# Mixed-state long-range order and criticality from measurement and feedback

Tsung-Cheng Lu<EMAIL_ADDRESS>Perimeter Institute for Theoretical
Physics, Waterloo, Ontario N2L 2Y5, Canada  Zhehao Zhang
<EMAIL_ADDRESS>Department of Physics, University of California, Santa
Barbara, CA 93106, USA  Sagar Vijay<EMAIL_ADDRESS>Department of
Physics, University of California, Santa Barbara, CA 93106, USA  Timothy H.
Hsieh<EMAIL_ADDRESS>Perimeter Institute for Theoretical
Physics, Waterloo, Ontario N2L 2Y5, Canada

###### Abstract

We propose a general framework for using local measurements, local unitaries,
and non-local classical communication to construct quantum channels which can
efficiently prepare mixed states with long-range quantum order or quantum
criticality. As an illustration, symmetry-protected topological (SPT) phases
can be universally converted into mixed-states with long-range entanglement,
which can undergo phase transitions with quantum critical correlations of
local operators and a logarithmic scaling of the entanglement negativity,
despite coexisting with volume-law entropy. Within the same framework, we
present two applications using fermion occupation number measurement to
convert (i) spinful free fermions in one dimension into a quantum-critical
mixed state with enhanced algebraic correlations between spins and (ii) Chern
insulators into a mixed state with critical quantum correlations in the bulk.
The latter is an example where mixed-state quantum criticality can emerge from
a gapped state of matter in constant depth using local quantum operations and
non-local classical communication.

###### Contents

  1. I Introduction
  2. II Non-local channel from local quantum operations and non-local classical communication
  3. III Entanglement structure of output mixed states
  4. IV Mixed-state long-range order by measuring SPT phases
    1. IV.1 Measuring 1d SPT
    2. IV.2 Duality transformation
    3. IV.3 Mixed-state quantum criticality
    4. IV.4 Generalization to other mixed-state long-range order
  5. V Mixed-state quantum criticality by measuring fermions
    1. V.1 Measuring 1d fermion
    2. V.2 Measuring Chern insulators
  6. VI Summary and discussion
  7. A Constraints on mixed-state entanglement
  8. B Measuring 1d SPT
    1. B.1 Derivation of operator duality
    2. B.2 Numerical data on volume-law entropy
    3. B.3 Other non-fixed-point SPT as input
    4. B.4 SPT under decoherence as input
  9. C Mixed state with topological order from measuring 2d SPT
  10. D Fermionic mixed state with Kitaev topological order in 1d
  11. E Fermionic mixed state with topological order in 2d
  12. F Purification of the mixed state from measuring 1d free-fermions

##  I Introduction

Interacting quantum matter may exhibit long-range entanglement that is
intimately connected to fascinating phenomena such as fractionalized
quasiparticles and criticality. Typically the scope for exploring long-range
entanglement has been limited to pure states, especially the ground states of
many-body Hamiltonians. However, realistic physical systems require a mixed-
state description due to constant exposure to an environment, thus motivating
the consideration of long-range entanglement in many-body mixed states. In
equilibrium, mixed states naturally arise as finite temperature Gibbs states;
however, long-range entanglement is typically fragile to thermal fluctuations
[1, 2, 3, 4, 5]. In contrast, out-of-equilibrium mixed states offer richer
possibilities for stabilizing long-range entanglement. For example, there has
been substantial recent progress in characterizing universal properties of
various non-trivial states of matter, e.g. symmetry-protected topological
(SPT) phases, quantum critical states, and topological order, subject to noise
channels [6, 7, 8, 9, 10, 11, 12, 13, 14]. Signatures of the non-trivial
nature of these mixed-states generally require probing non-linear observables
in the mixed-state density matrix, e.g. measures of quantum entanglement.

In this work, we introduce a novel route for realizing long-range entangled
mixed states. We provide a general framework for constructing quantum channels
which allow for the efficient realization of a large class of mixed-state
long-range entanglement, including GHZ, topological order, and quantum
criticality.

Our construction (Sec.II, see Fig.1 for schematic) relies on three
ingredients, namely, local projective measurement, local unitary operations,
and non-local classical communication. Starting with an input state, which we
extensively bipartition in two disjoint Hilbert spaces, $A$ and $B$, we
perform single-site measurements within $A$. Based on the measurement
outcomes, recorded as classical data, we perform appropriate unitary feedback
consisting of local unitary gates acting on $B$. We will allow the local
unitary gates to depend on the global classical data, thus requiring non-local
classical communication. Such a two-step protocol generically realizes a mixed
state, i.e. an ensemble of pure states associated with distinct measurement
outcomes. The resulting mixed state on $B$ ($\rho_{B}$) may exhibit various
long-range quantum orders and quantum criticality coexisting with extensive
classical entropy. We show that $\rho_{B}$ admits a purification that can be
obtained by performing a controlled unitary on the composite $AB$ system. As
such, our protocol may be regarded as implementing a controlled unitary that
generically cannot be realized using finite-depth unitary circuits. This
unitary description also provides a powerful handle allowing us to
characterize $\rho_{B}$ by analyzing the parent Hamiltonian of its
purification.

Figure 1: We consider a system with $A,B$ degrees of freedom, which may
correspond to sublattices or charge and spin of fermions, for example. We
devise a depth-2 quantum channel that can effectively implement a controlled
unitary, giving rise to various long-range quantum orders and criticality on
the $B$ system. The first layer consists of single-site measurements on $A$,
and the second layer consists of single-site unitaries on $B$ conditioned on
the outcomes $\\{\alpha\\}$ in the first layer. This protocol outputs a
density matrix $\rho_{B}$ on $B$ such that
$\rho_{B}(=\tr_{A}\ket{\psi}\bra{\psi})$ admits a purification
$\ket{\psi}=U\ket{\psi_{0}}$ with $U$ being a controlled unitary
$U=\sum_{\alpha}U_{\alpha}P_{\alpha}$ on an $AB$ composite system;
$P_{\alpha}$ projects $A$ to a specific product state on $A$, and $U_{\alpha}$
is a product of on-site unitaries acting on $B$.

Our framework is inspired by adaptive circuits, where the choice of the
applied local unitary gates depends on the global measurement outcomes in a
way that post-measurement pure state trajectories associated with different
measurement outcomes will be deterministically converted to the same target
pure state. This architecture has provided a powerful framework that enables
the efficient preparation of a large class of non-trivial pure states,
including gapped topological orders and gapless conformal critical states in
short times [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], which are impossible
using any local unitary protocols. The operations required by adaptive
circuits (mid-circuit measurements and feedback) are available in several
quantum hardwares, and indeed the adaptive, finite-depth preparation of
certain topological orders has been realized experimentally in recent works
[26, 27]. While our construction of quantum channels utilizes the same
ingredients (local quantum operations and non-local classical communication)
as in adaptive circuits, various post-measurement states generically do not
converge in our protocol, thereby leading to a mixed-state ensemble that
exhibits certain long-range quantum order and quantum criticality despite
having extensive entropy.

For the applications discussed in this work, the unitary feedback after local
measurements is a product of onsite unitary operations, and hence, the long-
range order in the resulting ensemble may equivalently be decoded from post-
measurement pure state trajectories via appropriate classical post-processing
without unitary feedback (see e.g. Ref.[28, 29]). However, our protocol can be
generalized, e.g. by extending onsite-unitary feedback to a finite-depth local
unitary circuit, or by considering multiple rounds of layers of measurement
and unitary. In both cases, it is unclear if the properties of the resulting
mixed state can be efficiently obtained by classical post-processing based on
local measurement data. Furthermore, not all classical post-processing can be
implemented efficiently as quantum channels.

In the rest of the introduction, we provide an overview of two classes of
applications. The first class of examples takes a symmetry-protected
topological (SPT) order as an input state (Sec.IV). These are short-range
entangled phases that can be prepared from product states using local unitary
in finite time only when breaking the protecting symmetry [30, 31]. We will
focus on SPT phases characterized by decorated domain wall constructions [32].
One class of examples are SPT phases in $d$-spatial dimensions which are
protected by $\mathbb{Z}_{2}$ $p$-form $\times\mathbb{Z}_{2}$ $q$-form
symmetry with $p+q=d-1$. This type of SPT can be diagnosed by the long-range
order in certain non-local operators, e.g. string operators in 1d or membrane
operators in 2d. Based on these non-local operators, we show how to employ
measurement and feedback to prepare a mixed state with $\mathbb{Z}_{2}$ long-
range order coexisting with volume-law entropy. For instance, $\mathbb{Z}_{2}$
cat-state order and $\mathbb{Z}_{2}$ topological order can be prepared in 1
and 2 space dimensions in a mixed state, both of which cannot occur in
equilibrium thermal states.

Notably, the convertibility to a long-range order is a universal property of
the SPT: any pure state in the same SPT phase always leads to a long-range-
ordered (and generically mixed) state characterized by the same universal
properties that are related to the SPT order. In addition, the output state is
a reduced density matrix of a ground state of Hamiltonian $H$ that is “dual”
to the parent Hamiltonian $H_{0}$ of the input SPT. Most interestingly, when
$H_{0}$ is tuned to criticality, $H$ is critical as well, and the
corresponding mixed state possesses a critical entanglement structure
quantified by the entanglement negativity [33, 34, 35, 36, 37]. For example,
our protocol in 1d gives rise to a mixed state with volume-law scaling of von
Neumann entropy, but logarithmic scaling of the entanglement negativity with
subsystem size. Such mixed-state quantum criticality presents several
unconventional features as we will discuss in this work.

Our application of converting a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT to
long-range order is directly inspired by Ref.[15, 16, 20]. In particular,
Ref.[20] argues that measuring these SPTs will generically lead to certain
long-range order in post-measurement pure state trajectories. While this can
be shown analytically for fixed-point SPTs [15, 16], the fate of non-fixed-
point SPTs upon measurement was inconclusive111For non-fixed point SPTs,
Ref.[20] provides numerical evidence that measuring the Heisenberg spin-1
chain, an SPT protected by $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry of
$\pi$ rotations, leads a GHZ-type long-range order.. In contrast, our channel-
based approach allows us to show that the emergence of long-range orders in a
mixed state is a universal property that persists throughout the entire SPT
phase. In addition, these orders can be efficiently probed through certain
linear observables in the mixed-state ensemble that results from our quantum
channel.

The second class of examples focuses on spinful fermionic systems (Sec.V). We
consider extensive single-site fermion occupation number measurement followed
by unitary feedback according to the measurement outcomes. In contrast to the
single-site projective Pauli measurement, which trivializes the measured
qubit, the occupation number for spinful fermions allows for richer
phenomenology due to the possibility of a residual spin-1/2 degree of freedom
in the subspace of a singly-occupied site. Indeed, it is known that starting
with non-interacting spinful fermions, performing a Gutzwiller projection
(projecting onto single-occupation subspace) effectively induces interactions
that may lead to various exotic quantum spin liquids in two and higher spatial
dimensions (see Ref. [38, 39] for review). The Gutzwiller projection cannot be
implemented as a fermion occupation-number measurement, without post-selecting
the measurement outcome. In contrast, our scheme does not rely on post-
selection; for each post-measurement trajectory, we apply a depth-1 local
unitary feedback, so that the mixed state, formed from the ensemble of these
trajectories, exhibits certain non-trivial features. We also note that the
fermion-occupation number measurement is readily experimentally available via
quantum gas microscopes (see Ref.[40] for a review), which allows for the
implementation of our protocols. Below we briefly summarize two applications,
in one and two space dimensions respectively.

In 1d, we start from free spinful fermions (Sec.V.1) with nearest neighbor
hopping. In this case, it is known that spin-spin correlations decay
algebraically with the separation between two sites with an exponent $2$. We
devise a protocol that outputs a mixed state with an enhanced critical
correlation characterized by a smaller exponent, namely, $1$. The essential
idea is to notice that there is a long-range string order in the 1d free
fermions [41], and measurement and feedback can convert this hidden string
order into a truly long-range critical order. The resulting mixed state that
describes correlations in the spin sector can be regarded as a reduced density
matrix by tracing out the charge sector in a ground state of a parent
Hamiltonian that we derive. Notably, this Hamiltonian describes strongly
interacting spinful fermions and therefore, our measurement-feedback channel
may be viewed as a novel mean to effectively engineer interactions. We note
that our protocol of boosting critical correlations is also applicable to
interacting fermionic systems characterized by Luttinger liquids by exploiting
their hidden string order.

The 2d example (Sec.V.2) takes Chern insulators (see e.g.[42] for an
introduction) as an input. These are gapped invertible topological phases,
which feature trivial bulks and non-trivial chiral edge modes on the boundary.
Despite Chern insulators having exponentially decaying correlations in the
bulk, applying a measurement-feedback channel leads to a mixed state with
algebraic correlations. The essential property we use is a non-local membrane
order parameter due to the topological response in Chern insulating states
[43, 44]. Such a non-local hidden order can be converted into algebraic
decaying spin-spin correlations in the resulting mixed state using a non-local
quantum channel. This therefore furnishes a remarkable example where quantum
criticality emerges from a gapped state of matter via a quantum channel
involving local quantum operations but non-local classical communication.

##  II Non-local channel from local quantum operations and non-local classical
communication

Here we introduce the protocol for constructing quantum channels based on
local quantum operations (measurement and unitary) and non-local classical
communication (Fig.1). Given a system with two types of degrees of freedom $A$
and $B$ initialized in the state $\rho_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$, one
performs simultaneous, extensive single-site measurement on every degree of
freedom in $A$. This leads to a particular pure state
$\ket{\psi_{\alpha}}=\frac{P_{\alpha}\ket{\psi_{0}}}{\sqrt{P_{\alpha}\ket{\psi_{0}}}}$
with probability $\bra{\psi_{0}}P_{\alpha}\ket{\psi_{0}}$, where $\alpha$
labels the measurement outcome on $A$, and $P_{\alpha}$ is the projector
associated with the measurement. For each post-measurement state, we apply a
unitary $U_{\alpha}$ acting on $B$ based on the outcome $\alpha$. Note that
this requires non-local classical communication since the choice of a local
unitary relies on distant measurement outcomes recorded as classical data. The
above measurement-feedback protocol leads to a mixed state

$\rho=\sum_{\alpha}U_{\alpha}P_{\alpha}\rho_{0}P_{\alpha}U_{\alpha}^{\dagger}.$
(1)

With an appropriate choice of measurement and unitary feedback, the subsystem
$B$ described by a reduced density matrix $\rho_{B}$ may exhibit various long-
range quantum orders and criticality.

The aforementioned protocol may be viewed as a way to effectively implement a
controlled unitary acting on the $AB$ composite system followed by tracing out
$A$. To see this, we notice that $\rho_{B}$ admits a purification $\ket{\psi}$
via $\rho_{B}=\tr_{A}\ket{\psi}\bra{\psi}$, where $\ket{\psi}$ is defined as

$\ket{\psi}=\sum_{\alpha}U_{\alpha}P_{\alpha}\ket{\psi_{0}}.$ (2)

$U=\sum_{\alpha}U_{\alpha}P_{\alpha}$ takes the form of a controlled unitary
with $A$ being the control and $B$ being the target. $U$ may not be realized
as local unitary circuits, thereby enabling significant changes in the
entanglement structure. In particular, $U$ provides a non-local transformation
on operators according to Heisenberg evolution. For instance, the expectation
of an operator $O_{B}$ supported on $B$ in the resulting mixed state
$\rho_{B}$ amounts to the expectation of the operator $U^{\dagger}O_{B}U$ in
the input pure state $\ket{\psi_{0}}$. As such, this non-local unitary
transformation provides a powerful way to convert hidden orders in the input
state into long-range order or criticality in the density matrix $\rho_{B}$,
as we will illustrate using various examples. Moreover, this viewpoint of
unitary transformation provides a useful way to describe the output
$\rho_{B}$; since the initial state $\ket{\psi_{0}}$ and the purified state
$\ket{\psi}$ are connected by a unitary $U$, the structure of $\rho_{B}$ can
be characterized based on the Hamiltonian $H$ of $\ket{\psi}$ through
$H=UH_{0}U^{\dagger}$, where $H_{0}$ is the Hamiltonian of the input state
$\ket{\psi_{0}}$.

##  III Entanglement structure of output mixed states

We first discuss general constraints on the entanglement properties of the
mixed states generated from such protocols.

Bound on entanglement: The protocols presented in this work belong to LOCC,
namely, local operations (onsite measurement and unitaries) and classical
communication, and therefore, mixed-state entanglement cannot increase under
these quantum channels (see Appendix.A for proof based on the entanglement of
formation [45], a faithful entanglement measure for mixed states). This
entanglement constraint provides a sharp distinction between the mixed-state
orders that may and may not be realized within our protocols. For example,
starting with an area-law entangled gapped state in 1d, our finite-depth
channels cannot output a mixed-state with $\log L$ scaling entanglement.
Namely, to output a quantum critical mixed state in 1d with $\log L$ scaling
entanglement, the initial state must be critical with entanglement $\gtrsim
O(\log L)$ as well, and Sec.IV.3 presents one such example. On the other hand,
if one starts with a gapped, area-law state in 2d, the entanglement constraint
does not rule out the possibility of realizing quantum critical mixed states
(recall gapless conformal critical states obey an area law as well), and
indeed this is realized in Sec.V.2.

Sufficient conditions for a nontrivial mixed state: Despite the above
constraint on the quantity of entanglement, our protocol can produce mixed
states $\rho$ that are long-range entangled in the precise sense that they
cannot be a mixture of trivial pure states [2]. Namely, $\rho$ is long-range
entangled if $\rho\neq\sum_{n}p_{n}\ket{\phi_{n}}\bra{\phi_{n}}$, where each
$\ket{\phi_{n}}$ is a short-range entangled state that can be connected to a
product state using a finite-time local unitary.

We prove that $\rho$ cannot be a mixture of short-range entangled states given
the following two conditions (which will apply to all examples considered in
this work). (i) Global symmetry: There exists a unitary operator $S$ with
unit-norm expectation value, i.e. $\tr(\rho S)=e^{i\theta}$ with
$\theta\in\mathbb{R}$. (ii) Long-range correlations for charged operators:
There exist charged operators $O_{1},O_{2}$ (with respect to the symmetry
$S$)222Namely, $SO_{j}S^{\dagger}=e^{iQ_{j}}O_{j}$ with $Q_{j}$ mod $2\pi\neq
0$ for $j=1,2$., whose correlation $\tr(\rho O_{1}O_{2})$ decay slower than
$e^{-d/\xi}$, with $\xi\geq 0$, and $d$ being the spatial separation between
$O_{1}$ and $O_{2}$. For example, $\tr(\rho O_{1}O_{2})$ could be constant or
follow a power-law decay with the separation $d$.

The claim above can be proved by contradiction. Assume
$\rho=\sum_{n}p_{n}\ket{\phi_{n}}\bra{\phi_{n}}$, i.e. a mixture of short-
range entangled pure states $\ket{\phi_{n}}$. Since $\tr(\rho S)=e^{i\theta}$,
it must be that $\langle\phi_{n}|S|\phi_{n}\rangle=e^{i\theta}$ for each
$\ket{\phi_{n}}$333This follows from the fact the eigenspectrum of the unitary
$S$ is located on the unit circle in the complex plane.. Further,
$\ket{\phi_{n}}$ being short-range entangled means the connected correlation
function decays exponentially:
$\langle\phi_{n}|O_{1}O_{2}|\phi_{n}\rangle-\langle\phi_{n}|O_{1}|\phi_{n}\rangle\langle\phi_{n}|O_{2}|\phi_{n}\rangle\sim
e^{-d/\xi}$ [46]. $\langle\phi_{n}|S|\phi_{n}\rangle=e^{i\theta}$ and the
assumption that each operator is charged with respect to $S$ imply that
$\langle\phi_{n}|O_{1}|\phi_{n}\rangle=\langle\phi_{n}|O_{2}|\phi_{n}\rangle=0$,
and thus $\langle\phi_{n}|O_{1}O_{2}|\phi_{n}\rangle\sim e^{-d/\xi}$. Since
$O_{1}O_{2}$ decays exponentially in each $\ket{\phi_{n}}$, $O_{1}O_{2}$ must
decay exponentially in the mixed-state ensemble $\rho$, which contradicts the
result that the decay of $\tr(\rho O_{1}O_{2})$ is slower than exponential. As
a result, the initial assumption must be false.

As a demonstration, when the expectation value of a global $\mathbb{Z}_{2}$
symmetry generator follows
$\expectationvalue{S}=\expectationvalue{\prod_{i}X_{i}}=1$ in the mixed state,
$Z_{i}Z_{j}$ having non-zero expectation value as the separation $d\to\infty$
implies a non-trivial mixed state with the GHZ-type order. Such an example
will be presented in Sec.IV.1. If $Z_{i}Z_{j}$ instead decays algebraically,
the corresponding non-trivial mixed state exhibits a quantum criticality with
several unconventional properties, which will be discussed in IV.3. One may
also generalize $S$ and charged operators $O_{j}$ to string-like operators,
which can witness mixed-state topological order in two space dimensions (see
Appendix.C for details).

##  IV Mixed-state long-range order by measuring SPT phases

In this section, we apply the general framework discussed in Sec.II to convert
SPT phases characterized by decorated domain-wall construction [32] to mixed-
state long-range orders. We will illustrate the main idea using 1d
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT. More general types of SPTs with
distinct symmetries or higher-space dimensions and the corresponding emergent
long-range order will be briefly outlined in Sec.IV.4 (with more details in
Appendix).

We note that Ref.[28, 29] also discussed a protocol that can convert certain
SPTs to a mixed-state ensemble with hidden long-range orders that can be
decoded via non-linear observables or classical post-processing. However, the
specific protocol is different from ours, and thus the type of long-range
order that can be stabilized also differs. For example, our setup can realize
GHZ-type order in 1d and $\mathbb{Z}_{2}$ topological order in 2d, both of
which are absent in those works. More technically, the protocol discussed in
Ref.[28, 29] amounts to introducing thermal fluctuations, which is a
qualitatively different perturbation than the ones we consider.

###  IV.1 Measuring 1d SPT

Figure 2: Starting with an input $\ket{\psi_{0}}$, which undergoes a
transition from a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT order in the $AB$
composite system to two independent GHZ-type symmetry breaking orders in $A,B$
respectively, a measurement-feedback protocol leads to a density matrix on $B$
(i.e. $\rho_{B}$), which undergoes a transition from a GHZ-type order to a
trivial mixed state. $\rho_{B}$ admits a purification $\ket{\psi}$ that can be
obtained from the input state $\ket{\psi_{0}}$ via a controlled unitary
$U=\sum_{\alpha}U_{\alpha}P_{\alpha}$, and the Hamiltonian of $\ket{\psi}$ can
be derived based on the transformation rules (Eq.6). The purification
therefore provides a useful handle to characterize the structure of the output
density matrix $\rho_{B}$.

We consider a 1d lattice with periodic boundary conditions, and the lattice
sites are labeled with the ordering $(a,1),(b,1),(a,2),(b,2),...,(a,L),(b,L)$.
The cluster state Hamiltonian is defined as
$-\sum_{i=1}^{L}(Z_{a,i}X_{b,i}Z_{a,i+1}+Z_{b,i}X_{a,i+1}Z_{b,i+1})$. The
ground state exhibits an SPT order protected by the global
$\mathbb{Z}_{2}\crossproduct\mathbb{Z}_{2}$ symmetry generated by
$\prod_{i=1}^{L}X_{a,i}$ on $A$ sublattice and $\prod_{i=1}^{L}X_{b,i}$ on $B$
sublattice. Under symmetric local perturbations without gap closing, the SPT
order is robust in the ground state $\ket{\psi_{0}}$, and can be diagnosed by
the long-range string order (see e.g. Ref.[47])

$\bra{\psi_{0}}Z_{b,i}\left(\prod_{k=i+1}^{j}X_{a,k}\right)Z_{b,j}\ket{\psi_{0}}=c$
(3)

with $0<c\leq 1$ as $\absolutevalue{i-j}\to\infty$ 444$c=1$ in the ground
state of cluster state Hamiltonian..

Now we show that based on the long-range string order, one can prepare a mixed
state with GHZ-like $\mathbb{Z}_{2}$ symmetry-breaking order using measurement
and unitary feedback in constant depth.

To start, we measure Pauli-X for every site in $A$ sublattice and denote the
measurement outcome of $X_{a,i}$ by $\alpha_{i}$. Defining
$\alpha=\\{\alpha_{i}\\}$ as the collection of outcomes, one obtains a post-
measurement state
$\frac{P_{\alpha}\ket{\psi_{0}}}{\sqrt{\bra{\psi_{0}}P_{\alpha}\ket{\psi_{0}}}}$
with probability $p_{\alpha}=\bra{\psi_{0}}P_{\alpha}\ket{\psi_{0}}$ and
projector $P_{\alpha}\equiv\prod_{i}\frac{1+\alpha_{i}X_{a,i}}{2}$. Without
recording the outcome, the system is described by a mixed state, i.e. an
ensemble of pure states corresponding to distinct measurement outcomes:
$\sum_{\alpha}P_{\alpha}\rho_{0}P_{\alpha}$ with
$\rho_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$. This measurement-induced ensemble
lacks any long-range order. For instance, the two-point functions on $B$
sublattice with respect to the mixed state is
$\bra{\psi_{0}}Z_{b,i}Z_{bj}\ket{\psi_{0}}$, which is nothing but the two-
point functions with respect to the initial SPT, therefore decaying
exponentially with the separation.

In contrast, applying unitary feedback based on the measurement outcomes leads
to a non-trivial mixed state. The essential idea is to choose a unitary such
that the two-point $Z_{b}Z_{b}$ operator evaluated in the mixed state amounts
to the string operator $Z_{b}X_{a}...X_{a}Z_{b}$ in the input SPT, thereby
taking a non-zero expectation value.

First, for a post-measurement pure state with outcome $\alpha$, we apply a
unitary $U_{\alpha}$ on $B$ sublattice:

$U_{\alpha}=\prod_{i}X_{b,i}^{\frac{1-\prod_{j=1,2,\cdots}^{i}\alpha_{j}}{2}}.$
(4)

In other words, $X_{b,i}$, a Pauli-X on $B$ sublattice, is applied when there
is an odd number of outcome $-1$ from the site $(a,1)$ to the site $(a,i)$. It
follows that $Z_{b,i}$ conjugated by $U_{\alpha}$ will acquire a $1~{}(-1)$
sign if there are even (odd) number of $-1$ measurement outcomes from the site
$(a,1)$ to the site $(a,i)$. Correspondingly, one finds
$U^{\dagger}_{\alpha}Z_{b,i}Z_{b,j}U_{\alpha}=Z_{b,i}\left(\prod_{k=i+1}^{j}\alpha_{k}\right)Z_{b,j}$.
The overall measurement and unitary operation lead to the mixed state
$\rho=\sum_{\alpha}U_{\alpha}P_{\alpha}\rho_{0}P_{\alpha}U_{\alpha}^{\dagger}$.
The long-range order can be diagnosed by the two-point $ZZ$ correlation on $B$
sublattice:

$\begin{split}\tr[\rho
Z_{b,i}Z_{b,j}]&=\sum_{\alpha}\bra{\psi_{0}}P_{\alpha}U_{\alpha}^{\dagger}Z_{b,i}Z_{b,j}U_{\alpha}P_{\alpha}\ket{\psi_{0}}\\\
&=\sum_{\alpha}\bra{\psi_{0}}P_{\alpha}Z_{b,i}\left(\prod_{k=i+1}^{j}\alpha_{k}\right)Z_{b,j}P_{\alpha}\ket{\psi_{0}}\\\
&=\bra{\psi_{0}}Z_{b,i}\left(\prod_{k=i+1}^{j}X_{a,k}\right)Z_{b,j}\ket{\psi_{0}}\end{split}$
(5)

where we have used the fact that the measurement outcome $\alpha_{k}$ can be
replaced with $X_{a,k}$ due to the projector $P_{\alpha}$, and
$\sum_{\alpha}P_{\alpha}=1$. Therefore, the two-point function in $\rho$ is
exactly the string order in the initial SPT order state, and hence
$\expectationvalue{Z_{b,i}Z_{b,j}}=c=O(1)>0$. As discussed in Sec.III, this
non-decaying two-point function together with the $\mathbb{Z}_{2}$ symmetry on
$B$ sublattice, i.e. $\expectationvalue{\prod_{i}X_{b,i}}=1$, indicates that
$\rho_{B}(=\tr_{A}\rho)$ is a non-trivial mixed state with a GHZ-type quantum
long-range order.

###  IV.2 Duality transformation

As discussed in our general framework (Sec.II), our protocol can be viewed as
realizing a controlled unitary $U=\sum_{\alpha}P_{\alpha}U_{\alpha}$ acting on
the $AB$ composite system followed by tracing out $A$. Specifically, starting
from the input $\ket{\psi_{0}}$ with the Hamiltonian $H_{0}$, there exists a
purification $\ket{\psi}$ of the output state
$\rho_{B}(=\tr_{A}\ket{\psi}\bra{\psi})$ with $\ket{\psi}=U\ket{\psi_{0}}$.
This allows us to derive the parent Hamiltonian $H$ of $\ket{\psi}$ through
$H=UH_{0}U^{\dagger}$, so the structure of the output mixed state $\rho_{B}$
can be characterized. Fig.2 provides a summary for this subsection.

With $U=\sum_{\alpha}P_{\alpha}U_{\alpha}$ ($U_{\alpha}$ defined in Eq.4), one
derives the transformation rule for operators under the conjugation of $U$
(see Appendix.B.1 for details).

$\begin{split}&X_{a,i}\to X_{a,i},\quad X_{b,i}\to X_{b,i},\\\
&Z_{a,i}Z_{a,i+1}\to Z_{a,i}X_{b,i}Z_{a,i+1},\\\ &Z_{b,i}Z_{b,i+1}\to
Z_{b,i}X_{a,i+1}Z_{b,i+1}.\end{split}$ (6)

Pauli-X is invariant since $U$ is diagonal in X basis. On the other hand,
neighboring $ZZ$ on one sublattice (note that Pauli-Zs must appear in pairs
due to the $\mathbb{Z}_{2}$ symmetry) is attached with a Pauli-X on another
sublattice in between two Pauli-Zs. This is quite intuitive since the unitary
feedback is designed to transform the product of two Pauli-Zs on one
sublattice with a sign that depends on the product of measurement outcomes (on
another lattice) between these two Pauli-Zs.

The above duality mapping provides a powerful tool to characterize the
structure of the output mixed state; one can perturb the fixed-point SPT
Hamiltonian by adding any symmetry-preserving local terms and derive the
parent Hamiltonian of the purification of the output mixed state $\rho_{B}$.
As an application, we consider the ground state of the following Hamiltonian
as an input:

$\begin{split}H_{0}=&-\sum_{i}Z_{a,i}X_{b,i}Z_{a,i+1}-\sum_{i}Z_{b,i}X_{a,i+1}Z_{b,i+1}\\\
&-g\sum_{i}Z_{a,i}Z_{a,i+1}-g\sum_{i}Z_{b,i}Z_{b,i+1}.\end{split}$ (7)

The phase diagram of $H_{0}$ can be completely determined. To see this, by
conjugating a product of controlled-Z gate: $U_{\text{CZ}}=\prod
CZ_{(a,i),(b,i)}CZ_{(b,i),(a,i+1)}$, one obtains

$\begin{split}&-\sum_{i}X_{b,i}-\sum_{i}X_{a,i}\\\
&-g\sum_{i}Z_{a,i}Z_{a,i+1}-g\sum_{i}Z_{b,i}Z_{b,i+1},\end{split}$ (8)

i.e. two decoupled Ising chains on two sublattices, where
$\absolutevalue{g}<1$ and $\absolutevalue{g}>1$ correspond to trivial phase
and spontaneous symmetry breaking (SSB) phase with GHZ $\mathbb{Z}_{2}$ orders
on $A$ sublattice and $B$ sublattice. This implies the ground state of $H_{0}$
belongs to SPT and SSB phase with GHZ order for $\absolutevalue{g}<1$ and
$\absolutevalue{g}>1$, respectively.

Using the transformation rule in Eq.6, one finds the measurement-feedback
channel leads to a mixed state $\rho_{B}$ on $B$ sublattice, which is a
reduced density matrix of the ground state $\ket{\psi}$ of the following
Hamiltonian

$\begin{split}H=&-\sum_{i}Z_{a,i}Z_{a,i+1}-\sum_{i}Z_{b,i}Z_{b,i+1}\\\
&-g\sum_{i}Z_{a,i}X_{b,i}Z_{a,i+1}-g\sum_{i}Z_{b,i}X_{a,i+1}Z_{b,i+1}.\end{split}$
(9)

Comparing Eq.7 and Eq.9, one finds they are dual to each other, and the phase
of $H$ can be determined analogously: $A$ and $B$ sublattices individually
exhibit a GHZ order for $\absolutevalue{g}<1$, and the $A\bigcup B$ together
exhibits SPT order for $\absolutevalue{g}>1$. Consequently, $\rho_{B}$
possesses a mixed-state GHZ order for $\absolutevalue{g}<1$, and $\rho_{B}$
becomes a trivial mixed state for $\absolutevalue{g}>1$ (since the subsystem
of the SPT gives a trivial mixed state). In addition, since $\ket{\psi}$ can
be obtained by applying extensive controlled-Z gates across $A,B$ sublattices
(i.e. $U_{\text{CZ}}=\prod CZ_{(a,i),(b,i)}CZ_{(b,i),(a,i+1)}$) on two
decoupled Ising chains on A/B sublattice, one expects
$\rho_{B}=\tr_{A}\ket{\psi}\bra{\psi}$ has volume-law entropy for any non-zero
$g$ ($U_{\text{CZ}}$ acts trivially at $g=0$). This is indeed the case based
on our Exact Diagonalization calculation (see Appendix.B.2).

Here we emphasize again that the emergent long-range cat-state order is a
universal property of the input SPT phase. While here we only consider one
type of perturbations, in Appendix.B.3 we discuss other types of perturbation,
including independently tunable perturbation strength in
$Z_{a,i}Z_{a,i+1},Z_{b,i}Z_{b,i+1}$ as well as onsite Pauli-X perturbation.
Interestingly, when perturbing the fixed-point SPT using onsite Pauli-Xs, the
corresponding output $\rho_{B}$ is exactly the (pure) ground state of the
transverse-field Ising chain in the symmetry-broken phase in the subspace with
$\prod_{i}X_{b,i}=1$.

###  IV.3 Mixed-state quantum criticality

Our protocol can also output a mixed state with volume-law entropy coexisting
with critical (algebraic) long-range order. This occurs when applying our
measurement-feedback channel to a critical state, i.e. the ground state of
$H_{0}$ in Eq.7 at $g=1$. In this case, the output will be a mixed state
$\rho_{B}$ on $B$ sublattice by tracing out $A$ sublattice for the ground
state $\ket{\psi}$ of $H$ in Eq.9 at $g=1$ 555In fact, at the critical point
$g=1$, $H=UH_{0}U^{\dagger}=H_{0}$, so $\ket{\psi_{0}}$ is the same as
$\ket{\psi}$ defined in Eq.10., where $\ket{\psi}$ reads

$\ket{\psi}=U_{\text{CZ}}\ket{\text{CFT}}_{A}\otimes\ket{\text{CFT}}_{B}.$
(10)

$\ket{\text{CFT}}_{A/B}$ denotes the ground state of a transverse-field Ising
chain (on $A/B$ sublattice) at a critical point, characterized by the 1+1D
Ising CFT, and $U_{\text{CZ}}=\prod CZ_{(a,i),(b,i)}CZ_{(b,i),(a,i+1)}$. The
corresponding mixed state $\rho_{B}=\tr_{A}\ket{\psi}\bra{\psi}$ exhibits
quantum criticality diagnosed by certain operators. For example, since
$U_{\text{CZ}}$ commutes with Pauli-Zs, the two-point $ZZ$ function is given
by the single Ising critical chain, which exhibits an algebraic decay:
$\expectationvalue{Z_{b,i}Z_{b,j}}=\bra{\text{CFT}}_{B}Z_{b,i}Z_{b,j}\ket{\text{CFT}_{B}}\sim\frac{1}{\absolutevalue{i-j}^{\eta}}$
with $\eta=1/4$ being a critical exponent in 1+1D Ising CFT. On the other
hand, the disorder operator $X_{b,i}X_{b,i+1}...X_{b,j}$ also exhibits an
algebraic decay:

$\begin{split}&\left\langle X_{b,i}...X_{b,j}\right\rangle\\\
&=\bra{\text{CFT}}_{A}\bra{\text{CFT}}_{B}U_{CZ}^{\dagger}X_{b,i}...X_{b,j}U_{CZ}\ket{\text{CFT}}_{A}\ket{\text{CFT}}_{B}\\\
&=\bra{\text{CFT}}_{A}\bra{\text{CFT}}_{B}Z_{a,i}X_{b,i}...X_{b,j}Z_{a,j+1}\ket{\text{CFT}}_{A}\ket{\text{CFT}}_{B}\\\
&=\left\langle Z_{a,i}Z_{a,j+1}\right\rangle_{CFT,A}\left\langle
X_{b,i}...X_{b,j}\right\rangle_{CFT,B}\\\ &\sim\frac{1}{\lvert
i-j\rvert^{2\eta}},\end{split}$ (11)

where we used the Kramers-Wannier duality to convert
$X_{b,i}X_{b,i+1}...X_{b,j}$ to $Z_{b,i}Z_{b,j+1}$. Importantly, we note that
the disorder operator is distinct from a single pure CFT, where
$X_{b,i}X_{b,i+1}...X_{b,j}\sim\frac{1}{\lvert i-j\rvert^{\eta}}$.

Figure 3: Entanglement negativity $E_{N}$ between two complementary segments
of sizes $x$ and $L-x$ on $B$ sublattice. Data denoted by circles are for the
critical mixed state resulting from measurement and feedback on the SPT
critical point (equivalently, $\rho_{B}=\tr_{A}\ket{\psi}\bra{\psi}$ with
$\ket{\psi}$ defined in Eq.10). The linear fit of the data indicates
$\rho_{B}$ has an entanglement structure like that of a 1+1D CFT. As a
comparison, triangles correspond to the pure 1+1D Ising critical state
$\ket{\text{CFT}}_{B}$ on $B$.

The mixed state $\rho_{B}$ exhibits genuine quantum criticality. Specifically,
based on the two conditions, i.e. algebraic decay in $ZZ$ two-point function
and $\prod_{i}X_{b,i}=1$, $\rho_{B}$ cannot be an ensemble of short-range
entangled pure states as discussed in Sec.III. To better characterize the
entanglement signature for the mixed-state quantum criticality, we bipartite
$B$ sublattice into two intervals $B_{1}$ and $B_{2}$, and quantify their
entanglement using entanglement negativity $E_{N}$ [33, 34, 35, 36, 37], an
entanglement measure of mixed states:
$E_{N}(\rho_{B})=\ln\Big(\lvert\lvert\rho_{B}^{T_{B_{1}}}\rvert\rvert_{1}\Big{missing})$
, where the upper script $T_{B_{1}}$ denotes the partial transpose with
respect to the subsystem $B_{1}$ and $\lvert\lvert~{}\cdot~{}\rvert\rvert_{1}$
denotes the trace norm. Using Exact Diagonalization (ED), we find (see Fig.3)
negativity follows a universal scaling form as in the 1+1D CFT [48, 49]:

$E_{N}(x,L)=\alpha\ln(\frac{L}{\pi}\sin(\frac{\pi x}{L}))+\beta$ (12)

where $x,L$ are the sizes of the subsystem $B_{1}$ and the entire $B$
sublattice, and $\alpha\approx 0.15$. As a comparison, the finite-size
numerics for a pure critical Ising chain reports $\alpha\approx 0.30$666For a
pure 1+1D CFT, $\alpha$ is a universal number related to central charges, and
$\alpha=\frac{1}{4}$ for Ising CFT [50]. The mismatch between $\frac{1}{4}$
and $0.30$ is attributed to finite-size effects.. This indicates that the
amount of long-range entanglement on $B$ sublattice decreases when coupling to
$A$ sublattice. Since the prefactor $\alpha$ is generically a universal number
that relates to the number of low-energy degrees of freedom, one
interpretation is that coupling between $A,B$ sublattices diminishes the low-
energy degrees of freedom that carry long-distance entanglement on $B$
sublattice. This is also consistent with the fact that the disorder operator
decays faster after coupling to $A$ sublattice (Eq.11).

Finally, we remark that our protocol of converting hidden order is also
applicable when the input state is mixed. One immediate application is that
when the input pure SPT is subject to a symmetry-preserving noise channel, in
which case the string order survives, the output of our measurement-feedback
channel will remain long-range ordered. In addition, with the critical state
($g=1$ in Eq.7) under a noise channel as an input, the output mixed state
exhibits a log scaling of bipartite entanglement negativity (as in Eq.12)
while the log-scaling prefactor decreases continuously as increasing the noise
rate. See Appendix.B.4 for a detailed discussion.

###  IV.4 Generalization to other mixed-state long-range order

Here we briefly summarize several classes of mixed-state long-range order that
can be realized based on our general framework. In all these examples, we
derive an exact duality between the Hamiltonian of the input SPT and the
Hamiltonian of a purified state whose sublattice encodes the long-range order,
which allows us to establish the correspondence between phase diagram of the
input state and that of the output state.

$\mathbb{Z}_{2}$ topological order in higher space dimensions: our protocol
can be straightforwardly generalized by considering input states as
$\mathbb{Z}_{2}$ $p$-form $\times\mathbb{Z}_{2}$ $q$-form SPTs in $d$-space
dimension with $p+d=d-1$. Using measurement-feedback channels, one can prepare
a mixed state with $\mathbb{Z}_{2}$ topological order at $d\geq 2$ space
dimensions. In Appendix.C we detail the realization of mixed-state
$\mathbb{Z}_{2}$ topological order in $2d$.

$\mathbb{Z}_{2}$ fermionic topological order: Another straightforward
application is the preparation of mixed states with $\mathbb{Z}_{2}$ fermionic
topological order. This can be achieved by taking
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{f}$ SPTs [51, 52, 53, 54, 20] as input,
where the first $\mathbb{Z}_{2}$ symmetry acts on qubits and the second
$\mathbb{Z}_{2}^{f}$ is the parity symmetry on fermions. Based on certain non-
local hidden orders in these SPTs, we devise a protocol that outputs fermionic
mixed states with long-range order. In Appendix.D we present the protocol that
realizes the fermionic mixed state in 1d with the same long-range order as in
the topological phase of the 1d Kitaev chain. In Appendix.E we present the
protocol that realizes the fermionic mixed states in 2d with intrinsic
topological order.

##  V Mixed-state quantum criticality by measuring fermions

In this section, we construct quantum channels based on fermion occupation
number measurement in spinful fermions. This is naturally motivated by
Gutzwiller projection [55], a standard approach to construct exotic states of
matter by projecting spinful non-interacting fermions into the subspace of
single occupation number per lattice site, yielding a spin-1/2 wavefunction.
For example, Gutzwiller projecting free fermions at half-filling in 1d leads
to a critical ground state of the Haldane-Shastry model [56, 57], a long-range
interacting Heisenberg antiferromagnet with $1/r^{2}$ exchange coupling. In
2d, certain spin liquids with topological order can be constructed similarly
(see Ref. [38, 39] for review). More broadly, Gutzwiller projection may be
viewed as an application of parton construction [58], a well-known approach to
constructing non-trivial states by imposing certain constraints on non-
interacting particles.

Gutzwiller projection/parton construction provides inspiration for using
measurement to implement the projection. Indeed, Ref.[23] has presented a
protocol to prepare the ground state of the Kitaev honeycomb model [59], where
notably, the desired projection can be achieved by measuring fermions and
applying a depth-1 local unitary feedback. However, beyond this specific
model, starting with a generic state, post-measurement states with unwanted
measurement outcomes may not converge to the same target Gutzwiller projected
state using finite-depth unitaries. This presents a difficulty in realizing
Gutzwiller projection with measurement-based protocols. Here our framework
avoids this conundrum. We will show that occupation number measurement
followed by appropriate unitary feedback enables the realization of non-
trivial mixed states with certain long-distance quantum correlations. A field-
theoretic understanding of the resulting mixed-states and their correlations
will be presented in forthcoming work [60].

###  V.1 Measuring 1d fermion

Taking 1d spinful free fermions as input, below we will present a finite-depth
protocol for realizing quantum-critical mixed states with critical
correlations distinct from the input states.

Consider non-interacting, spinful fermions on a 1d lattice, where the
annihilation fermion operators at site $i$ are denoted by
$c_{i,s},c_{i,s}^{\dagger}$ with site indices $i=1,2,\cdots,L$ and spin
indices $s=\uparrow,\downarrow$, we define a tight-binding Hamiltonian
$H=-\sum_{i,s}(c_{i+1,s}^{\dagger}c_{i,s}+\text{h.c.})$, whose ground state
$\ket{\psi_{0}}$ takes the following form:
$\prod_{k}c_{k,\uparrow}^{\dagger}\prod_{k}c^{\dagger}_{k,\downarrow}\ket{0}$
with $k$ being the momentum indices. $\ket{\psi_{0}}$ exhibits an algebraic
correlation between spins:

$\expectationvalue{S^{z}_{i}S^{z}_{j}}\sim\frac{1}{(j-i)^{2}},$ (13)

with the spin operator
$S^{z}_{i}=\frac{1}{2}[n_{i,\uparrow}-n_{i,\downarrow}]$, and
$n_{i}=n_{i,\uparrow}+n_{i,\downarrow}$ is the number operator at site $i$.

Interestingly, by decorating with a fermion parity string, the spin-spin
correlation decays slower (see Ref.[41] for details):

$\expectationvalue{S^{z}_{i}\prod_{l=i+1}^{j-1}(-1)^{n_{l}}S^{z}_{j}}\sim\frac{1}{j-i}$
(14)

In other words, the fermion-parity string reveals a more ordered, long-range
correlation that is hidden in free fermions. Conceptually the insertion of the
fermion string has the effect of removing the charge fluctuations, thereby
enhancing the order in the spin sector. The physics is akin to the spin-1 AKLT
chain [61] (belonging to the Haldane SPT phase [62]), where the ground state
is a superposition of a class of product states that have a staggered pattern
of $+1$ and $-1$ with an arbitrary number of $0$s in between, e.g.
$...+00-0+000-+...$. While the two-point function $S^{z}_{i}S^{z}_{j}$ decays
exponentially due to the position disorder of $0$, inserting the string
$\prod_{l=i+1}^{j-1}(-1)^{S^{z}_{l}}$ in between $S^{z}_{i}$ and $S^{z}_{j}$
effectively removes the disorder, thereby revealing the hidden
antiferromagnetic order [63]. The process of removing certain disorders by
inserting string operators is dubbed the squeezed space construction in
Ref.[41].

More technically, the string order may be understood as the following: the
free-fermion state consists of two decoupled fermion chains with opposite spin
flavors:
$\ket{\psi_{0}}=\ket{\psi_{0}}_{\uparrow}\otimes\ket{\psi_{0}}_{\downarrow}$,
Each fermion chain exhibits a string order
$c^{\dagger}_{i,s}\prod_{l=i+1}^{j-1}(-1)^{n_{l,s}}c_{j,s}\sim\frac{1}{\sqrt{\absolutevalue{j-i}}}$
with $s\in\\{\uparrow,\downarrow\\}$. Considering a product of two string
operators, one finds
$S_{i}^{\dagger}\prod_{l=i+1}^{j-1}(-1)^{n_{l}}S_{j}^{-}\sim\frac{1}{j-i}$,
where $S_{i}^{\pm}$ is the spin-raising/lowering operator. Using the spin
rotational symmetry, one then recovers the string order in Eq.14.

Given the above free-fermion state $\ket{\psi_{0}}$, measuring the fermion
occupation number at each site gives the outcome
$n=\\{n_{1},n_{2},\cdots,n_{L}\\}$ with each $n_{i}\in\\{0,1,2\\}$.
Correspondingly, the post-measurement state becomes
$\frac{P_{n}\ket{\psi_{0}}}{\sqrt{\bra{\psi_{0}}P_{n}\ket{\psi_{0}}}}$ with
probability $p_{n}=\bra{\psi_{0}}P_{n}\ket{\psi_{0}}$. The measurement-only
protocol leads to a mixed state $\rho_{m}=\sum_{n}P_{n}\rho_{0}P_{n}$, i.e. an
ensemble of pure states corresponding to distinct measurement outcomes. Since
spin operators commute with fermion number operators, the spin-spin
correlation remains unchanged:
$\tr\left[\rho_{m}S^{z}_{i}S^{z}_{j}\right]=\bra{\psi_{0}}S^{z}_{i}S^{z}_{j}\ket{\psi_{0}}\sim\frac{1}{(j-i)^{2}}.$

Now we show that with appropriate unitary feedback, one can obtain a mixed
state with enhanced long-range correlation. Specifically, the string order of
the original input state $\ket{\psi_{0}}$ will be transformed into spin-spin
correlation with $S^{z}_{i}S^{z}_{j}\sim\frac{1}{j-i}$ in the resulting mixed
state. This is in strong contrast to
$S^{z}_{i}S^{z}_{j}\sim\frac{1}{(j-i)^{2}}$ in the case without unitary
feedback.

The protocol is as follows. First, we measure the fermion occupation number on
every site. For each post-measurement state with an outcome labeled by $n$, we
apply a local unitary $U_{n}$. This depth-2 protocol leads to a mixed state:
$\rho=\sum_{n}U_{n}P_{n}\rho_{0}P_{n}U^{\dagger}_{n}$. Now we look for the
unitary $U_{n}$ that transforms $S_{i}^{z}$ to $S_{i}^{z}(-1)^{\sum_{j\leq
i}n_{j}}$ for all $i$. This can be achieved by choosing

$U_{n}=\prod_{i=1}^{L}\left(\mathcal{S}_{i}^{x}\right)^{\sum_{j\leq i}n_{j}},$
(15)

where $\mathcal{S}_{i}^{x}$ is a spin-flip operator:
$\mathcal{S}_{i}^{x}=2S_{i}^{x}=c_{i,\uparrow}^{\dagger}c_{i,\downarrow}+c_{i,\downarrow}^{\dagger}c_{i,\uparrow}$
in the subspace of $n_{i}=1$, and $\mathcal{S}_{i}^{x}=1$ (i.e. it acts
trivially) in the subspace of $n_{i}=0,2$. In other words, the operator
$\mathcal{S}_{i}^{x}$ is applied at site $i$ when there is an odd number of
fermions in the interval $[1,i]$.

Under the transformation by $U_{n}$, the spin-spin correlator becomes
$U_{n}^{\dagger}S^{z}_{i}S^{z}_{j}U_{n}=-S_{i}^{z}(-1)^{\sum_{l=i+1}^{j-1}n_{l}}S_{j}^{z}$,
which acquires a sign that depends on the number of fermions between sites $i$
and $j$. A straightforward calculation shows that

$\begin{split}\tr\left[\rho
S^{z}_{i}S^{z}_{j}\right]&=\sum_{n}\bra{\psi_{0}}P_{n}U_{n}^{\dagger}S^{z}_{i}S^{z}_{j}U_{n}P_{n}\ket{\psi_{0}}\\\
&=-\bra{\psi_{0}}S_{i}^{z}(-1)^{\sum_{l=i+1}^{j-1}n_{j}}S_{j}^{z}\ket{\psi_{0}}\\\
&\sim\frac{1}{j-i}.\end{split}$ (16)

Therefore, the hidden string order of the input state $\ket{\psi_{0}}$ is
converted to a critical correlation shared among spins in the output mixed
state $\rho$. In particular, the reduced density matrix of $\rho$ that
describes spins, obtained by tracing out charge fluctuations, can be purified
into a ground state of a local Hamiltonian that describes interacting spinful
fermions (see Appendix.F).

One may also consider the spin-spin correlation in other orientations. The
global $SU(2)$ spin-rotation symmetry of the input state $\ket{\psi_{0}}$
implies
$\bra{\psi_{0}}S^{\nu}_{i}S^{\nu}_{j}\ket{\psi_{0}}\sim\frac{1}{(j-i)^{2}}$
and
$\bra{\psi_{0}}S^{\nu}_{i}\prod_{l=i+1}^{j-1}(-1)^{n_{l}}S^{\nu}_{j}\ket{\psi_{0}}\sim\frac{1}{j-i}$
for $\nu=x,y,z$. Since the feedback unitary consists of $\mathcal{S}_{i}^{x}$,
which anticommutes with $S_{i}^{y}$, the channel also boosts the correlation
in $y$ component, i.e. $S^{y}_{i}S^{y}_{j}$ is enhanced from
$\frac{1}{(j-i)^{2}}$ to $\frac{1}{j-i}$ in the resulting mixed state. On the
other hand, correlation in $x$ component remains $\frac{1}{(j-i)^{2}}$ since
the spin operator in $x$ component transforms trivially under the unitary
feedback.

The mixed state arising from our protocol is non-trivial in the sense that it
cannot be written as a mixture of short-range entangled states. To see this,
we first note that due to the global spin-flip symmetry of the input state,
i.e. $\prod_{i}\mathcal{S}_{i}^{x}\ket{\psi_{0}}=\ket{\psi_{0}}$, the output
mixed state $\rho$ will have $\prod_{i}\mathcal{S}_{i}^{x}=1$ in the
expectation value as well (since this operator commutes with both measurement
operators and unitary correction). Based on the algebraic decay
$S_{i}^{z}S_{j}^{z}$ and $\prod_{i}\mathcal{S}_{i}^{x}=1$, one can again use
the proof technique in Sec.III to show that $\rho$ cannot be a mixture of
short-range entangled states. This in turn suggests certain long-distance
entanglement in the mixed state. Specifically, in light of the well-known
$\log L$ entanglement scaling of the input free fermions in 1d, the output
state will likely exhibit a $\log L$ scaling mixed-state entanglement (e.g.
quantified by entanglement negativity).

Our protocol is also applicable to an interacting spinful fermionic system
characterized by a Luttinger liquid. As shown in Ref.[41], given a Luttinger
liquid with the Luttinger parameters $K_{s}$ and $K_{c}$ that characterize the
stiffness of spins and charges, with repulsive charge interaction (implying
$0<K_{c}<1$) and global $SU(2)$ spin-rotation symmetry (implying $K_{s}=1$),
the leading order spin-spin correlation behaves as
$S^{z}_{i}S^{z}_{j}\sim\frac{1}{(j-i)^{K_{c}+K_{s}}}$ and the string operator
behaves as
$S^{z}_{i}\prod_{l=i+1}^{j-1}(-1)^{n_{l}}S^{z}_{j}\sim\frac{1}{(j-i)^{K_{s}}}$.
Using the same measurement-feedback channel, one can then transform this
string order to a critical spin-spin correlation
$S^{z}_{i}S^{z}_{j}\sim\frac{1}{(j-i)^{K_{s}}}$ in the resulting mixed state,
which is in contrast to the initial exponent $K_{c}+K_{s}$.

###  V.2 Measuring Chern insulators

Here, we show that starting with a two-dimensional gapped state with short-
range correlations in the bulk, measurement and feedback can remarkably lead
to a mixed state with long-range critical correlations.

Our starting point is the Chern insulator, a gapped state of matter which
features integer-value quantized Hall conductance and gapless edge states.
Despite the short-range correlation in the bulk, i.e. exponential decay of
two-point functions, there exists a non-local operator whose correlation
decays algebraically. Denoting a fermion creation operator by
$c^{\dagger}(\mathbf{x})$ with $\mathbf{x}=(x,y)$ being the space coordinate,
one defines a dressed fermion creation operator
$\tilde{c}^{\dagger}(\mathbf{x})=\eta(\mathbf{x})c^{\dagger}(\mathbf{x})$,
where $\eta(\mathbf{x})$ is a non-local operator:

$\eta(\mathbf{x})=e^{i\int
d^{2}x^{\prime}c^{\dagger}(\mathbf{x}^{\prime})c(\mathbf{x}^{\prime})\text{arg}(\mathbf{x}-\mathbf{x}^{\prime})},$
(17)

with $\text{arg}(\mathbf{x})$ being the polar angle of $\mathbf{x}$.

In a Chern insulator $\ket{\nu=1}$ with associated Hall conductance
$\sigma_{H}=\nu(e^{2}/h)$, the dressed operators exhibit a critical
correlation [44]:

$\bra{\nu=1}\tilde{c}(\mathbf{x})\tilde{c}^{\dagger}(\mathbf{x^{\prime}})\ket{\nu=1}\sim\frac{1}{\absolutevalue{\mathbf{x}-\mathbf{x^{\prime}}}^{\alpha}},$
(18)

where $\alpha$ is a non-universal constant related to the $U(1)$ response of
Chern insulators.

Figure 4: Spin-up and spin-down fermions are initialized in Chern insulating
states with opposite Chern numbers. Such an initial state can be converted to
a mixed state with algebraic correlations in the bulk through a two-step
protocol consisting of single-site fermion occupation number measurement and
unitary.

As discussed in Ref.[44], $\eta(\mathbf{x})$ can be understood as the operator
associated with the infinitesimal time-evolution given by inserting a 2+1D
$U(1)$ monopole, and the algebraic long-range order results from the
topological response to such monopole insertion in a Chern insulator. Note
that the above critical order presents an off-diagonal long-range order
(ODLRO), which was first discovered by Girvin and MacDonald [43] in fractional
quantum hall states at filling fraction $\nu=1/m$. In particular, these
quantum hall states may be understood as a condensate of composite bosons by
attaching fluxes to fermions, so the bare fermions need to be dressed by an
appropriate non-local operator Eq. 17 in order to reveal the hidden ODLRO. For
quantum hall states described by Laughlin’s wave function at $\nu=1$, $\alpha$
is analytically found to be $1/2$ [43].

Below we will discuss a protocol that realizes a critical mixed state with
algebraic two-point correlations based on the hidden order in Chern
insulators. To start, we consider an initial state
$\ket{\psi_{0}}=\ket{\psi_{0}}_{\uparrow}\otimes\ket{\psi_{0}}_{\downarrow}$,
where $\ket{\psi_{0}}_{\uparrow}$ is the Chern insulator with $\nu=1$
consisting of spin-up fermions, and $\ket{\psi_{0}}_{\downarrow}$ is the Chern
insulator with $\nu=-1$ consisting of spin-down fermions. As a result, spin-up
and spin-down fermions respectively exhibit an algebraic hidden order:

$\bra{\psi_{0}}_{s}\tilde{c}_{s}(\mathbf{x})\tilde{c}_{s}^{\dagger}(\mathbf{x^{\prime}})\ket{\psi_{0}}_{s}\sim\frac{1}{\absolutevalue{\mathbf{x}-\mathbf{x^{\prime}}}^{\alpha}},$
(19)

where $s=1,-1$ for spin-up and spin-down fermions, and the dressed fermion
operators are
$\tilde{c}^{\dagger}_{s}(\mathbf{x})=\eta_{s}(\mathbf{x})c^{\dagger}_{s}(\mathbf{x})$,
with

$\eta_{s}(\mathbf{x})=e^{is\int
d^{2}x^{\prime}c_{s}^{\dagger}(\mathbf{x}^{\prime})c_{s}(\mathbf{x}^{\prime})\text{arg}(\mathbf{x}-\mathbf{x}^{\prime})}.$
(20)

Note the opposite sign in the exponent is due to the opposite sign of Chern
numbers for spin-up and spin-down fermions.

The hidden critical order in the Chern insulators can be detected through
spin-raising/lowering operators
$S^{+}(\mathbf{x})=c^{\dagger}_{\uparrow}(\mathbf{x})c_{\downarrow}(\mathbf{x})$
and
$S^{-}(\mathbf{x})=c^{\dagger}_{\downarrow}(\mathbf{x})c_{\uparrow}(\mathbf{x})$
by dressing them with the appropriate non-local operators. Specifically, let
$\hat{n}_{\uparrow}(\mathbf{x}),\hat{n}_{\downarrow}(\mathbf{x})$ be the
number operator of spin-up/down fermions, and define
$\hat{\phi}(\mathbf{x})=\int
d^{2}x^{\prime}\hat{n}(\mathbf{x})\text{arg}(\mathbf{x}-\mathbf{x}^{\prime})$
with
$\hat{n}(\mathbf{x})=\hat{n}_{\uparrow}(\mathbf{x})+\hat{n}_{\downarrow}(\mathbf{x})$.
Then the non-local operator
$e^{i\hat{\phi}(\mathbf{x})}S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})e^{-i\hat{\phi}(\mathbf{x}^{\prime})}$
exhibits an algebraic decay:

$\begin{split}&\bra{\psi_{0}}e^{i\hat{\phi}(\mathbf{x})}S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})e^{-i\hat{\phi}(\mathbf{x}^{\prime})}\ket{\psi_{0}}\\\
&=\bra{\psi_{0}}\tilde{c}^{\dagger}_{\uparrow}(\mathbf{x})\tilde{c}_{\downarrow}(\mathbf{x})\tilde{c}^{\dagger}_{\downarrow}(\mathbf{x}^{\prime})\tilde{c}_{\uparrow}(\mathbf{x}^{\prime})\ket{\psi_{0}}\\\
&=-\bra{\psi_{0}}_{\uparrow}\tilde{c}_{\uparrow}(\mathbf{x}^{\prime})\tilde{c}^{\dagger}_{\uparrow}(\mathbf{x})\ket{\psi_{0}}_{\uparrow}\bra{\psi_{0}}_{\downarrow}\tilde{c}_{\downarrow}(\mathbf{x})\tilde{c}^{\dagger}_{\downarrow}(\mathbf{x}^{\prime})\ket{\psi_{0}}_{\downarrow}\\\
&\sim\frac{1}{\absolutevalue{\mathbf{x}-\mathbf{x}^{\prime}}^{2\alpha}}.\end{split}$
(21)

Now we show how to convert this hidden non-local order to a critical order in
the two-point function $S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})$ using
measurement and feedback. Starting with the input state $\ket{\psi_{0}}$, we
measure the fermion occupation number on every site and then provide unitary
feedback. The resulting density matrix takes the form:

$\rho=\sum_{n}U_{n}P_{n}\ket{\psi_{0}}\bra{\psi_{0}}P_{n}U^{\dagger}_{n},$
(22)

where $P_{n}$ is the projector to a definite fermion number configuration, and
$U_{n}$ is the unitary feedback

$U_{n}=\prod_{\mathbf{x}}e^{-i\phi_{n}(\mathbf{x})S^{z}(\mathbf{x})}$ (23)

with
$S^{z}(\mathbf{x})=\frac{1}{2}(\hat{n}_{\uparrow}(\mathbf{x})-\hat{n}_{\downarrow}(\mathbf{x}))$
and the phase $\phi_{n}(\mathbf{x})=\int
d^{2}x^{\prime}n(\mathbf{x})\text{arg}(\mathbf{x}-\mathbf{x}^{\prime})$
depending on the measurement outcome. The resulting density matrix $\rho$
exhibits a critical two-point correlation inherited from the non-local hidden
order of the input $\ket{\psi_{0}}$:

$\begin{split}&\tr\left[\rho
S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})\right]\\\
&=\bra{\psi_{0}}e^{i\hat{\phi}(\mathbf{x})}S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})e^{-i\hat{\phi}(\mathbf{x}^{\prime})}\ket{\psi_{0}}\sim\frac{1}{\absolutevalue{\mathbf{x}-\mathbf{x}^{\prime}}^{2\alpha}}.\end{split}$
(24)

To see this, one first note that the spin operators under the transformation
by $U_{n}$ will be dressed by an appropriate phase, namely,

$\begin{split}&U_{n}^{\dagger}S^{+}(\mathbf{x})U_{n}=e^{i\phi_{n}(\mathbf{x})}S^{+}(\mathbf{x})\\\
&U_{n}^{\dagger}S^{-}(\mathbf{x})U_{n}=e^{-i\phi_{n}(\mathbf{x})}S^{-}(\mathbf{x}).\end{split}$
(25)

This implies
$P_{n}U_{n}^{\dagger}S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})U_{n}P_{n}=P_{n}e^{i\hat{\phi}(\mathbf{x})}S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})e^{-i\hat{\phi}(\mathbf{x}^{\prime})}$,
where the phase $\phi_{n}(\mathbf{x})$ has been promoted to an operator
$\hat{\phi}(\mathbf{x})$ under the projector $P_{n}$. Using this result, it is
then straightforward to derive Eq.24 with the density matrix in Eq.22.

Similarly, one may compute $S^{-}(\mathbf{x})S^{+}(\mathbf{x}^{\prime})$,
which turns out to be equal to $S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})$
in the operator expectation value, hence decaying algebraically. These two
results together imply
$S_{x}(\mathbf{x})S_{x}(\mathbf{x}^{\prime})+S_{y}(\mathbf{x})S_{y}(\mathbf{x}^{\prime})=\frac{1}{2}(S^{+}(\mathbf{x})S^{-}(\mathbf{x}^{\prime})+S^{-}(\mathbf{x})S^{+}(\mathbf{x}^{\prime}))\sim\frac{1}{\absolutevalue{\mathbf{x}-\mathbf{x}^{\prime}}^{2\alpha}}$,
where $S_{x}(\mathbf{x})=\frac{1}{2}(S^{+}(\mathbf{x})+S^{-}(\mathbf{x}))$,
$S_{y}(\mathbf{x})=\frac{1}{2i}(S^{+}(\mathbf{x})-S^{-}(\mathbf{x}))$, i.e.
the spin operator in $x$ and $y$ component. In particular, since the initial
input state $\ket{\psi_{0}}$ satisfies
$\prod_{\mathbf{x}}\mathcal{S}^{z}(\mathbf{x})\ket{\psi_{0}}=\ket{\psi_{0}}$,
where $\mathcal{S}^{z}(\mathbf{x})$ is the $\pi$ rotation about $z$ axis, the
expectation value of $\mathcal{S}^{z}(\mathbf{x})$ is one in the resulting
mixed state. With the proof technique used in Sec.III,
$\mathcal{S}^{z}(\mathbf{x})=1$ together with algebraic decay of
$S_{x}(\mathbf{x})S_{x}(\mathbf{x}^{\prime})+S_{y}(\mathbf{x})S_{y}(\mathbf{x}^{\prime})$
indicate that the critical mixed state $\rho$ cannot be a mixture of short-
range entangled state.

##  VI Summary and discussion

In this work, we present a general framework for realizing mixed-state quantum
order and quantum criticality using finite-depth quantum channels consisting
of local measurement, local unitary feedback, and non-local classical
communication. As an illustration, our protocol universally converts certain
SPT phases to mixed states with long-range (topological) orders. In addition,
when the input SPT is tuned to a critical point, our protocol outputs a
quantum critical mixed state diagnosed by volume-law classical entropy,
algebraic decay of correlations, as well as logarithmically scaling bipartite
mixed-state entanglement. Within the same theoretical framework, we also show
how to transform the correlation exponent in constant depth by considering 1d
spinful free fermions. More interestingly, our protocol can convert Chern
insulators to a mixed state with an algebraically decaying correlation in the
bulk. This furnishes a notable example where mixed-state quantum criticality
can emerge from gapped states of matter in finite depth.

Our work motivates several questions that are worth further exploration.
First, while we provide several examples of realizing non-trivial mixed states
with critical correlations, a deeper understanding of the entanglement
structure of these novel mixed-state quantum criticality is lacking. Second,
since this work only focuses on the utility of depth-2 protocols with onsite
measurement and onsite unitary feedback, it would be interesting to generalize
the current protocol, which may enable the realization of more types of exotic
mixed-state orders and criticality. For instance, the onsite unitary feedback
may be generalized to finite-depth local unitary circuits, and the current
depth-2 protocols may be extended to multiple rounds of measurement and
unitary layers.

Finally, it remains unclear what are the limitations of local measurement,
local unitary, and non-local classical communication for realizing non-trivial
quantum orders and criticality. Answering this question would provide sharp
distinctions between the states that can and cannot be realized in finite
depth. While entanglement provides one such constraint as discussed in
Sec.III, it would be desirable to explore the limitations from various aspects
in the future (see Ref.[64, 25] for progress along this direction).

###### Acknowledgements.

The authors thank the Kavli Institute for Theoretical Physics (KITP), where
this research was initiated and partly performed. The KITP is supported, in
part, by the National Science Foundation under Grant No. NSF PHY-1748958. S.V.
and Z.Z. thank Matthew Fisher, Ali Lavasani, Andreas Ludwig, and Cenke Xu for
helpful discussions. T.-C.L. and T.H. thank Tarun Grover, Shengqi Sang, Liujun
Zou, and Yijian Zou for helpful discussions. T.-C.L. and T.H.’s research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Economic Development Canada and by
the Province of Ontario through the Ministry of Colleges and Universities.

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##  Appendix A Constraints on mixed-state entanglement

Here we first review the basic notion of entanglement of formation, which we
will use to constrain the entanglement structure of output states from finite-
depth quantum channels.

Entanglement of formation: given a density matrix $\rho_{C\overline{C}}$
acting on the bipartite Hilbert space
$\mathcal{H}_{C}\otimes\mathcal{H}_{\overline{C}}$, the entanglement between
$C$ and $\overline{C}$ can be quantified via entanglement of formation $E_{f}$
[45], which is defined as follows: one may decompose a density matrix into a
convex sum of pure states, i.e.
$\rho_{C\overline{C}}=\sum_{i}p_{i}\ket{\psi_{i}}\bra{\psi_{i}}$ with
$p_{i}\geq 0$ and $\sum_{i}p_{i}=1$. Then
$E_{f}\equiv\text{Min}\left\\{\sum_{i}p_{i}S_{C}(\tr_{\overline{C}}\ket{\psi_{i}}\bra{\psi_{i}})\right\\}$,
where $S_{C}(\rho_{C})$ is the von-Neumann entropy of $\rho_{C}$. Namely,
$E_{f}$ is the average of bipartite entanglement entropy minimized over all
possible ways of decomposing $\rho$ as a mixture of pure states. When
$\rho_{C\overline{C}}$ is pure, $E_{f}$ reduces to the von-Neumann
entanglement entropy. To demonstrate the idea of entanglement of formation,
one may consider two qubits in a maximally mixed state, i.e.
$\rho=\mathbb{I}/4$ with $\mathbb{I}$ being the identity operator. Certainly,
the mixed state can be written as a uniform mixture of four Bell states
$\frac{1}{\sqrt{2}}(\ket{00}\pm\ket{11}),\frac{1}{\sqrt{2}}(\ket{01}\pm\ket{10})$,
so the average entanglement is $\log 2$. However, the mixed state can also be
written as a uniform mixture of four product states,
$\ket{00},\ket{01},\ket{10},\ket{11}$, in which case one achieves the minimal
average entanglement entropy, i.e. zero. Therefore $E_{f}=0$ for this
maximally mixed state.

Bounds on entanglement: in the main text, we construct quantum channels from
LOCC, i.e. single-site measurement, single-site unitaries, and classical
communication. As such, the mixed-state entanglement quantified by $E_{f}$
must be non-increasing under such quantum channels. To see this, starting with
a state $\rho_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$, performing single-site
measurement followed by single-site unitaries leads to a mixed state $\rho$, a
mixture of pure states
$\ket{\psi_{\alpha}}=\frac{U_{\alpha}P_{\alpha}\ket{\psi_{0}}}{\sqrt{P_{\alpha}\ket{\psi_{0}}}}$
with corresponding probability
$p_{\alpha}=\bra{\psi_{0}}P_{\alpha}\ket{\psi}$. Being an entanglement
measure, $E_{f}$ cannot increase on average under LOCC [45, 65], indicating
$E_{f}(\rho_{0})\geq\sum_{\alpha}p_{\alpha}E_{f}(\ket{\psi_{\alpha}})=\sum_{\alpha}p_{\alpha}S_{C}(\tr_{\overline{C}}\ket{\psi_{\alpha}}\bra{\psi_{\alpha}})$.
Meanwhile, one has
$\sum_{\alpha}p_{\alpha}S_{C}(\tr_{\overline{C}}\ket{\psi_{\alpha}}\bra{\psi_{\alpha}})\geq
E_{f}(\rho=\sum_{\alpha}p_{\alpha}\ket{\psi_{\alpha}}\bra{\psi_{\alpha}})$
because $E_{f}$ is the average entanglement entropy minimized over all
possible pure state realizations. As a result, one finds $E_{f}(\rho_{0})\geq
E_{f}(\rho)$.

When generalizing onsite unitary operations to multi-site, geometrically local
unitary gates, the protocol no longer belongs to LOCC, and mixed-state
entanglement may increase. To bound the entanglement growth, one may consider
the Renyi-$n$ entanglement of formation [66] defined as
$R_{n}\equiv\text{Min}\left\\{\sum_{i}p_{i}S^{(n)}_{C}(\tr_{\overline{C}}\ket{\psi_{i}}\bra{\psi_{i}})\right\\}$,
where $S^{(n)}_{C}$ is the Renyi-$n$ entropy. Starting with a pure state,
after a finite-depth protocol with local measurement and unitary gates (acting
on $O(1)$ number of neighboring sites), one obtains an ensemble $\rho$ of pure
state trajectories $\ket{\psi_{\alpha}}$ with probability $p_{\alpha}$. For
each pure state trajectory $\ket{\psi_{\alpha}}$, the increase of Renyi-0
entanglement
$S^{(0)}_{C}(\tr_{\overline{C}}\ket{\psi_{\alpha}}\bra{\psi_{\alpha}})$, also
dubbed max-entropy or Hartley entropy, is upper bounded by
$D\absolutevalue{\partial A}$, where $D$ is the depth of the channel and
$\absolutevalue{\partial A}$ is the bipartition boundary area. This is because
Hartley entropy is simply bounded by the number of local measurements and
unitary gates that act on the boundary. Therefore, the increase of Renyi-0
entanglement of formation $R_{0}$ is also bounded by the area law. One
immediate consequence is that if one starts with a pure product state (i.e.
with $R_{0}=0$), then after a finite-depth channel, the Renyi entanglement of
formation of any Renyi index will follow an area law since $R_{n}\leq R_{0}$
for any $n>0$.

##  Appendix B Measuring 1d SPT

###  B.1 Derivation of operator duality

Here we derive the transformation rules for operators under the controlled
unitary $U=\sum_{\alpha}P_{\alpha}U_{\alpha}$. $U_{\alpha}$ is the unitary
feedback defined as
$U_{\alpha}=\prod_{i}X_{b,i}^{\frac{1-\prod_{j=1,2,\cdots}^{i}\alpha_{j}}{2}}$.
$P_{\alpha}$ is a projector defined as $P_{\alpha}=\ket{\alpha}\bra{\alpha}$,
which projects $A$ sublattice to a product state in Pauli-X basis. Below we
provide two distinct approaches for derivation.

Operator-based approach: First, the controlled unitary can be written as
$U=\prod_{i}X_{b,i}^{\frac{1-\prod_{j=1}^{i}{X_{a,j}}}{2}}$. Since $U$ is
diagonal in the Pauli-X basis, Pauli-X operators are invariant under the
conjugation by $U$. Next, we consider $UZ_{b,i}Z_{b,i+1}U^{\dagger}$ (note
that Pauli-Zs come in pairs due to the global $\mathbb{Z}_{2}$ symmetry). This
operator can be simplified as $uZ_{b,i}Z_{b,i+1}u^{\dagger}$, where

$u=X_{b,i}^{\frac{1-\prod_{j=1}^{i}{X_{a,j}}}{2}}X_{b,i+1}^{\frac{1-\prod_{j=1}^{i+1}{X_{a,j}}}{2}}.$
(26)

In the subspace that $\prod_{j=1}^{i}X_{a,j}=1$, one finds
$UZ_{b,i}Z_{b,i+1}U^{\dagger}=X_{b,i+1}^{\frac{1-X_{a,i+1}}{2}}Z_{b,i}Z_{b,i+1}X_{b,i+1}^{\frac{1-X_{a,i+1}}{2}}$.
For $X_{a,i+1}=1$, one has $UZ_{b,i}Z_{b,i+1}U^{\dagger}=Z_{b,i}Z_{b,i+1}$
while for $X_{a,i+1}=-1$ one has
$UZ_{b,i}Z_{b,i+1}U^{\dagger}=-Z_{b,i}Z_{b,i+1}$. Combining the cases of
$X_{a,i+1}=\pm 1$ leads to the result
$UZ_{b,i}Z_{b,i+1}U^{\dagger}=Z_{b,i}X_{a,i+1}Z_{b,i+1}$. Similarly, one finds
$UZ_{a,i}Z_{a,i+1}U^{\dagger}=Z_{a,i}X_{b,i}Z_{a,i+1}$.

Here we remark that as we see in the main text, the controlled unitary
$U=\prod_{i}X_{b,i}^{\frac{1-\prod_{j=1}^{i}{X_{a,j}}}{2}}$ maps a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT to two $\mathbb{Z}_{2}$ symmetry-
breaking (GHZ) ordered states on $A$ and $B$ sublattices. This is akin to the
Kennedy-Tasaki transformation [67, 68], which transforms a Haldane spin-1
chain [62] with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT order to two
spontaneous $\mathbb{Z}_{2}$ symmetry-breaking orders. Indeed, Kennedy-Tasaki
transformation can also be realized as a controlled unitary [69], which
therefore fits into our general framework. In particular, we can take a
Haldane spin-1 chain in the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT phase,
and perform a layer of onsite measurement followed by a layer of onsite
unitaries to realize a mixed state with $\mathbb{Z}_{2}$ GHZ-like order.

Wave-function based approach: starting with a global
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetric input state $\ket{\psi_{0}}$,
in Pauli-X basis $\ket{\psi_{0}}$ can be written as

$\ket{\psi_{0}}=\sum^{\prime}_{\alpha,\beta}(-1)^{\chi(\alpha,\beta)}\psi(\alpha,\beta)\ket{\alpha,\beta}.$
(27)

$\alpha=\\{\alpha_{i}=\pm 1\\},\beta=\\{\beta_{i}=\pm 1\\}$ denote the product
state in Pauli-X basis in the sublattice $A$ and sublattice $B$. The
$\mathbb{Z}_{2}$ symmetry on each sublattice implies that the allowed
$\alpha,\beta$ must satisfy $\prod_{i}\alpha_{i}=\prod_{i}\beta_{i}=1$. We use
$\Sigma^{\prime}$ to denote that only those $\alpha,\beta$ that satisfy this
constraint are considered.

Due to the symmetry, $\alpha_{i}=-1$ must come in pairs, which can be regarded
as the two endpoints of a string. Correspondingly, various $\alpha$
configurations can be understood as various $A$-strings in $A$ sublattice, and
similarly, $\beta$ configurations can be represented by $B$-strings in $B$
sublattice. Within the string representations, $\chi$ is the number of times
that $A$-strings intersect (braid) with $B$-strings. When $\ket{\psi_{0}}$ is
a fixed-point cluster SPT with $ZXZ$ stabilizers, $\psi(\alpha,\beta)$ is a
constant independent of $\alpha,\beta$ (see Fig.5).

Figure 5: 1d fixed-point $\mathbb{Z}_{2}\crossproduct\mathbb{Z}_{2}$ SPT (with
parent Hamiltonian
$H_{0}=-\sum_{i}Z_{a,i}X_{b,i}Z_{a,i+1}-\sum_{i}Z_{b,i}X_{a,i+1}Z_{b,i+1}$) as
a condensate of two types of strings with a braiding sign structure in the
Pauli-X basis. A product state $\ket{\alpha,\beta}$ in X basis with
$\prod_{i}\alpha_{i}=\prod_{i}\beta=1$ corresponds to a configuration of blue
strings and red strings whose endpoints indicate $\alpha_{i}=-1$ and
$\beta_{i}=-1$ respectively. The braiding number $\chi(\alpha,\beta)$ counts
the number of times that two types of strings intersect.

Given Eq.27, measuring Pauli-X on $A$ sublattice projects $A$ sublattice to a
particular $\alpha$ configuration:
$P_{\alpha}\ket{\psi_{0}}=\ket{\alpha}\otimes\sum^{\prime}_{\beta}(-1)^{\chi(\alpha,\beta)}\psi(\alpha,\beta)\ket{\beta}$.
After the measurement, applying the unitary feedback
$U_{\alpha}(=\prod_{i}X_{b,i}^{\frac{1-\prod_{j=1,2,\cdots}^{i}\alpha_{j}}{2}}$)
removes the braiding phase:
$U_{\alpha}P_{\alpha}\ket{\psi_{0}}=\ket{\alpha}\otimes\sum^{\prime}_{\beta}\psi(\alpha,\beta)\ket{\beta}$.
It follows that the output state on $B$ sublattice reads

$\begin{split}\rho_{B}&=\tr_{A}\left(\sum_{\alpha}U_{\alpha}P_{\alpha}\ket{\psi_{0}}\bra{\psi_{0}}P_{\alpha}U^{\dagger}_{\alpha}\right)\\\
&=\sum^{\prime}_{\beta,\beta^{\prime}}\ket{\beta}\bra{\beta^{\prime}}\sum^{\prime}_{\alpha}\psi(\alpha,\beta)\psi^{*}(\alpha,\beta^{\prime}).\end{split}$
(28)

$\rho_{B}$ is simply the reduced density matrix on $B$ of the following pure
state:

$\ket{\psi}=\sum^{\prime}_{\alpha,\beta}\psi(\alpha,\beta)\ket{\alpha,\beta}.$
(29)

One consequence is that the entropy of the mixed state $\rho_{B}$ results from
the entanglement entropy between $A$ and $B$ in $\ket{\psi}$, i.e. it is the
residual quantum fluctuation when removing the braiding phase in the initial
input SPT $\ket{\psi_{0}}$.

Comparing Eq.27 and Eq.29, it is obvious that $\ket{\psi}=U\ket{\psi_{0}}$
with $U$ is a diagonal matrix whose entries encode the braiding phase:

$U=\sum^{\prime}_{\alpha,\beta}\ket{\alpha,\beta}\bra{\alpha,\beta}(-1)^{\chi(\alpha,\beta)},$
(30)

Although $U$ is not unitary in the entire Hilbert space, it is unitary in the
symmetric subspace specified by $\prod_{i}X_{a,i}=\prod_{i}X_{b,i}=1$. Note
that this form of unitary has appeared in the literature [70] as a way to
reveal the hidden long-range order of certain SPTs. We also note that in the
symmetric subspace, $U$ is the same as the controlled unitary
$\sum_{\alpha}P_{\alpha}U_{\alpha}=\prod_{i}X_{b,i}^{\frac{1-\prod_{j=1}^{i}{X_{a,j}}}{2}}$
discussed in the operator-based approach.

With
$U=\sum^{\prime}_{\alpha,\beta}(-1)^{\chi(\alpha,\beta)}\ket{\alpha,\beta}\bra{\alpha,\beta}$,
one can derive the parent Hamiltonian of $\ket{\psi}$ from
$H=UH_{0}U^{\dagger}$, where the following symmetry constraint is further
imposed on $H$: $\prod_{i}X_{a,i}=\prod_{i}X_{b,i}=1$777More specifically,
based on the condition that $P\ket{\psi_{0}}=\ket{\psi_{0}}$ and
$P\ket{\psi}=\ket{\psi}=U\ket{\psi_{0}}$, where $P$ is a projector to the
subspace with $\prod_{i}X_{a,i}=\prod_{i}X_{b,i}=1$, the eigenequation
$H_{0}\ket{\psi_{0}}=E_{g}\ket{\psi_{0}}$ implies
$PUHU^{\dagger}P\ket{\psi}=E_{g}\ket{\psi_{0}}$, so the parent Hamiltonian of
$\ket{\psi}$ is $PUHU^{\dagger}P$..

Now we present the derivation of operator mapping:

$\begin{split}&UZ_{b,i}Z_{b,i+1}U^{\dagger}\\\
&=\sum^{\prime}_{\alpha^{\prime},\beta^{\prime},\alpha,\beta}\ket{\alpha^{\prime},\beta^{\prime}}\bra{\alpha,\beta}\\\
&\bra{\alpha^{\prime},\beta^{\prime}}(-1)^{\chi(\alpha^{\prime},\beta^{\prime})}Z_{b,i}Z_{b,i+1}(-1)^{\chi(\alpha,\beta)}\ket{\alpha,\beta}.\end{split}$
(31)

While $Z_{b,i}Z_{b,i+1}$ has no effect on $\ket{\alpha}$ (i.e.
$\alpha^{\prime}=\alpha$), this operator makes the transition from $\beta$ to
$\beta^{\prime}$, where $\beta$ and $\beta^{\prime}$ only differ in the site
$(b,i)$ and site $(b,i+1)$. It is not hard to find that the product of the
braiding phase
$(-1)^{\chi(\alpha^{\prime},\beta^{\prime})}(-1)^{\chi(\alpha,\beta)}$, i.e.
the braiding sign change induced by $Z_{b,i}Z_{b,i+1}$ is simply equal to
$\alpha_{i+1}$. In other words,
$\bra{\alpha^{\prime},\beta^{\prime}}(-1)^{\chi(\alpha^{\prime},\beta^{\prime})}Z_{b,i}Z_{b,i+1}(-1)^{\chi(\alpha,\beta)}\ket{\alpha,\beta}$

$\begin{split}&=\bra{\alpha^{\prime},\beta^{\prime}}Z_{b,i}\alpha_{i+1}Z_{b,i+1}\ket{\alpha,\beta}\\\
&=\bra{\alpha^{\prime},\beta^{\prime}}Z_{b,i}X_{a,i+1}Z_{b,i+1}\ket{\alpha,\beta}.\end{split}$
(32)

Therefore one finds $Z_{b,i}Z_{b,i+1}\to Z_{b,i}X_{a,i+1}Z_{b,i+1}$. With a
similar calculation, one finds $Z_{a,i}Z_{a,i+1}\to Z_{a,i}X_{b,i}Z_{a,i+1}$.

Figure 6: von Neumann entropy $S$ of the mixed state
$\rho_{B}(=\tr_{A}\ket{\psi}\bra{\psi})$ with $\ket{\psi}$ being the ground
state of $H$ in Eq.9. Upper panel: $S$ as a function of the tuning parameter
$g$. Lower panel: the scaling of $S$ with the system size $L$ of sublattice
$B$. We find $S$ scales linearly with $L$ for any non-zero $g$.

###  B.2 Numerical data on volume-law entropy

Here we report the data on von Neumann entropy of the mixed state $\rho_{B}$
from the ground state of $H$ (Eq.9). The entropy is defined as
$S(\rho_{B})=-\tr_{B}\big{[}\rho_{B}\ln(\rho_{B})\big{]}$. The ED calculation
for various system sizes is shown in Fig.6. Our data suggest that
$S(\rho_{B})$ scales linearly with $L$, i.e. volume-law scaling, for any non-
zero $g$. In particular, this means that at the critical point $g=1$, $B$
sublattice is a mixed state with volume-law classical entropy coexisting with
critical long-range entanglement.

###  B.3 Other non-fixed-point SPT as input

Here we consider various types of symmetric perturbation on the fixed-point
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT, and discuss the structure of the
corresponding output state.

On-site X perturbation: here the input is the ground state of the following
Hamiltonian:

$\begin{split}H_{0}=&-\sum_{i}Z_{a,i}X_{b,i}Z_{a,i+1}-\sum_{i}Z_{b,i}X_{a,i+1}Z_{b,i+1}\\\
&-g\sum_{i}(X_{a,i}+X_{b,i}).\end{split}$ (33)

The ground state of $H_{0}$ belongs to $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
SPT for $g<1$, and belongs to a trivial phase for $g>1$[71, 72]. Using the
transformation rule in Eq.6, one obtains a Hamiltonian of two decoupled
transverse-field Ising chain: $H=H_{A}+H_{B}$, where
$H_{A}=-\sum_{i}Z_{a,i}Z_{a,i+1}-g\sum_{i}X_{a,i}$ and
$H_{B}=-\sum_{i}Z_{b,i}Z_{b,i+1}-g\sum_{i}X_{b,i}$. The state $\rho_{B}$ from
the measurement-feedback channel is the reduced density matrix of the ground
state of $H$ by tracing out $A$. Since $A$ and $B$ are decoupled, $\rho_{B}$
is in fact a pure state, namely, the ground state of the transverse-field
Ising chain, whose long-range order exists for $\absolutevalue{g}<1$ (i.e. the
regime where the input state is an SPT). In particular, $\rho_{B}$ being pure
means that all post-measurement state trajectories are transformed to the same
state on $B$ with the unitary feedback.

Two-body ZZ perturbation: here the input is the ground state of the
Hamiltonian with independently tunable two-body $ZZ$ perturbation on $A$ and
$B$ sublattices:

Figure 7: Phase diagram of $H_{0}$ and $H$ defined in Eq.34 and Eq.35. Taking
the ground state of $H_{0}$ as an input, the measurement-feedback channel
prepares a mixed state on $B$ sublattice, which is the reduced density matrix
of the ground state of $H$.

$\begin{split}H_{0}=&-\sum_{i}Z_{a,i}X_{b,i}Z_{a,i+1}-\sum_{i}Z_{b,i}X_{a,i+1}Z_{b,i+1}\\\
&-g_{A}\sum_{i}Z_{a,i}Z_{a,i+1}-g_{B}\sum_{i}Z_{b,i}Z_{b,i+1}\end{split}$ (34)

The measurement-feedback channel will prepare a state $\rho_{B}$ on $B$
sublattice with the parent Hamiltonian:

$\begin{split}H=&-\sum_{i}Z_{a,i}Z_{a,i+1}-\sum_{i}Z_{b,i}Z_{b,i+1}\\\
&-g_{A}\sum_{i}Z_{a,i}X_{b,i}Z_{a,i+1}-g_{B}\sum_{i}Z_{b,i}X_{a,i+1}Z_{b,i+1}.\end{split}$
(35)

$H_{0}$ and $H$ are dual to each other, and their phase diagrams are presented
in Fig.7. Interestingly, one notices that with measurement-feedback channel
acting on a gapless state, we can prepare a non-critical mixed state with
long-range order. To see this, we consider $g_{B}=1$ and $g_{A}<1$. In this
case, $H_{0}$ exhibits the gapless SPT order [73] since the $\mathbb{Z}_{2}$
charges on $A$ are used to decorate the domain walls in $B$ sublattice that is
tuned to a critical point between a trivial phase and a $\mathbb{Z}_{2}$ SSB
phase. It follows that by measuring the gapped degrees of freedom on $A$
followed by unitary feedback, one obtains mixed state $\rho_{B}$ on $B$
sublattice that exhibits a $\mathbb{Z}_{2}$ SSB order.

More generally, one may consider an input state of the form
$\ket{\psi_{0}}=U\ket{\psi_{A}}_{A}\ket{\psi_{B}}_{B}$ with $U=\prod
CZ_{(a,i),(b,i)}CZ_{(b,i),(a,i+1)}$ being a product of controlled-Z that
entangles $A$ and $B$ sublattices. The measurement-feedback protocol will
always lead to a mixed state on $B$ with long-range order, as long as
$\ket{\psi_{0}}$ satisfies the following two conditions (in particular
$\ket{\psi_{0}}$ does not need to be an SPT): (1) $\ket{\psi_{A}}$ is trivial,
where domain-walls condense (i.e. the product of $X$ along an open string
takes a non-zero expectation value $c$). (2) The input state on $B$
sublattice, i.e. $\ket{\psi_{B}}$, is invariant under the global
$\mathbb{Z}_{2}$ symmetry: $\prod_{i\in
B}X_{i}\ket{\psi_{B}}_{B}=\ket{\psi_{B}}_{B}$ (e.g. $\ket{\psi_{B}}$ can be
trivial, spontaneously breaking the symmetry, or critical). The first
condition implies that a long-range string operator
$\bra{\psi_{0}}Z_{B}X_{A}X_{A}\cdots X_{A}Z_{B}\ket{\psi_{0}}=c$; measuring
$X_{A}$ on $A$ sublattice followed by a unitary feedback acting on $B$ leads
to a mixed state $\rho_{B}$ on $B$ with a long-range $ZZ$ two-point function.
(2) The second condition implies that $\tr[\rho\prod_{i\in B}X_{i}]=1$. As a
result, combining these two conditions already guaranteed the existence of
long-range order in the mixed state $\rho_{B}$. Practically speaking, it is of
best interest of our protocol to consider $\ket{\psi_{B}}_{B}$ as a trivial
state (in which case $\ket{\psi_{0}}$ is an SPT), since it furnishes an
example of obtaining long-range order only after implementing the protocol.

###  B.4 SPT under decoherence as input

In the main text, we discussed a measurement-feedback channel that transforms
the string order of the input pure SPT to a long-range order. Specifically,
the main result is

$\tr[\rho
Z_{b,i}Z_{b,j}]=\tr[\rho_{0}Z_{b,i}\left(\prod_{k=i+1}^{j}X_{a,k}\right)Z_{b,j}]$
(36)

where $\rho_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$ is the input state. In fact,
with a straightforward derivation, one finds the equation holds true for
arbitrary input density matrix $\rho_{0}$. Namely, $\rho_{0}$ needs not be a
pure state. As an application, the input state $\rho_{0}$ may be a mixed state
obtained by subjecting an SPT to certain decoherence channels. If the string
order is robust under the channel, the output state of our protocol will
remain long-range ordered. Below we will first discuss the decohered SPTs and
the corresponding emergent long-range order. We will then discuss the fate of
those decohered SPTs when tuned to criticality.

First, we consider a dephasing channel in Pauli-X basis. A local channel is
given by $\mathcal{E}_{i}:\rho\to(1-p)\rho+pX_{i}\rho
X_{i}=\sum_{\sigma=0,1}p_{\sigma}K^{\dagger}_{\sigma}\rho K_{\sigma}$ with
$K_{\sigma}=X_{i}^{\sigma_{i}}$, and $p_{\sigma}=p,1-p$ for $\sigma=1,0$
respectively. This can be understood as applying $X_{i}$ operator with
probability $p$, so $p=1/2$ corresponds to maximal dephasing. We consider the
local channels on every lattice site, so the overall dephasing channel
$\mathcal{E}$ is the composition of the local noise channels:
$\mathcal{E}=\mathcal{E}_{1}\circ\mathcal{E}_{2}\circ\cdots\circ\mathcal{E}_{2L}$,
and

$\mathcal{E}[\rho]=\sum_{\bm{\sigma}}P(\bm{\sigma})K^{\dagger}_{\bm{\sigma}}\rho
K_{\bm{\sigma}},$ (37)

where $\bm{\sigma}=\\{\sigma_{i}\\}$,
$P(\bm{\sigma})=\prod_{i=1}^{2L}p_{\sigma_{i}}$, and
$K_{\bm{\sigma}}=\prod_{i=1}^{2L}X^{\sigma_{i}}_{i}$. Given a pure SPT,
$\rho_{\text{SPT}}=\ket{\text{SPT}}\bra{\text{SPT}}$, the dephasing channel
leads to $\rho_{0}=\mathcal{E}[\rho_{\text{SPT}}]$, and an operator $O$ w.r.t.
$\rho_{0}$ becomes

$\expectationvalue{O}_{\rho_{0}}=\sum_{\bm{\sigma}}P(\bm{\sigma})\bra{\text{SPT}}K_{\bm{\sigma}}OK^{\dagger}_{\bm{\sigma}}\ket{\text{SPT}}.$
(38)

One immediate consequence is that if $[O,K_{\bm{\sigma}}]=0$, then
$\expectationvalue{O}_{\rho_{0}}=\bra{\text{SPT}}O\ket{\text{SPT}}$, i.e. the
expectation value of $O$ remains invariant under the dephasing channel.
Therefore, $\expectationvalue{\prod_{i=1}^{L}X_{b,i}}_{\rho_{0}}=1$ as in
$\ket{\text{SPT}}$. On the other hand, for the string operator
$\mathcal{S}\equiv Z_{b,i}\left(\prod_{k=i+1}^{j}X_{a,k}\right)Z_{b,j}$, since
only $Z_{b,i}$ and $Z_{b,j}$ may anticommute with the noise operator
$K_{\bm{\sigma}}$, $\expectationvalue{\mathcal{S}}_{\rho_{0}}$ can be
simplified as

$\sum_{\sigma_{b,i}\sigma_{b,j}}p_{\sigma_{b,i}}p_{\sigma_{b,j}}\bra{\text{SPT}}X_{b,i}^{\sigma_{b,i}}X_{b,j}^{\sigma_{b,j}}\mathcal{S}X_{b,i}^{\sigma_{b,i}}X_{b,j}^{\sigma_{b,j}}\ket{\text{SPT}}.$
(39)

Since the string operator $\mathcal{S}$ conjugated by
$X_{b,i}^{\sigma_{b,i}}X_{b,j}^{\sigma_{b,j}}$ acquires a $+1$, $-1$ sign when
$\sigma_{b,i}=\sigma_{b,j}$, $\sigma_{b,i}\neq\sigma_{b,j}$ respectively, one
finds

$\begin{split}&\expectationvalue{\mathcal{S}}_{\rho_{0}}\\\
&=\left[\text{Prob.}(\sigma_{b,i}=\sigma_{b,j})-\text{Prob.}(\sigma_{b,i}\neq\sigma_{b,j})\right]\bra{\text{SPT}}\mathcal{S}\ket{\text{SPT}}\\\
&=(1-2p)^{2}\bra{\text{SPT}}\mathcal{S}\ket{\text{SPT}}.\end{split}$ (40)

Therefore, under the X-dephasing channel, the string order survives for any
non-maximal dephasing (i.e. $p<1/2$). Given this dephased SPT mixed state, one
can then use the measurement-feedback channel to obtain a mixed state with
GHZ-like long-range order.

If one instead considers the symmetry-breaking dephasing channel based on
Pauli-Z noise, with a similar calculation, one finds the resulting dephased
state cannot have both $Z_{B}X_{A}\cdots X_{A}Z_{B}\sim O(1)$ and
$\prod_{i}X_{b,i}=1$. Specifically, due to the Pauli-Z noise on $A$
sublattice, one finds $Z_{B}X_{A}\cdots
X_{A}Z_{B}\sim(1-2p)^{\absolutevalue{S}}$, i.e. exponentially decaying with
the length of the string $\absolutevalue{S}$ for any non-zero dephasing. On
the other hand, the Pauli-Z noise on $B$ sublattice results in
$\prod_{i}X_{b,i}=(1-2p)^{L}$, i.e. exponentially decaying with sublattice
size of $B$ for any non-zero dephasing. As a result, the measurement-feedback
channel cannot lead to a mixed state with the GHZ-like long-range order based
on such dephased state.

At criticality: when $\ket{\text{SPT}}$ is tuned to a critical point (e.g.
$g=1$ in Eq.7), the string operator exhibits critical order
$Z_{b,i}(\prod_{k=i+1}^{j}X_{a,k})Z_{b,j}\sim\frac{1}{\absolutevalue{i-j}^{\eta}}$.
Under the Pauli-X dephasing, with a similar analysis as above, one obtains a
mixed state $\rho_{0}$ with
$\expectationvalue{Z_{b,i}(\prod_{k=i+1}^{j}X_{a,k})Z_{b,j}}_{\rho_{0}}=(1-2p)^{2}\expectationvalue{Z_{b,i}(\prod_{k=i+1}^{j}X_{a,k})Z_{b,j}}_{\text{SPT}}\sim\frac{1}{\absolutevalue{i-j}^{\eta}}$.
Therefore, the measurement-feedback channel discussed in Sec.IV.1 gives rise
to a mixed state $\rho_{B}$ with the critical correlation
$Z_{b,i}Z_{b,j}\sim\frac{1}{\absolutevalue{i-j}^{\eta}}$. On the other hand,
the disorder operator $X_{b,i}X_{b,i+1}...X_{b,j}$ commutes with both the
dephasing channel and the measurement-feedback channel, and therefore,
$\expectationvalue{X_{b,i}X_{b,i+1}...X_{b,j}}_{\rho_{0}}=\expectationvalue{X_{b,i}X_{b,i+1}...X_{b,j}}_{\text{SPT}}\sim\frac{1}{\absolutevalue{i-j}^{2\eta}}$.
The algebraic order in these operators suggests that $\rho_{B}$ exhibits
critical scaling of entanglement. This is verified with our ED calculation,
where the bipartite entanglement negativity follows the universal scaling form
as in a 1+1D CFT (see Fig.8 Upper panel):

$E_{N}(\alpha,\beta)=\alpha\ln(\frac{L}{\pi}\sin(\frac{\pi x}{L}))+\beta.$
(41)

In particular, the prefactor $\alpha$ decreases as increasing the strength of
dephasing.

Figure 8: Entanglement negativity $E_{N}$ between two complementary segments
of size $x$ and $L-x$ on $B$ sublattice for the mixed state $\rho_{B}$. The
input state is a mixed state $\rho_{0}$ obtained by acting the input SPT
$\psi_{0}$ with a decoherence channel. Upper panel: negativity of $\rho_{B}$
obtained by the measurement-feedback channel. Lower panel: negativity averaged
over mixed-state trajectories obtained by measuring $A$ sublattice for the
input dephased SPT $\rho_{0}$.

Another natural question is, given an input dephased SPT $\rho_{0}$, what are
the typical entanglement structures of the post-measurement (mixed) state
trajectories right after measuring $A$ sublattice? To address this question,
we numerically compute the entanglement negativity averaged over these
trajectories. As indicated by Fig.8 lower panel, this quantity also exhibits
the critical scaling in entanglement, indicating the pattern of critical
entanglement in $\rho_{B}$ may be attributed to the post-measurement state
trajectories that constitute the ensemble. Note that for a given dephasing
strength $p$, the trajectory-average negativity is always smaller than the
negativity in $\rho_{B}$. This is consistent with the intuition that some of
the long-distance entanglement is diminished by classical fluctuations in the
ensemble of those trajectories.

##  Appendix C Mixed state with topological order from measuring 2d SPT

Here we discuss a finite-depth protocol that converts higher dimensional SPTs
to mixed-state topological orders. In particular, we will consider the input
state as a 2d SPT protected by the $\mathbb{Z}_{2}$ 0-form $\times$
$\mathbb{Z}_{2}$ 1-form symmetry, in which case the resulting mixed state
exhibits a $Z_{2}$ topological order.

Consider a 2d lattice with every vertex $v$ and every edge $e$ accommodating a
qubit, the fixed-point SPT Hamiltonian reads

$H_{0}=-\sum_{v}X_{v}\prod_{e\ni v}Z_{e}-\sum_{e}X_{e}\prod_{v\in e}Z_{v}.$
(42)

The first term is a product of a Pauli-X on vertex $v$ and four neighboring
Pauli-Zs on edges, and the second term is a product of a Pauli-X on edge $e$
and two neighboring Pauli-Zs on vertices. This Hamiltonian can be obtained
from a trivial paramagnet $-\sum_{v}X_{v}-\sum_{e}X_{e}$ by a depth-1 unitary
circuit $U_{CZ}=\prod_{\expectationvalue{v,e}}CZ_{v,e}$, i.e. a product of
controlled-Z gates, each of which acts on a pair of neighboring vertex and
edge. The ground state is the 2d cluster state, a non-trivial SPT protected by
$\mathbb{Z}_{2}$ 0-form $\times$ $\mathbb{Z}_{2}$ 1-form symmetry, where the
$\mathbb{Z}_{2}$ 0-form symmetry is given by $\prod_{v}X_{v}$ on vertices and
the $\mathbb{Z}_{2}$ 1-form symmetry is given by $\prod_{e\in\gamma}X_{e}$
acting on edges along any closed loop $\gamma$ (including non-contractible
ones).

The non-trivial SPT order can be diagnosed by the membrane operator
$M_{\tilde{\gamma}}=\prod_{e\in\tilde{\gamma}}Z_{e}\prod_{v\in
A_{\tilde{\gamma}}}X_{v}$, where $\tilde{\gamma}$ is a loop in the dual
lattice, and $v\in A_{\tilde{\gamma}}$ denotes all vertices (in the primary
lattice) in the area enclosed by the loop $\tilde{\gamma}$. For instance,

$M_{\tilde{\gamma}}=\includegraphics[width=76.82234pt,valign={c}]{2d_membrane.pdf}$
(43)

For the fixed-point SPT, $\expectationvalue{M_{\tilde{\gamma}}}=1$ for any
loops $\tilde{\gamma}$. Away from the fixed-point limit, $M_{\tilde{\gamma}}$
exhibits a long-range correlation in the form of a perimeter law:
$\expectationvalue{M_{\tilde{\gamma}}}\sim
e^{-c\absolutevalue{\tilde{\gamma}}}$. This follows from the decorated domain
wall description of this type of SPT; starting with a trivial paramagnet of
Ising spins on vertices, the $\mathbb{Z}_{2}$ domain walls condense, implying
the perimeter law of a domain-wall creation operator $\prod_{v\in
A_{\tilde{\gamma}}}X_{v}\sim e^{-c\absolutevalue{\tilde{\gamma}}}$, where
$\tilde{\gamma}$ is a closed loop in the dual lattice 888This can also be
understood in a dual picture, where the domain-wall creation operator in the
trivial phase of Ising model correspond to a Wilson loop in the deconfined
phase of the $2+1$D $\mathbb{Z}_{2}$ gauge theory, thereby following a
perimeter law.. We then introduce $\mathbb{Z}_{2}$ charge on edges to decorate
the domain walls by applying $U_{CZ}$. This leads to an SPT with
$M_{\tilde{\gamma}}=\prod_{e\in\tilde{\gamma}}Z_{e}\prod_{v\in
A_{\tilde{\gamma}}}X_{v}\sim e^{-c\absolutevalue{\tilde{\gamma}}}$ because
$M_{\tilde{\gamma}}$ is obtained by conjugating the domain wall operator
$\prod_{v\in A_{\tilde{\gamma}}}X_{v}$ with $U_{CZ}$.

Now we can utilize the presence of long-range membrane order to devise a
protocol for preparing a mixed state with $\mathbb{Z}_{2}$ topological order.
Given an SPT $\ket{\psi_{0}}$, one measures $X_{v}$ on all vertices with
outcome $\alpha=\\{\alpha_{v}\\}$. For each post-measurement state trajectory
with outcome $\alpha$, one applies a corresponding local unitary $U_{\alpha}$
acting on edges. This 2-step protocol leads to the ensemble
$\rho=\sum_{\alpha}U_{\alpha}P_{\alpha}\rho_{0}P_{\alpha}U_{\alpha}^{\dagger}$.
Using a unitary $U_{\alpha}$ such that
$U_{\alpha}^{\dagger}\prod_{e\in\tilde{\gamma}}Z_{e}U_{\alpha}=\prod_{e\in\tilde{\gamma}}Z_{e}\left(\prod_{v\in
A_{\tilde{\gamma}}}\alpha_{v}\right)$, (i.e. the dual loop operator acquires a
sign that depends on the measurement outcome enclosed by the loop), the
expectation value of loop operators $\prod_{e\in\tilde{\gamma}}Z_{e}$ in
$\rho$ will exactly equal the expectation value of the membrane operator
$M_{\tilde{\gamma}}=\prod_{e\in\tilde{\gamma}}Z_{e}\prod_{v\in
A_{\tilde{\gamma}}}X_{v}$ w.r.t the initial input SPT, which exhibits a
perimeter-law scaling. In addition, Since the symmetry sector is fixed with
$\prod_{e\in\gamma}X_{e}=1$, the perimeter law scaling indicates a
$\mathbb{Z}_{2}$ topological order on edges in the output mixed state, as in
the deconfined phase of a 2+1D $\mathbb{Z}_{2}$ gauge theory.

Here we discuss the choice of $U_{\alpha}$. Due to the global $\mathbb{Z}_{2}$
symmetry on vertices in the input SPT $\ket{\psi_{0}}$, the measurement
outcomes on vertices satisfy $\prod_{v}\alpha_{v}=1$, i.e. the $-1$ outcomes
will occur in pairs. It turns out $U_{\alpha}$ is consisting of string
operators of Pauli-Xs on edges; each string is deformable and connects two
$\alpha_{v}=-1$ outcomes on vertices. Physically, the $\alpha_{v}=-1$ outcomes
may be regarded as $e$ particles in the $\mathbb{Z}_{2}$ topological order,
and $U_{\alpha}$ is the string operator that annihilates those anyon
excitations.

Figure 9: On a 2d lattice of size $L\times L$ on the surface of a 2-torus (by
imposing periodic boundary conditions), the perimeter law decay of the Wilson
loops in the dual lattice and the 1-form symmetry in the primary lattice
together imply a non-trivial mixed state.

Nontrivialness of the mixed state: now we show that the 1-form symmetry
$\prod_{e\in\gamma}X_{e}=1$ together with the perimeter law of the Wilson loop
in the dual lattice $\prod_{e\in\tilde{\gamma}}Z_{e}\sim
e^{-c\absolutevalue{\tilde{\gamma}}}$ implies that the mixed state on edges
($\rho_{B}$) is non-trivial; it cannot be a mixture of short-range entangled
pure states. This can be proved by contradiction, as discussed in Sec.III. We
first assume $\rho_{B}=\sum_{n}p_{n}\ket{n}\bra{n}$, where each $\ket{n}$ is a
trivial short-range entangled state that can be connected to a product state
using a finite-depth local unitary. We define two Wilson loop operators
$W_{Z}(\tilde{\gamma}_{i})=\prod_{e\in\tilde{\gamma}_{i}}Z_{e}$ for $i=1,2$,
where $\tilde{\gamma}_{1},\tilde{\gamma}_{2}$ are non-contractible loops
around the vertical direction, and they are horizontally separated by a scale
$O(d)$, see Fig.9. For each $\ket{n}$, the connected correlator is
$\bra{n}W_{Z}(\tilde{\gamma}_{1})W_{Z}(\tilde{\gamma}_{2}){\ket{n}}_{c}=\bra{n}W_{Z}(\tilde{\gamma}_{1})W_{Z}(\tilde{\gamma}_{2}){\ket{n}}-\bra{n}W_{Z}(\tilde{\gamma}_{1}){\ket{n}}\bra{n}W_{Z}(\tilde{\gamma}_{1}){\ket{n}}$.
Due to the 1-form symmetry $W_{X}(\gamma)=\prod_{e\in\gamma}X_{e}=1$ with
$\gamma$ winding around torus horizontally,
$\bra{n}W_{Z}(\tilde{\gamma}_{i}){\ket{n}}=0$ for both $i=1,2$. On the other
hand, the connected correlator in the trivial state $\ket{n}$ decays
exponentially with $d$ 999For example, in the ground state of a
$\mathbb{Z}_{2}$ gauge theory, such an operator follows an area-law decay as
$e^{-uLd}$ with $u$ being an $O(1)$ constant., and correspondingly
$\bra{n}W_{Z}(\tilde{\gamma}_{1})W_{Z}(\tilde{\gamma}_{2}){\ket{n}}$ decays
with the separation $d$ as well. This in turn implies the
$W_{Z}(\tilde{\gamma}_{1})W_{Z}(\tilde{\gamma}_{2})$ decays with $d$ in the
ensemble of $\ket{n}$, which contradicts the perimeter law scaling
$W_{Z}(\tilde{\gamma}_{1})W_{Z}(\tilde{\gamma}_{2})\sim e^{-\alpha L}$, which
does not decay with $d$ in the output mixed state. Therefore, the assumption
that $\rho_{B}$ is a mixture of trivial states must be false.

Structure of mixed state topological order: Here we discuss the structure of
the topologically ordered mixed state using a wave-function perspective. This
allows us to derive a duality transformation that facilitates the
characterization of the phase diagram of the mixed state as varying the input
state.

First, we express the input state in Pauli-X basis:

$\ket{\psi_{0}}=\sum^{\prime}_{\alpha,\beta}(-1)^{\chi(\alpha,\beta)}\psi(\alpha,\beta)\ket{\alpha,\beta}$
(44)

$\alpha=\\{\alpha_{v}\\}$ and $\beta=\\{\beta_{e}\\}$ denote the
configurations in $X$ basis for qubits on vertices and edges, respectively.
$\sum_{\alpha,\beta}^{\prime}$ denotes summing all symmetry-allowed
$\alpha,\beta$ configurations: the global symmetry on vertices implies
$\alpha_{v}=-1$ occurs in pairs, so every allowed $\alpha$ configuration
corresponds to the configuration of open strings whose end-points label the
location of $\alpha_{v}=-1$. On the other hand, the 1-form symmetry on edges
indicates the edge with $\beta_{e}=-1$ must form a closed loop in the dual
lattice, which equivalently can be regarded as the boundary of an open
membrane. $\chi(\alpha,\beta)$ is the number of times that the
$\alpha$-strings pierce through the $\beta$-membranes. When $\ket{\psi_{0}}$
is a fixed-point SPT, $\psi(\alpha,\beta)$ is independent of $\alpha,\beta$
(see Fig.10).

Figure 10: Wave function of the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ fixed-
point SPT, i.e. the ground state of $H_{0}=-\sum_{v}X_{v}\prod_{e\ni
v}Z_{e}-\sum_{e}X_{e}\prod_{v\in e}Z_{v}$. The fixed-point state is understood
as a condensate of open blue strings and open red membranes with a non-trivial
braiding sign structure in Pauli-X basis. The wave function
$(-1)^{\chi(\alpha,\beta)}$ takes the value 1 (-1) corresponding to the even
(odd) number of times that blue strings pierce through red membranes.

With the input state $\ket{\psi_{0}}$, measuring vertices ($A$ sublattice)
projects to a particular $\alpha$ configuration, and the follow-up unitary
correction $U_{\alpha}$ removes the braiding phase. One then finds the
measurement-feedback protocol leads to a mixed state $\rho_{B}$ on the edges
($B$ sublattice), which is the reduced density matrix (by tracing out $A$
sublattice) of the following state:

$\ket{\psi}=\sum^{\prime}_{\alpha,\beta}\psi(\alpha,\beta)\ket{\alpha,\beta}.$
(45)

Therefore, the input state and the purification of the output $\rho_{B}$ can
be connected through $\ket{\psi}=U\ket{\psi_{0}}$, where
$U=\sum^{\prime}_{\alpha,\beta}\ket{\alpha,\beta}\bra{\alpha,\beta}(-1)^{\chi(\alpha,\beta)}$
is unitary in the symmetric subspace. Under the conjugation of $U$, operators
transform as

$\begin{split}X_{v}\to X_{v},&\quad X_{e}\to X_{e},\\\ \prod_{e\ni v}Z_{e}\to
X_{v}\prod_{e\ni v}Z_{e},&\quad\prod_{v\in e}Z_{v}\to X_{e}\prod_{v\in
e}Z_{v}.\end{split}$ (46)

This allows us to derive the parent Hamiltonian $H$ of $\ket{\psi}$ from the
Hamiltonian $H_{0}$ of the input state via $H=UH_{0}U^{\dagger}$. Also, note
that $\ket{\psi}$ lives in the subspace given by 0-form symmetry
$\prod_{v}X_{v}=1$ and $\mathbb{Z}_{2}$ 1-form symmetry
$\prod_{e\in\gamma}X_{e}=1$ (acting on edges along any closed loop $\gamma$),
these two constraints need to be further imposed in $H$.

As an application, we may consider perturbing the fixed-point SPT by onsite
Pauli-Xs: $H_{0}=-\sum_{v}X_{v}\prod_{e\ni v}Z_{e}-\sum_{e}X_{e}\prod_{v\in
e}Z_{v}-g\sum_{e}X_{e}-g\sum_{v}X_{v}$, the corresponding $H$ is the decoupled
Ising model on vertices and 2d toric code, both of which are subject to the
onsite transverse field: $H=-\sum_{v}\prod_{e\ni v}Z_{e}-\sum_{e}\prod_{v\in
e}Z_{v}-g\sum_{e}X_{e}-g\sum_{v}X_{v}$, where the constraints
$\prod_{v}X_{v}=1$ and $\prod_{e\in\gamma}X_{e}=1$ are further imposed.
Therefore, the measurement-feedback protocol leads to a pure state $\rho_{B}$
on edges with $\mathbb{Z}_{2}$ topological order. This also indicates all
possible post-measurement states by measuring the input state SPT
$\ket{\psi_{0}}$ are deterministically converted to the same pure state with
topological order.

To discuss a non-trivial mixed state topological order on edges, we consider
the following form of input Hamiltonian:

$\begin{split}H_{0}=&-\sum_{v}X_{v}\prod_{e\ni
v}Z_{e}-\sum_{e}X_{e}\prod_{v\in e}Z_{v}\\\ &-g\sum_{v}\prod_{e\ni
v}Z_{e}-g\sum_{e}\prod_{v\in e}Z_{v}.\end{split}$ (47)

Using the transformation rule in Eq.46, one finds the corresponding $H$

$\begin{split}H=&-\sum_{v}\prod_{e\ni v}Z_{e}-\sum_{e}\prod_{v\in e}Z_{v}\\\
&-g\sum_{v}X_{v}\prod_{e\ni v}Z_{e}-g\sum_{e}X_{e}\prod_{v\in
e}Z_{v},\end{split}$ (48)

where the 0-form symmetry ($\prod_{v}X_{v}=1$) and 1-form symmetry
($\prod_{e\in\gamma}X_{e}=1$) are further imposed. To understand the phase
diagram for the mixed state $\rho_{B}$ defined on edges, we transform $H$ by
$U_{CZ}=\prod_{\expectationvalue{v,e}}{CZ}_{v,e}$. This leads to

$-\sum_{v}\prod_{e\ni v}Z_{e}-\sum_{e}\prod_{v\in
e}Z_{v}-g\sum_{v}X_{v}-g\sum_{e}X_{e},$ (49)

i.e., a transverse-field Ising model on vertices and transverse-field toric
code on edges. Their phase diagrams are well-understood: the global Ising
symmetry breaking order persists to $g=g_{\text{ising}}\approx 3.04438$ [74],
above which the state on vertices become trivial. On the other hand, using the
(Kramer-Wannier) duality between transverse-field Ising and transverse-field
toric code, the topological order persists up to
$g=g_{\text{toric}}=1/g_{\text{ising}}\approx 0.32847$, above which the state
on edges become trivial. As a result, starting with the ground state
$\ket{\psi_{0}}$ of Eq.47 for $g<g_{\text{toric}}$ (the regime where
$\ket{\psi_{0}}$ is an SPT), the measurement-feedback channel gives a mixed
state $\rho_{B}$ on edges with $\mathbb{Z}_{2}$ topological order. Similar to
the discussion in 1d, one expects $\rho_{B}$ to have volume-law entropy due to
the coupling between $A,B$ sublattices in Eq.48.

Finally, we note that mixed-state topological order encoded in $\rho_{B}$ may
be diagnosed by entanglement negativity, an entanglement measure for mixed
states. In particular, one expects a long-range entanglement structure that
manifests in the universal subleading contribution in negativity as discussed
in Ref.[4, 5, 11].

##  Appendix D Fermionic mixed state with Kitaev topological order in 1d

Here we discuss the finite-depth preparation of a mixed state in 1d that
exhibits a topological order of the Kitave Majorana chain [75, 76] by taking a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{f}$ SPT $\ket{\psi_{0}}$ [51, 52, 53, 54,
20] as an input. Here we closely follow Ref.[20] to construct the fixed-point
SPT, and then we will consider certain symmetric perturbation and apply
measurement and unitary feedback to prepare a non-trivial fermionic
topological mixed state.

To start, we consider a 1d lattice of size $2L$ initialized in a product
state, see Fig.11(a): every odd site accommodates a qubit initiated in the
$\ket{+}$ state, and every even site accommodates a fermion initiated in the
empty state, i.e. parity operator
$P=-i\gamma\gamma^{\prime}=1-2c^{\dagger}c=1$ with left Majorana and right
Majorana defined as $\gamma=c+c^{\dagger},\gamma^{\prime}=-i(c-c^{\dagger})$.
Define a Majorana hopping operator
$S_{2n}=i\gamma^{\prime}_{2n-2}\gamma_{2n}$, we define a controlled gate

$CS_{2n-1}=\ket{\uparrow}\bra{\uparrow}+\ket{\downarrow}\bra{\downarrow}S_{2n};$
(50)

the Majorana hops from the site $2n-2$ to site $2n$ only when the qubit at
$2n-1$ is in the spin-down state. Using $CS$ gate, one introduces a depth-1
local unitary circuit

$U_{\text{SPT}}=\prod_{n=1}^{L}CS_{2n-1}.$ (51)

Applying $U_{\text{SPT}}$ on the initial product state will then lead to the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{f}$ fixed-point SPT $\ket{\psi_{0}}$.
Equivalently, $\ket{\psi_{0}}$ is uniquely specified by the stabilizers

$\begin{split}U_{\text{SPT}}X_{2n-1}U_{\text{SPT}}^{\dagger}&=i\gamma_{2n-2}^{\prime}X_{2n-1}\gamma_{2n}\\\
U_{\text{SPT}}P_{2n}U_{\text{SPT}}^{\dagger}&=Z_{2n-1}P_{2n}Z_{2n+1}.\end{split}$
(52)

The SPT $\ket{\psi_{0}}$ is protected by the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{f}$ symmetry, where $\mathbb{Z}_{2}$
action is given by the product of $X_{n}$ on odd sites, and
$\mathbb{Z}^{f}_{2}$ action is given by the product of $P_{n}$ on even sites.
The state has long-range string order diagnosed by
$i\gamma_{2n-2}^{\prime}X_{2n-1}\gamma_{2n}i\gamma_{2n}^{\prime}X_{2n+1}\gamma_{2n+2}\cdots
i\gamma_{2m-2}^{\prime}X_{2m-1}\gamma_{2m}=1$ (this is simply the product of
stabilizers). Being a non-trivial SPT, the string order is robust under weak
symmetric perturbations, and it approaches a finite constant $c$ with $0<c<1$
in the limit $\absolutevalue{m-n}\to\infty$.

Figure 11:  (a) 1d lattice with every odd site accommodating a qubit and every
even site accommodating two Majorana fermions. Given an initial 0-form
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{f}$ SPT, measurement and unitary feedback
lead to a fermionic mixed state with the topological phase of the Kitaev
chain. (b) 2d lattice with every vertex accommodating a qubit and every edge
accommodating two Majorana fermions. Given an initial $\mathbb{Z}_{2}$ 0-form
$\times\mathbb{Z}_{2}^{f}$ 1-form SPT, measurement and unitary feedback leads
to a fermionic mixed state with $\mathbb{Z}_{2}$ topological order.

Given a non-fixed point SPT, as in the discussion for 1d
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ bosonic SPT, we measure Pauli-Xs on all
spins on odd sites and record the outcome $\alpha=\\{\alpha_{i}|\text{odd
}i\\}$. For each post-measurement state, one applies unitary feedback, i.e. a
product of on-site fermionic parity operators:

$U_{\alpha}=\prod_{n=1}^{L}P_{2n}^{\frac{1-\prod_{m=1,2,\cdots}^{n}\alpha_{2m-1}}{2}},$
(53)

namely, a parity operator at site $2n$ is applied when there is an odd number
of $-1$ outcomes among $\alpha_{1},\alpha_{3},...,\alpha_{2n-1}$. With this,
one obtains a mixed state of Majorana fermions with the long-range order
$i\gamma_{2n-2}^{\prime}\gamma_{2n}i\gamma_{2n}^{\prime}\gamma_{2n+2}\cdots
i\gamma_{2m-2}^{\prime}\gamma_{2m}=c$, which is a universal feature in the
topological phase of a 1d Kitaev chain [75].

Wave function perspective - It is useful to discuss the above result from the
wave function perspective. As in the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
fixed-point bosonic SPT, the $\mathbb{Z}_{2}\times\mathbb{Z}^{f}_{2}$ fixed-
point SPT can also be understood as the condensate of strings with a braiding
structure. Specifically, the fixed-point $\ket{\psi_{0}}$ can be written as

$\ket{\psi_{0}}=\sum^{\prime}_{\alpha,\beta}(-1)^{\chi(\alpha,\beta)}\ket{\alpha,\beta},$
(54)

where $\alpha=\\{\alpha_{i}\in\\{\pm 1\\}\\}$ denotes the product state in
Pauli-X basis for qubits and $\beta=\\{\beta_{i}\in\\{\pm 1\\}\\}$ denotes the
product state basis with definite fermion parity on even sites. The
$\mathbb{Z}_{2}$ symmetry and $\mathbb{Z}_{2}^{f}$ symmetry implies
$\prod_{\text{odd }i}\alpha_{i}=1$ and $\prod_{\text{even }i}\beta_{i}=1$
respectively. As $\alpha_{i}$ comes in pairs, an $\alpha$ configuration may be
represented by open strings ($A$-strings) whose boundary points label
$\alpha_{i}=-1$. Similarly one can define $B$-strings to represent product
state basis $\beta$ for fermions. $\chi(\alpha,\beta)$ is the number of times
that $A$-strings and $B$-strings intersect.

Away from the fixed-point, one can write
$\ket{\psi_{0}}=\sum^{\prime}_{\alpha,\beta}(-1)^{\chi(\alpha,\beta)}\psi(\alpha,\beta)\ket{\alpha,\beta}$
and the corresponding measurement-feedback channel gives the mixed states of
fermion $\rho_{B}$, which is the reduced density matrix of
$\ket{\psi}=\sum^{\prime}_{\alpha,\beta}\psi(\alpha,\beta)\ket{\alpha,\beta}$
by tracing out qubits on odd sites.

$\ket{\psi}$ and $\ket{\psi_{0}}$ can be connected by the unitary
$U=\sum^{\prime}_{\alpha,\beta}\ket{\alpha,\beta}\bra{\alpha,\beta}(-1)^{\chi(\alpha,\beta)}$101010$U$
is a unitary in the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{f}$ symmetric
subspace., which has the action that

$\begin{split}&X_{2n-1}\to X_{2n-1},\quad P_{2n}\to P_{2n},\\\
&Z_{2n-1}Z_{2n+1}\to Z_{2n-1}P_{2n}Z_{2n+1},\\\
&\quad\gamma^{\prime}_{2n-2}\gamma_{2n}\to\gamma^{\prime}_{2n-2}X_{2n-1}\gamma_{2n}.\end{split}$
(55)

As an application, we consider the initial Hamiltonian
$H_{0}=-\sum_{n}(i\gamma_{2n-2}^{\prime}X_{2n-1}\gamma_{2n}+Z_{2n-1}P_{2n}Z_{2n+1})-g\sum_{n}(i\gamma_{2n-2}^{\prime}\gamma_{2n}+Z_{2n-1}Z_{2n+1})$.
To understand the phase diagram, we may use the depth-1 unitary circuit
(Eq.51) and find
$UH_{0}U^{\dagger}=-\sum_{n}(X_{2n-1}+P_{2n})-g\sum_{n}(i\gamma_{2n-2}^{\prime}\gamma_{2n}+Z_{2n-1}Z_{2n+1})$,
which is a transverse-field Ising chain on odd sites, and fermionic Kitaev
chain on even sites. Consequently, for $\absolutevalue{g}<1$, both sublattices
are trivial, while for $\absolutevalue{g}>1$, odd sites exhibit a
$\mathbb{Z}_{2}$ symmetry-breaking order and even sites belong to the
topological phase of Kitaev chain. This implies the ground state of $H_{0}$
exhibits the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{f}$ SPT order for
$\absolutevalue{g}<1$, and for $\absolutevalue{g}>1$, odd sites exhibit
$\mathbb{Z}_{2}$ symmetry-breaking order and even sites belong to the
fermionic topological phase.

With the measurement-feedback channel, one obtains a state $\rho_{B}$ of
fermions, which is the reduced density matrix of the ground state $\ket{\psi}$
of $H$:

$\begin{split}H=&-\sum_{n}(i\gamma_{2n-2}^{\prime}\gamma_{2n}+Z_{2n-1}Z_{2n+1})\\\
&-g\sum_{n}(i\gamma_{2n-2}^{\prime}X_{2n-1}\gamma_{2n}+Z_{2n-1}P_{2n}Z_{2n+1}).\end{split}$
(56)

$H$ is exactly dual to $H_{0}$. As a result, the measurement-feedback channel
acting on the pure state of the SPT phase (Kitaev topological phase) leads to
the Kitaev topological phase (trivial phase) in the mixed state defined on
even sites. Note that for $\ket{\psi_{0}}$ belonging to the Kitaev topological
phase, $\ket{\psi}$ is a non-trivial SPT, but tracing out odd sites leads to a
trivial reduced density matrix on even sites.

##  Appendix E Fermionic mixed state with topological order in 2d

One may generalize the discussion above to higher space dimensions, which
allows for the existence of (non-invertible) topological order in fermionic
systems111111This is in contrast to invertible fermionic topological order in
Kitaev chain, which does not host fractionalized excitations in the bulk, and
can be trivialized by stacking another copy of Kitaev chain..

Consider a 2d lattice with every vertex accommodating a qubit and every edge
accommodating two Majorana fermions (see Fig.11(b)), we can construct a fixed-
point SPT with $\mathbb{Z}_{2}~{}$0-form$\times\mathbb{Z}_{2}^{f}~{}$1-form
symmetry as follows. We introduce Majorana hopping operators
$S_{v,\rightarrow}$ which take the form $i\gamma^{\prime}\gamma$, where
$\gamma^{\prime},\gamma$ are the Majoranas in the left and right of the vertex
$v$. Similarly, the operators $S_{v,\uparrow}$ take the form
$i\gamma^{\prime}\gamma$, where $\gamma^{\prime},\gamma$ are the Majoranas
right above and below the vertex $v$. Using the Majorana hopping operators,
one defines a controlled gate

$CS_{v}=\ket{\uparrow}\bra{\uparrow}+\ket{\downarrow}\bra{\downarrow}S_{v,\rightarrow}S_{v,\uparrow}.$
(57)

Starting with a product state, where qubits are in $\ket{+}$ and fermions are
in product states with definite fermion parity $P=1$, simultaneously applying
$CS_{v}$ on all vertices leads to the SPT $\ket{\psi_{0}}$, which is uniquely
specified by the stabilizers $ZP_{e}Z$ (the product of a fermion parity on the
edge $e$ and two Pauli-Z on two vertices on the boundary of $e$), and
$X_{2n-1}\gamma^{\prime}\gamma^{\prime}\gamma\gamma$ (i.e. the product of and
a Pauli-X at the vertex $v$ and four Majoranas around $v$ counter-clockwise).
The SPT is protected by a $\mathbb{Z}_{2}^{f}~{}$1-form symmetry ($\prod_{e\in
C}P_{e}$, i.e. the product of fermion parity along any loops) and
$\mathbb{Z}_{2}~{}$0-form symmetry ($\prod_{v}X_{v}$). In particular, the non-
trivialness of the SPT manifests in the membrane operator
$M_{\tilde{C}}=\prod_{e\in\tilde{C}}\tilde{\gamma}\prod_{v\in
A_{\tilde{C}}}X_{v}$ =1, where $\prod_{e\in\tilde{C}}\tilde{\gamma}$ is the
product of Majoranas ($\tilde{\gamma}$ could be $\gamma$ or $\gamma^{\prime}$)
along a loop in the dual lattice, and $\prod_{v\in A_{\tilde{C}}}X_{v}$ is the
product of $X_{v}$ in the area enclosed by the loop $\tilde{C}$. For instance,

$M_{\tilde{C}}=\includegraphics[width=76.82234pt,valign={c}]{fermion_membrane}.$
(58)

For non-fixed-point SPT, the membrane operator will survive in the form of a