}[\rho]}}\sum_{f}\lambda_{j}|\psi^{*}_{j}\rangle_{u}\otimes|\psi_{j}\rangle_{l}$
(112)

where $\rho=\sum_{j}\lambda_{j}|\psi_{j}\rangle\langle\psi_{j}|$. The Choi
state automatically respects the following symmetry:

$\displaystyle\textrm{SWAP}^{*}\equiv{\cal C}\circ\textrm{SWAP}$ (113)

where the SWAP symmetry exchanges ${\cal H}_{u}$ and ${\cal H}_{l}$, and
${\cal C}$ is the complex-conjugation symmetry. This corresponds to Hermitian
conjugation in the matrix language.

Now, for any state $|\Psi\rangle$ defined on ${\cal H}_{d}$ to be a Choi
state, it should have the following form of Schmidt decomposition between
upper and lower Hilbert spaces ${\cal H}_{u}$ and ${\cal H}_{l}$:

$\displaystyle|\Psi\rangle$
$\displaystyle=\sum_{i}\lambda_{i}|\psi^{*}_{i}\rangle_{u}\otimes|\psi_{i}\rangle_{l}$
(114)

where $\lambda_{i}\geq 0$. Note that, if we reshape a Choi state into the
matrix form, the SWAP∗ symmetry guarantees that it is Hermitian. Accordingly,
the above condition simply means that the eigenvalues of this matrix form
should be non-negative. This imposes a strong condition on valid Choi states.

Let $G$ be the symmetry of the original system. The Choi state would enjoy a
doubled symmetry group $G_{u}\times G_{l}$. We remark that the symmetry
representation of $g\in G$ in the upper Hilbert space ${\cal H}_{u}$ is
defined as a complex conjugated version of the original representation,
$U^{*}(g)$. Accordingly,

$\displaystyle|\Psi\rangle\mapsto(U^{*}(g_{u})\otimes
U(g_{l}))|\Psi\rangle\quad\forall g_{u}\cdot g_{l}\in G_{u}\times G_{l}$ (115)

In general, Choi states mapped from density matrices respect the symmetry
group $G_{d}\times\mathbb{Z}^{*}_{2}$, where $G_{d}$ is the diagonal subgroup
of $G_{u}\times G_{d}$ and $\mathbb{Z}^{*}_{2}$ is the SWAP∗ symmetry. On the
other hand, Choi states mapped from pure state density matrices respect the
larger symmetry group

$\displaystyle(G_{u}\times G_{l})\rtimes\mathbb{Z}^{*}_{2}$ (116)

where the SWAP symmetry exchanges $G_{u}\leftrightarrow G_{l}$.

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