# Mixed-state Quantum Phases: Renormalization and Quantum Error Correction

Shengqi Sang  Yijian Zou  Timothy H. Hsieh

###### Abstract

Open system quantum dynamics can generate a variety of long-range entangled
mixed states, yet it has been unclear in what sense they constitute phases of
matter. To establish that two mixed states are in the same phase, as defined
by their two-way connectivity via local quantum channels, we use the
renormalization group (RG) and decoders of quantum error correcting codes. We
introduce a real-space RG scheme for mixed states based on local channels
which ideally preserve correlations with the complementary system, and we
prove this is equivalent to the reversibility of the channel’s action. As an
application, we demonstrate an exact RG flow of finite temperature toric code
in two dimensions to infinite temperature, thus proving it is in the trivial
phase. In contrast, for toric code subject to local dephasing, we establish a
mixed state toric code phase using local channels obtained by truncating an
RG-type decoder and the minimum weight perfect matching decoder. We also
discover a precise relation between mixed state phase and decodability, by
proving that local noise acting on toric code cannot destroy logical
information without bringing the state out of the toric code phase.

###### Contents

  1. I Introduction
  2. II Local channel transformations and definition of mixed-state phases
    1. II.1 Local channel transformations
    2. II.2 Definition of mixed-state phase equivalence
  3. III Real-space RG of quantum mixed-states
    1. III.1 From pure-state RG to mixed-state RG
    2. III.2 Correlation-preserving map
    3. III.3 Ideal mixed-state RG
    4. III.4 From mixed-state RG to mixed-state quantum phases
  4. IV Overview of examples
  5. V Noisy GHZ states
    1. V.1 Bit-flip noise
    2. V.2 Phase-flip noise
  6. VI Thermal toric code state
    1. VI.1 Review of the toric code model
    2. VI.2 RG of the thermal toric code state
  7. VII Noisy toric code state
    1. VII.1 Logical information and long-range entanglement
    2. VII.2 RG of the dephased toric code state
    3. VII.3 Truncated minimal weight perfect matching channel
  8. VIII Discussion and outlook
  9. A Various short proofs
    1. A.1 Relation between pure- and mixed- state phase equivalence
    2. A.2 Derivation of Eq.(12)
    3. A.3 Derivation of Eq.(23)
    4. A.4 Derivation of Eq.(40)
    5. A.5 Derivation of Eq.(80)
  10. B Convergence of real-space RGs
    1. B.1 Classical statistical mechanics models
    2. B.2 Noisy GHZ state in Sec.V
    3. B.3 Thermal toric code state in Sec.VI.2
    4. B.4 Noisy toric code state in Sec.VII
  11. C RG of a mixed $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT state

##  I Introduction

Understanding quantum phases of matter is a central task of quantum many-body
physics. The traditional focus is on pure states, which are typically ground
states of local Hamiltonians. However, in many physical contexts ranging from
finite temperature systems to open system dynamics Diehl  _et al._ (2008);
Verstraete  _et al._ (2009), one is required to deal with mixed states.
Recently, in the context of non-equilibrium quantum simulators and computers,
there has been significant progress in constructing many different examples of
nontrivial mixed states from the effect of local decoherence on symmetry-
protected topological and long-range entangled pure states de Groot  _et al._
(2022); Ma and Wang (2023); Fan  _et al._ (2023); Bao  _et al._ (2023); Lee
_et al._ (2023); Zou  _et al._ (2023); Ma  _et al._ (2023); Wang  _et al._
(2023); Chen and Grover (2023) or from protocols involving measurement and
feedback Lu  _et al._ (2023); Zhu  _et al._ (2022); Lee  _et al._ (2022).

Given the increasing wealth of examples, it is thus desirable to have a
general framework of mixed-state phases and in particular a notion of
renormalization for distilling universal long-range properties of a phase.
Furthermore, in the class of mixed states obtained by decohering an error-
correcting code, there are remarkable instances Fan  _et al._ (2023) in which
mixed state entanglement measures undergo a transition at the same point at
which encoded information is lost. This motivates us to understand the precise
connection between mixed state phase transitions and error correction
thresholds. In this work we will address these questions.

One way of defining pure-state phases is via local unitary (LU) circuits Chen
_et al._ (2010): two states are in the same phase if there is a short-depth LU
circuit that connects them. This is based on the physical intuition that
phases should be defined by long-range properties and representatives only
differ in their local properties. For mixed states, an analogous definition
was proposed by Coser and Perez-Garcia Coser and Pérez-García (2019): two
mixed states $\rho_{1}$ and $\rho_{2}$ are in the same phase if there exists a
pair of short-time evolution with local Lindbladians from $\rho_{1}$ to
$\rho_{2}$ and from $\rho_{2}$ to $\rho_{1}$. The major difference with the LU
definition of pure state phases is that two-way connections are needed since
channels are not in general reversible. Another difference is that unlike pure
states of interest, there is often no notion of Hamiltonian, gap, or adiabatic
path to furnish the local transformations required to connect two mixed states
(see however Ref. Rakovszky  _et al._ (2023) for recent developments). These
make establishing the existence of a mixed state phase much more challenging.

We draw inspiration from the real-space renormalization group (RG), which has
played a major role in statistical mechanics and quantum many-body physics.
The idea dates back to Wilson Wilson (1975) and Kadanoff Kadanoff (1966) who
proposed that under block spin transformations, statistical mechanical systems
flow to fixed points whose properties are easier to characterize. In the
context of quantum many-body systems, real-space RG has led to the development
of powerful numerical algorithms, including density matrix renormalization
group (DMRG) White (1992), multiscale entanglement renormalization ansatz
(MERA) Vidal (2007), as well as theoretical tools, including matrix product
states (MPS) Fannes  _et al._ (1992), projected entangled pair states (PEPS)
Verstraete and Cirac (2004), etc. However, thus far, real-space RG has
predominantly been applied to coarse-grain pure quantum states.

In this work, we define a real-space RG scheme for mixed states involving
local channel (LC) transformations to establish the existence of mixed-state
phases. We define an “ideal” RG to consist of local channels acting on blocks
which preserve correlations between different blocks, and we prove that the
actions of such correlation-preserving channels can be reversed by another
channel, thus establishing the phase equivalence of the fine-grained and
coarse-grained states. As an example, we construct an ideal RG for the two-
dimensional toric code at finite temperature and show that the temperature
monotonically increases under coarse graining and thus the state does not
possess topological order.

We also consider mixed states obtained by applying local decoherence to
quantum error correction codes. There is a notion that the logical information
in topological codes is protected by long-range entanglement. With a
definition of mixed-state phases, we can make its relation to error correction
precise. We prove that short-range correlated noise (represented by a local
quantum channel) cannot destroy logical information without also transitioning
out of the mixed state topologically ordered phase. We illustrate these
connections in the example of toric code subject to local dephasing noise, for
which we demonstrate the existence of a toric code mixed state phase by
constructing (1) a real-space RG scheme based on the Harrington decoder
Harrington (2004) and (2) (quasi-)local channels based on truncating the
minimum weight perfect matching (MWPM) algorithm. Our local version of MWPM
can potentially be used for efficiently detecting the toric code mixed-state
phase in experiments Semeghini  _et al._ (2021); Satzinger  _et al._ (2021).

We mention that several prior related works Swingle and McGreevy (2016); Lin
_et al._ (2021) developed a mixed state RG scheme based on purification of the
mixed state, which is generally different from our scheme but in some cases
can furnish the local channels required in our scheme. Refs. de Alarcón  _et
al._ (2022); Cirac  _et al._ (2017) defined an RG fixed point condition for
one-dimensional matrix product density operators and related them to
boundaries of two-dimensional topological order. Refs. Lake  _et al._ (2022);
Cong  _et al._ (2022) demonstrated how quantum convolutional neural networks
Cong  _et al._ (2019) can furnish RG schemes for detecting non-trivial pure
state phases.

This paper is structured as follows. In Sec. II we define LC transformations
and mixed state phase equivalence. In Sec. III we formulate the real-space RG
for mixed-states and discuss its implications. In Secs. IV to VII we analyze
several examples, including the dephased GHZ state, thermal toric code, and
dephased toric code. In Sec. VII.1 we prove a relation between decodability
and mixed-state phases.

Figure 1:  (a) Definition of mixed-state phase equivalence adopted in this
work. Two many-body mixed states $\rho_{1}$ and $\rho_{2}$ are in the same
phase if there is a pair of low-depth spatially local quantum channels
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ such that
$\rho_{2}\approx\mathcal{C}_{1}(\rho_{1})$ and
$\rho_{1}\approx\mathcal{C}_{2}(\rho_{2})$. (b) Illustration of the
correlation-preserving criterion in Def.3. For a given bipartite mixed state
$\rho_{AB}$, a quantum channel $\mathcal{E}$ acting on one party is
correlation-preserving if it leaves the mutual information between two parties
invariant. Thm.2 shows that $\mathcal{E}$ is correlation-preserving if and
only if its action can be reversed by another channel $\mathcal{D}$. (c)
Mixed-state RG consists of local channels ($\mathcal{E}$s) which coarse-grain
degrees of freedom within a block. After iterating, all short-range
correlations of the input state are discarded and only long-range ones remain.
If all coarse-graining channels satisfy the correlation-preserving criterion,
then the whole RG process can be reversed, by running from top to bottom and
replacing each $\mathcal{E}$ with its recovery map $\mathcal{D}$. (d) Phase
diagrams and RG flows of 4 exemplary mixed states studied in the Sec.IV. All 4
states come from perturbing a long-range entangled pure state in an incoherent
way: In examples (i, ii) the pure state is the GHZ state, and in (iii, iv) it
is the toric code state. In examples (i, ii, iv) the incoherent perturbation
is a dephasing noise with strength $p$ acted upon the state, while in (iii)
the perturbation is a non-zero temperature. The mixed-state phase
corresponding to the GHZ state and the toric code state are denoted by [GHZ]
and [T.C.], respectively.

##  II Local channel transformations and definition of mixed-state phases

###  II.1 Local channel transformations

We define local channel transformations following the proposal in Hastings
(2011).

######  Definition 1 (Local channel (LC) transformation).

On a given lattice of linear dimension $L$, a range-$r$ LC transformation is a
quantum channel composed of the following steps:
(1) Adding qubits to each lattice site, all initialized in the $\ket{0}$
state;
(2) Applying a range-$r$ unitary circuit $U$ on the lattice;
(3) Tracing out some qubits on each lattice site.

The range of a circuit is defined as the maximal range of each unitary gate
times the depth of the circuit. Henceforth if $r$ is not specified for an LC
transformation, it is assumed that $r/L\rightarrow 0$ in the thermodynamic
limit.

The major difference between local channel transformations and local unitary
ones (LU) Chen  _et al._ (2010) is step (3). In local unitary transformations,
a qubit can be discarded only when it is disentangled from the rest of the
system. In that context, (1) and (3) are inverse operations, and hence LU
transformations are invertible. In contrast, LC transformations allow
discarding a qubit that is still entangled with the rest of the system, i.e.
$\rho_{i,\bar{i}}\neq\rho_{i}\otimes\rho_{\bar{i}}$ with $i$ being the qubit
to be discarded and $\bar{i}$ being the rest of the system. As a result, LC
transformations are generically non-invertible.

Figure 2: A circuit of local channel gates represented as an LC
transformation.

LC transformations constitute a broad class of operations including any
circuit composed of local channel gates, i.e. channels that only act on local
domains of sites. To show this, one needs the Stinespring dilation theorem:
any quantum channel $\mathcal{E}_{X\rightarrow Y}$ can be rewritten as:

$\mathcal{E}(\cdot)=\text{tr}_{A^{\prime}}\left(U((\cdot)\otimes\ket{0}\bra{0}_{A})U^{\dagger}\right)$
(1)

where $U$ is a unitary map from $X\cup A$ to $Y\cup A^{\prime}$. In other
words, any quantum channel can be implemented by adding some degrees of
freedom, applying a unitary on the joint system, and discarding some degrees
of freedom. Applying the theorem to a circuit of channel gates, one can first
replace each channel gate with its Stinesping dilation form. Then one can move
forward all the ancillae addition to the beginning of the circuit and postpone
all the tracing-out to the end of the circuit. We graphically illustrate this
in Fig.2. Furthermore, any finite-time local Lindbladian evolution can also be
approximated by LC transformations by trotterizing the continuous dynamics.

###  II.2 Definition of mixed-state phase equivalence

When studying (pure) ground states of gapped local Hamiltonians, two many-body
states are defined to be in the same phase if one can be turned into the other
through a LU transformation Chen  _et al._ (2010). The definition reflects the
idea that phases of matter should be characterized by long-range properties of
the state, and should remain unchanged under reversible local modifications.

LC transformations, albeit local, are generally not reversible and can destroy
long-range correlations. As an example, if one starts from an arbitrary state
$\ket{\psi}$ and applies the amplitude damping channel $\mathcal{E}_{\rm
damping}(\cdot):=\text{tr}(\cdot)\ket{0}\bra{0}$ to each qubit in the system,
the resulting state would be a product state $\ket{0}^{\otimes L}$ without any
non-trivial long-range correlation. On the other hand, an LC transformation’s
ability to create correlations is no stronger than LU ones. This follows from
the fact that an LC transformation is some LU followed by discarding some
degrees of freedom.

Thus the connectivity under LC transformations induces a partial order
relation among mixed-states. States are ordered according to the amount of
long-range correlation they possess: if $\rho_{2}=\mathcal{C}(\rho_{1})$ for
some LC transformation $C$, then $\rho_{1}$ has at least as much long-range
correlation as $\rho_{2}$. This naturally leads to the following definition of
mixed-state phase equivalence 111The definition resembles the one taken in
Coser and Pérez-García (2019), where a pair of LC transformations is replaced
by a pair of (quasi-)local Lindbladian evolutions. A similar definition also
appears in Ma and Wang (2023) when defining mixed-state symmetry-protected
topological orders.

######  Definition 2 (Mixed-state phase equivalence).

On a given lattice, two many-body mixed states $\rho_{1}$ and $\rho_{2}$ are
in the same phase if there exists a pair of LC transformations $C_{1}$ and
$C_{2}$ such that $C_{1}(\rho_{1})\approx\rho_{2}$ and
$C_{2}(\rho_{2})\approx\rho_{1}$.

Several clarifications regarding the definition:

  * •

Mixed-states of interest: Though the definition above does not assume any
restrictions on states $\rho_{1,2}$, we are interested in physically relevant
mixed states such as local Hamiltonian Gibbs states at finite temperature,
gapped ground states subject to decoherence, and steady states of local
Lindbladians.

  * •

The precise meaning of ‘$\approx$’: This requires some distance measure of
mixed states. For instance, we could define two mixed states
$\rho\approx\sigma$ if and only if $F(\sigma,\rho)>1-\epsilon$ for some small
$\epsilon>0$, where $F(\sigma,\rho):=||\sqrt{\sigma}\sqrt{\rho}||_{1}$ is the
(Uhlmann) fidelity.

  * •

Ranges of LC transformations $C_{1,2}$: In general we only require the range
to be much smaller than the linear size of the lattice $\rho_{1,2}$ is defined
on. But as we will see later, it will be sufficient to have a range $r=O({\rm
polylog}(L/\epsilon))$ when $\rho_{1,2}$ have finite correlation length.

The definition is a natural generalization of pure-state phase equivalence
defined through LU transformations. When restricting to pure many-body states,
one can show that two states $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ are of the
same mixed-state phase if and only if $\ket{\psi_{1}}$ and
$\ket{\psi_{2}}\otimes\ket{\phi}$ are of the same pure-state phase for some
invertible state $\ket{\phi}$. We provide a proof in App.A.1.

Product states, e.g. $\ket{0}^{\otimes L}$, are states without any long-range
correlations. This is also reflected by the partial order relation under LC
circuits: any state can be turned into the product state by a LC
transformation which only consists of the amplitude damping channel. Thus we
identify the trivial phase as the set of states that can be LC transformed
from the product state. In other words, a mixed state is in the trivial phase
if it can be written as $\rho_{\rm trivial}=\mathcal{C}\left[\ket{0}^{\otimes
L}\bra{0}^{\otimes L}\right]$ for some LC transformation $\mathcal{C}$. This
is equivalent to requiring that the state can be locally purified into a
short-range entangled pure state.

We comment that the above definition treats quantum and classical correlations
on the same footing. As an example, the state
$\rho=\frac{1}{2}(\ket{0^{\otimes L}}\bra{0^{\otimes L}}+\ket{1^{\otimes
L}}\bra{1^{\otimes L}})$ is a classical ensemble of $L$ spins which is non-
trivial under the above definition, because it has classical long-range
correlation. To single out states that contain long-range classical
correlation only, we can define a state to be in a classical phase if it can
be written as $\rho_{\rm
classical}=\mathcal{C}(\rho_{\mathrm{Pr}(\textbf{s}))})$ for some LC
transformation $\mathcal{C}$. Here
$\rho_{\mathrm{Pr}(s)}:=\sum_{\textbf{s}}\mathrm{Pr}(\textbf{s})\ket{\textbf{s}}\bra{\textbf{s}}$
is a classical distribution $\mathrm{Pr}(\textbf{s})$ of product states
$\\{\ket{\textbf{s}}:\textbf{s}\in\\{0,1\\}^{L}\\}$ represented as a density
matrix.

##  III Real-space RG of quantum mixed-states

To answer whether two given states $\rho_{1}$ and $\rho_{2}$ are in the same
phase, we need to either construct a pair of local channel transformations or
prove their nonexistence.

Recall that when studying pure-state phases of ground states, adiabatic paths
between Hamiltonians provide a convenient way of obtaining phase equivalence.
Let $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ be ground-states of local
Hamiltonians $H_{1}$ and $H_{2}$. If there is a path from $H_{1}$ to $H_{2}$
in the space of local Hamiltonians such that the energy gap remains $O(1)$
throughout the path, there is a standard way to construct a LU transformation
connecting $\ket{\psi_{1}}$ to $\ket{\psi_{2}}$ which establishes the phase
equivalence Hastings and Wen (2005). For mixed states, there is generally no
counterpart to adiabatic paths.

In this section, we introduce the mixed-state real-space renormalization group
(RG) as an alternative way to find LC connections and identify mixed-state
phases.

###  III.1 From pure-state RG to mixed-state RG

Conceptually, RG transformation in classical and quantum statistical mechanics
is an iterative coarse-graining process that discards short-range degrees of
freedom while preserving long-range ones. The idea of using real-space
renormalization to study zero-temperature physics of lattice quantum systems
(‘numerical RG’) was pioneered by Wilson when considering impurity problems
Wilson (1975), and was later generalized and developed into a series of
powerful RG-based numerical methods including DMRG White (1992), entanglement
RG Vidal (2007), etc. We refer to all of them as pure-state RGs, in contrast
to the mixed-state RG we introduce in this work. 222Another aspect of pure-
state RG is that it preserves the low-energy physics of the system. But here
we emphasize this less because for mixed-states, there may not be a notion of
energy or Hamiltonian.

For the sake of presentation, we restrict our attention to one-dimensional
systems and only focus on tree-like RG circuits (see Fig. 3). All the main
ideas can be easily generalized to more sophisticated RG circuit structures,
e.g. entanglement RG circuits Vidal (2007), as well as higher dimensional
systems.

Figure 3:  Real-space RG transformation of pure states– Circuit representation
of two iterations of pure state RG transformation. At the $\ell$-th iteration,
the coarse-graining isometry $w^{(\ell)}$ is determined by the level’s input
state $\ket{\psi^{(\ell)}}$ using Eq.(2). By applying the circuit from bottom
to top (red arrows), all the short-range features of the initial UV state are
gradually discarded, and only long-range ones are kept in the IR state
$\rho^{(\ell\rightarrow\infty)}$. By applying the circuit from top to bottom
(blue arrows), the circuit generates the UV state $\ket{\psi^{(1)}}$.

Pure-state RG, in its simplest form, involves partitioning the lattice into
consecutive blocks each with size $b$ and applying a coarse-graining map
$w^{\dagger}_{B}$ to each block $B$ of a pure state $\ket{\psi}$. More
specfically, coarse-graining involves truncating the Hilbert space, and
$w_{B}$ is an isometry satsifying $w^{\dagger}_{B}w_{B}=\mathbb{I}$. As
proposed White (1992) for the density matrix renormalization group (DMRG)
algorithm, the optimal choice of $w_{B}$ that preserves all correlations
between $B$ and its complement is given by

$\mathsf{supp}~{}w_{B}w^{\dagger}_{B}=\mathsf{supp}~{}\rho_{B}$ (2)

where $\rho_{B}:=\text{tr}_{\bar{B}}(\ket{\psi}\bra{\psi})$ is the reduced
density matrix of the block $B$. $\mathsf{supp}~{}K$ of a positive semi-
definite matrix $K$ means the subspace spanned by $K$’s eigenstates with
positive eigenvalues. If the original state $\ket{\psi}$ has area law
entanglement $S_{B}\equiv-\text{tr}[\rho_{B}\log\rho_{B}]=O(1)$, then each
block is efficiently coarse-grained into a constant dimensional Hilbert space
independent of the original block size 333Rigorously speaking, it is only
proven that the ground state of a gapped local Hamiltonian satisying the area
law can be represented as an MPS with a bond dimension that grows sublinearly
with the system size Arad  _et al._ (2013). However, in pratice, it is usually
true that a finite bond dimension suffices to reproduce an accurate
wavefunction for arbitrary large (even infinite) system sizes. .

Now we turn to 1D mixed-states. In contrast to the pure state case, physical
mixed states (e.g. ones mentioned below Def.2) typically have volume-law
scaling of $S_{B}$, leading to inefficient compression using the $w_{B}$
selected according to Eq.(2). This is because $S_{B}$ results from not only
correlations between $B$ and the complementary system $\bar{B}$ but also
between $B$ and a purifying environment $E$ of the mixed state. The latter is
the non-universal information that should be discarded. We thus need a new
criterion for finding the coarse-graining map.

To motivate the criterion we introduce, we observe that Eq.(2) can be
interpreted as the solution to the optimization problem:

$\displaystyle\mathrm{argmin}_{w_{B}}\,\dim_{\rm out}(w^{\dagger}_{B})$ (3)
$\displaystyle s.t.{}{}{}$ $\displaystyle
I_{B:\bar{B}}(w_{B}^{\dagger}\ket{\psi})=I_{B:\bar{B}}(\ket{\psi}),$

where $\dim_{\rm out}(w^{\dagger}_{B})$ is the output dimension of
$w^{\dagger}_{B}$ and $I_{X:Y}:=S_{X}+S_{Y}-S_{XY}$ is the quantum mutual
information, a measure of correlations between two parties $X$ and $Y$. The
constraint has clear physical meaning in the context of RG: by preserving
$I_{B:\bar{B}}$, it preserves all the long-distance correlation within
$\ket{\psi}$. We thus use ‘$I_{B:\bar{B}}$ preserving’ condition as a
guideline to generalize Eq.(2) to mixed-states.

###  III.2 Correlation-preserving map

We make the argument above more precise:

######  Definition 3 (correlation-preserving maps).

For a given bipartite quantum state $\rho=\rho_{AB}$, a quantum channel
$\mathcal{E}_{A\rightarrow A^{\prime}}$ acting on $A$ is correlation-
preserving with respect to $\rho_{AB}$ if it satisfies

$I_{A^{\prime}:B}(\mathcal{E}_{A\rightarrow A^{\prime}}(\rho))=I_{A:B}(\rho)$

It is worth noting that a channel being correlation-preserving or not depends
on both the input state and the bipartition: the same map $\mathcal{E}$ that
is correlation-preserving with respect to one $(\rho,B)$ pair may not be so
with respect to another pair.

Recalling that the motivation for defining an RG scheme is to establish
equivalence between two mixed states by finding a local channel transformation
and its inverse, ideally we would like $\mathcal{E}$’s action on $\rho$ to be
reversible. Conveniently, the two desired properties (correlation-preserving
and reversibility) are equivalent, as we prove in the following theorem.

######  Theorem 1.

For a given bipartite quantum state $\rho=\rho_{AB}$, the map
$\mathcal{E}_{A\rightarrow A^{\prime}}$ is correlation-preserving if and only
if there exists another quantum channel $\mathcal{D}_{A^{\prime}\rightarrow
A}$, such that:

$\rho=\mathcal{D}_{A^{\prime}\rightarrow A}\circ\mathcal{E}_{A\rightarrow
A^{\prime}}(\rho)$

###### Proof.

(reversibility $\Rightarrow$ correlation-preserving) According to the quantum
data processing inequality, a channel acting only on $A$ cannot increase
correlations between $A$ and $B$:

$I_{A:B}(\rho)\geq I_{A^{\prime}:B}(\mathcal{E}(\rho))\geq
I_{A:B}(\mathcal{D}\circ\mathcal{E}(\rho))=I_{A:B}(\rho)$ (4)

Thus $I_{A:B}(\rho)=I_{A^{\prime}:B}(\mathcal{E}(\rho))$.

(correlation-preserving $\Rightarrow$ reversibility) Let $W$ be an isometry
from $A$ to $A^{\prime}\cup E$ that dilates the channel
$\mathcal{E}_{A\rightarrow A^{\prime}}$:

$\mathcal{E}_{A\rightarrow
A^{\prime}}(\cdot):=\text{tr}_{E}\left[W(\cdot)W^{\dagger}\right],$ (5)

where $E$ is an ancillary system, and let $\sigma_{A^{\prime}EB}=W\rho
W^{\dagger}$. Then we have the following relation:

$I_{A:B}(\rho)=I_{A^{\prime}E:B}(\sigma_{A^{\prime}EB})=I_{A^{\prime}:B}(\sigma_{A^{\prime}EB}).$
(6)

The second equality, due to the correlation-preserving property, implies that
$I_{B:E|A^{\prime}}(\sigma_{A^{\prime}EB})=0$ and $B-A^{\prime}-E$ forms a
quantum Markov chain. Thus there is a channel
$\mathcal{T}_{A^{\prime}\rightarrow A^{\prime}E}$ that reconstructs
$\sigma_{A^{\prime}EB}$ from $\mathcal{E}_{A\rightarrow
A^{\prime}}(\rho)=\sigma_{A^{\prime}B}=\text{tr}_{E}\sigma_{A^{\prime}EB}$
alone:

$\mathcal{T}_{A^{\prime}\rightarrow
A^{\prime}E}(\sigma_{A^{\prime}B})=\sigma_{A^{\prime}EB}.$ (7)

The map $\mathcal{T}_{A^{\prime}\rightarrow A^{\prime}E}$ is the Petz recovery
map Petz (1988):

$\mathcal{T}_{A^{\prime}\rightarrow
A^{\prime}E}(\cdot):=\sigma_{A^{\prime}E}^{1/2}\left(\sigma^{-1/2}_{A^{\prime}}(\cdot)\sigma^{-1/2}_{A^{\prime}}\otimes\mathbb{I}_{E}\right)\sigma_{A^{\prime}E}^{1/2}$
(8)

We can then choose the inverse channel $\mathcal{D}$ to be

$\mathcal{D}_{A^{\prime}\rightarrow
A}(\cdot)=\text{tr}_{R}\left(U_{W}^{\dagger}\mathcal{T}_{A^{\prime}\rightarrow
A^{\prime}E}(\cdot)U_{W}\right),$ (9)

where $U_{W}:~{}A\cup R\rightarrow A^{\prime}\cup E$ is a unitary operator
that ‘completes’ the isometry $W:~{}A\rightarrow A^{\prime}\cup E$, namely:

$W(\cdot)W^{\dagger}=U_{W}\left((\cdot)\otimes\ket{0}_{R}\bra{0}\right)U^{\dagger}_{W}.$
(10)

∎

We remark that the relation between correlation-preserving and reversibility
is robust in one direction. More precisely, if the channel
$\mathcal{E}_{A\rightarrow A^{\prime}}$ almost preserves correlation

$I_{A:B}(\rho)-I_{A^{\prime}:B}(\mathcal{E}_{A\rightarrow
A^{\prime}}(\rho))=\epsilon,$ (11)

then there exists an almost perfect recovery channel
$\mathcal{D}_{A^{\prime}\rightarrow A}$ such that

$F(\rho,\mathcal{D}\circ\mathcal{E}(\rho))\geq 2^{-\epsilon/2}.$ (12)

A proof, based on approximate quantum Markov chains Fawzi and Renner (2015),
can be found in App.A.2. The robustness property is desirable especially when
we would like to numerically search for the correlation-preserving channel
$\mathcal{E}$.

When using $\mathcal{E}$ for the purpose of coarse-graining, the target space
of the channel should be as small as possible. This corresponds to solving the
following optimization problem:

$\displaystyle\mathrm{argmin}_{\mathcal{E}_{A\rightarrow
A^{\prime}}}\,\dim\mathcal{H}_{A^{\prime}}$ (13) $\displaystyle s.t.{}{}{}$
$\displaystyle
I_{A:\bar{A}}(\rho)-I_{A^{\prime}:\bar{A}}(\mathcal{E}(\rho))\leq\epsilon$

with $\epsilon$ taken to be a small number or zero. The problem is analogous
to Eq.(3) for pure states, which has Eq.(2) as an explicit solution. The
current problem, in contrast, has no known explicit solution. In fact, the
problem is closely related to the mixed-state quantum data compression
problem, which is under active exploration in quantum information theory. We
refer interested readers to Refs. Anshu  _et al._ (2021); Khanian and Winter
(2022); Khanian (2022); Kato (2023) for recent discussions on the problem.

To search for a good coarse-graining map for a given state, one can either
numerically solve the optimization problem Eq.(13) (in this case the
robustness property is crucial for the purpose of estimating error), or try to
construct the channel analytically by exploiting the special structure of the
given state, as we do later when studying examples in Sec.IV.

We point out that two familiar coarse-graining schemes in 1D, one for quantum
ground states and one for classical statistical mechanics models, are in fact
correlation-preserving maps. The first one is the Hilbert space truncation
reviewed in Sec.III.1 using the rule Eq.(2). Since this scheme preserves the
entropy of a block, it satisfies the correlation-preserving condition (for a
pure state $\ket{\psi_{AB}}$, $S_{A}=\frac{1}{2}I_{A:B}$). The other example
is Kadanoff’s block spin decimation of classical spin chains. Consider the
Gibbs state of a classical spin chain with nearest-neighbor interaction, but
written as a quantum mixed-state:

$\rho_{\beta}\propto\sum_{\textbf{s}=s_{0}...s_{L}}\exp\left(-\beta\sum_{i}h_{i}(s_{i},s_{i+1})\right)\ket{\textbf{s}}\bra{\textbf{s}}$
(14)

The state is classical because it is diagonal in the computational basis
$\ket{\textbf{s}}=\ket{s_{1}...s_{L}}$. For each block
$B=\\{i_{1},...,i_{b}\\}$, the block spin decimation corresponds to a quantum
channel that traces out all spins in $B$ other than $i_{1}$. This operation is
correlation-preserving with respect to $B^{\prime}:=B\cup\\{i_{b+1}\\}$,
because:

$I_{B^{\prime}:\overline{B^{\prime}}}(\rho)=I_{\\{i_{1},i_{b+1}\\}:\overline{B^{\prime}}}(\rho)$
(15)

which is a consequence of the Markov property of the Gibbs distribution.

###  III.3 Ideal mixed-state RG

In this section, we formulate an ideal real-space RG scheme built from local
correlation-preserving channels.

Assume $\rho$ is a many-body mixed-state on a lattice with linear size $L$. We
further assume that we have constructed, either numerically or analytically, a
series of coarse-graining transformations
$\\{\mathcal{C}^{(0)},\mathcal{C}^{(1)},...,\mathcal{C}^{(\ell)},...\\}$
acting on $\rho$ sequentially. In 1D, each $\mathcal{C}^{(\ell)}$ may have one
of the structures shown in Fig. 3, or any other structure as long as it is
composed of at most $O(1)$-layer of local channels. This leads to an ‘RG flow’
of mixed-states:

$\rho=\rho^{(0)}\xrightarrow{\mathcal{C}^{(0)}}\rho^{(1)}\xrightarrow{\mathcal{C}^{(1)}}...\xrightarrow{\mathcal{C}^{(\ell-1)}}\rho^{(\ell)}\xrightarrow{\mathcal{C}^{(\ell)}}...$
(16)

along which the level of coarse-graining increases gradually.

Each state $\rho^{(\ell)}$ is supported on a coarse-grained lattice
$\mathcal{L}^{(\ell)}$ with an $L^{(\ell)}:=L/b^{\ell}$ linear size. The chain
has a length at most $\sim\log_{b}L$, after which the state is supported on
$O(1)$ number of sites.

We call this RG process ideal if every channel gate $\mathcal{E}$ within each
$\mathcal{C}^{(\ell)}$ is correlation-preserving with respect to its input and
the prescribed bipartition.

As a direct consequence of Thm.1, ideal RG is reversible. More specifically,
there exists a series of local ‘fine-graining’ transformations
$\\{\mathcal{F}^{(0)},\mathcal{F}^{(1)},...,\mathcal{F}^{(\ell)},...\\}$ that
recovers the original mixed-state from its coarse-grained version by gradually
adding local details:

$\rho^{(0)}\xleftarrow{\mathcal{F}^{(0)}}\rho^{(1)}\xleftarrow{\mathcal{F}^{(1)}}...\xleftarrow{\mathcal{F}^{(\ell-1)}}\rho^{(\ell)}\xleftarrow{\mathcal{F}^{(\ell)}}...$
(17)

where each $\mathcal{F}^{(\ell)}$ is the ‘reversed’ channel of
$\mathcal{C}^{(\ell)}$, obtained by replacing each channel $\mathcal{E}$
within $\mathcal{F}^{(\ell)}$ by its corresponding recovery map $\mathcal{D}$
(see Thm.1). In graphical notation, if

$\mathcal{C}^{(\ell)}=\left(\vbox{\hbox{\includegraphics[height=22.76228pt]{fig/fig_eq_forward}}}\right)$
(18)

then

$\mathcal{F}^{(\ell)}:=\left(\vbox{\hbox{\includegraphics[height=22.76228pt]{fig/fig_eq_backward}}}\right)$
(19)

In both plots, the state ($\rho^{(\ell)}$ for $\mathcal{C}^{(\ell)}$ and
$\rho^{(\ell+1)}$ for $\mathcal{F}^{(\ell)}$), is inserted from the bottom.

Coarse-graining and fine-graining maps also establish relations between
operators at different coarse-graining levels. Let $O^{(\ell)}$ be any
operator (not necessarily local) defined on the lattice
$\mathcal{L}^{(\ell)}$. Then the following relation holds:

$\displaystyle\text{tr}\left(\rho^{(\ell)}O^{(\ell)}\right)$
$\displaystyle=\text{tr}\left(\mathcal{F}^{(\ell)}\circ\mathcal{C}^{(\ell)}(\rho^{(\ell)})O^{(\ell)}\right)$
(20)
$\displaystyle=\text{tr}\left(\rho^{(\ell+1)}\mathcal{F}^{\dagger(\ell)}(O^{(\ell)})\right),$

where $\mathcal{F}^{\dagger}$ is $\mathcal{F}$’s dual map, defined through the
relation
$\text{tr}(A\cdot\mathcal{F}(B))\equiv\text{tr}(\mathcal{F}^{\dagger}(A)\cdot
B)$. Thus we can define the coarse-grained operator of $O^{(\ell)}$ through:

$O^{(\ell+1)}:=\mathcal{F}^{\dagger(\ell)}(O^{(\ell)})$ (21)

so that:

$\langle{O^{(\ell)}}\rangle^{(\ell)}=\langle{O^{(\ell+1)}}\rangle^{(\ell+1)}$
(22)

where
$\langle{~{}\cdot~{}}\rangle^{(\ell)}:=\text{tr}(\rho^{(\ell)}(~{}\cdot~{}))$.

It is worth noting that $\mathcal{F}^{\dagger}=\mathcal{F}^{(\ell)\dagger}$
does not preserve operator multiplication: if
$O^{(\ell)}=o_{1}^{(\ell)}o_{2}^{(\ell)}$ then it is possible that
$\mathcal{F}^{\dagger}(O^{(\ell)})=O^{(\ell+1)}\neq
o_{1}^{(\ell+1)}o_{2}^{(\ell+1)}=\mathcal{F}^{\dagger}(o_{1}^{(\ell)})\mathcal{F}^{\dagger}(o_{2}^{(\ell)})$.
However, if two operators $o_{1}$ and $o_{2}$ are spatially well-separated,
then the multiplication is preserved:

$\mathcal{F}^{\dagger}(o_{1}^{(\ell)}o_{2}^{(\ell)})=\mathcal{F}^{\dagger}(o_{1}^{(\ell)})\mathcal{F}^{\dagger}(o_{2}^{(\ell)})$
(23)

Here ‘well-separated’ means that the lightcones for $o_{1}$ and $o_{2}$, as
determined by the circuit structure of $\mathcal{C}^{(\ell)}$ (or equivalently
that of $\mathcal{F}^{(\ell)\dagger}$), are non-overlapping. We illustrate the
definition of the lightcone and a proof of the above equation in App.A.3. This
guarantees that long-distance behavior of all the $k$-point functions are
preserved along an ideal RG:

$\langle o_{1}^{(\ell)}o_{2}^{(\ell)}...o_{k}^{(\ell)}\rangle^{(\ell)}=\langle
o_{1}^{(\ell+1)}o_{2}^{(\ell+1)}...o_{k}^{(\ell+1)}\rangle^{(\ell+1)}$ (24)

where $\\{o_{i}\\}$ are mutually well-separated local operators.

###  III.4 From mixed-state RG to mixed-state quantum phases

In this section we discuss how to use RGs, both ideal and non-ideal ones, to
furnish the two-way LC transformations required to establish the phase
equivalence of two mixed states (Def.2).

Assume that for the state of interest $\rho$, we have found a (not necessarily
ideal) real-space RG process
$\\{\mathcal{C}^{(1)},\mathcal{C}^{(2)},...,\mathcal{C}^{(\ell)},...\\}$. We
further assume that the RG has a well-defined fixed-point state
$\rho^{(\infty)}$, whose mixed-state phase of matter is presumably easy to
identify.

Intuitively, this RG can be treated as an LC transformation connecting $\rho$
to $\rho^{(\infty)}$ 444When discussing LC transformations in this section, we
fix the reference lattice as the one that $\rho=\rho^{(0)}$ is defined upon.
The renormalized state $\rho^{(\ell)}$ can be considered as supported on a
sub-lattice with $L/b^{\ell}$ sites.. To rigorously show this according to
Defs.1 and 2, one needs to show that the sequence $\\{\rho^{(\ell)}\\}$
converges toward $\rho^{(\infty)}$ fast enough. More concretely, we need to
show that there exists $\ell^{*}$ such that:

  1. 1.

The channel
$\mathcal{C}^{(\ell^{*})}\circ...\circ\mathcal{C}^{(2)}\circ\mathcal{C}^{(1)}$
is an LC transformation. Noticing that $\mathcal{C}^{(\ell)}$ entails
$b^{\ell}$ range of non-locality while an LC transformation can have at most
$o(L)$ range, the condition is equivalent to requiring $\ell^{*}\lesssim\log
L^{\alpha}$, for some $\alpha<1$.

  2. 2.

The state after $\ell^{*}$ iterations is close enough to $\rho^{(\infty)}$,
namely $F(\rho^{(\ell^{*})},\rho^{(\infty)})>1-\epsilon$ for a small
$\epsilon$.

We find that such an $\ell^{*}$ does exist in many cases when the fixed-point
state $\rho^{(\infty)}$ has a finite correlation length. More specifically, in
such cases, the fidelity function satisfies the form:

$F(\rho^{(\ell)},\rho^{(\infty)})\simeq\exp(-\alpha\
\theta^{(\ell)}L^{(\ell)})$ (25)

for some $\alpha=O(1)$ and a positive coefficient $\theta^{(\ell)}$. Further,
$\theta^{(\ell)}$ displays a power-law iteration relation under each coarse-
graining step:

$\theta^{(\ell+1)}\lesssim(\theta^{(\ell)})^{\gamma}~{}~{}~{}{\rm
when}~{}~{}~{}\theta^{(\ell)}\rightarrow 0_{+}$ (26)

for some coefficient $\gamma>1$.

As detailed in the App.B, Eqs.(25),(26) guarantee that choosing

$\ell^{*}\sim\log\log(L/\epsilon)$ (27)

is sufficient to have $F(\rho^{(\ell)},\rho^{(\infty)})>1-\epsilon$. We remark
that one can let $\epsilon$ be as small as $(\mathrm{poly}L)^{-1}$ but still
guarantee that $\ell^{*}$ steps of RG is a $({\rm polylog}L)$ -range LC
transformation.

In the App.B we show that conditions Eq.(25) and Eq.(26) hold for:

  * •

1D pure-state RG of a matrix product state

  * •

Gibbs state of a classical statistical mechanics model flowing toward a non-
critical fixed point

  * •

All examples we study in Sec.IV

So far in this section, we have shown that RG can be viewed as an LC
transformation connecting $\rho$ to $\rho^{(\infty)}$. Recalling that the
phase equivalence is defined through two-way LC connections, we have to find
another LC channel connecting $\rho^{(\infty)}$ to $\rho$ to conclude that the
two states are in the same phase.

If the RG is an ideal one, it is composed of correlation-preserving channels
and thus reversible. In this case, the other direction comes from the
‘reversed’ RG process ${\rm
RG}^{-1}=\\{\mathcal{F}^{(1)},\mathcal{F}^{(2)},...\mathcal{F}^{(\ell)},...\\}$
which we discussed in Sec.III.3. Similar to the forward RG, there is the issue
of convergence concerning whether ${\rm RG}^{-1}$ can be treated as an LC
transformation. But the discussion is completely parallel to the one for the
forward RG. Thus in this case, the LC bi-connection is established as

$\rho~{}\xrightarrow{{\rm RG}}~{}\rho^{(\infty)}~{}\xrightarrow{{\rm
RG}^{-1}}~{}\rho$ (28)

and we can conclude $\rho$ and $\rho^{(\infty)}$ are in the same phase.

Next we discuss what we can learn from a non-ideal RG. One class of mixed
states of significant interest is a long-range entangled pure-state
$\ket{\psi}$ subject to local decoherence represented by an LC transformation,
and an important question is whether or not the decohered state in the same
phase as $\ket{\psi}$. In this setting, one direction of the connection is
already given by the decoherence. Therefore, if the RG (ideal or not) has
$\ket{\psi}$ as the fixed point:

$\ket{\psi}\xrightarrow{\rm decohere}\rho\xrightarrow{\rm
RG}\rho^{(\infty)}=\ket{\psi}\bra{\psi},$ (29)

then an LC bi-connection is established and $\rho$ and $\ket{\psi}$ are in the
same phase. But on the other hand, if the fixed-point is not in the same phase
as $\ket{\psi}$, then we cannot determine $\rho$’s phase because no bi-
connection is identified.

##  IV Overview of examples

In the remaining sections we use our formalism to understand the quantum
phases of several many-body mixed-states of recent interest.

In all the examples, the mixed-state is obtained by ‘perturbing’ a long-range
entangled pure state, either through incoherent noise or finite temperature.
The question we address is whether the states before and after the
perturbation are in the same phase.

The long-range entangled pure state is chosen to be either the
Greenberger–Horne–Zeilinger (GHZ) state or Kitaev’s toric code state. In most
examples, the LC circuits for identifying phases take the form of RG. The
coarse-graining maps therein are either constructed according to the
correlation-preserving criterion (Def.3), or inspired by decoders of quantum
error correcting codes.

In the App.C we include an example of a mixed symmetry-protected topological
(SPT) state and its associated mixed-state RG.

##  V Noisy GHZ states

The many-body GHZ state, defined as

$\ket{\mathrm{GHZ}_{L}}:=\frac{1}{\sqrt{2}}\left(\ket{0^{\otimes
L}}+\ket{1^{\otimes L}}\right),$ (30)

has long-range entanglement, i.e. it can not be generated from a product state
using any one-dimensional LU (or LC) transformation from a product state. In
this section, we study the effect of dephasing noise on this state.

For convenience of analysis, we let $L=b^{\ell_{\rm max}}$ for some integer
$\ell_{\rm max}$ and an odd integer $b$. The state can be rewritten as:

$\ket{\mathrm{GHZ}_{L}}=w_{b}^{\otimes(\ell_{\rm max}-1)}\cdot
w_{b}^{\otimes(\ell_{\rm max}-2)}...w_{b}^{\otimes 1}\ket{+}$ (31)

where $w_{b}=\ket{0^{\otimes b}}\bra{0}+\ket{1^{\otimes b}}\bra{1}$ is an
isometry and $\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$. This provides a
tree tensor network representation of the state (see Fig.4), as well as a way
of blocking sites when performing RG.

Figure 4: Tree tensor network of a GHZ state with $L=b^{\ell}=9$, $b=3$,
$\ell=2$. Each triangle represents an isometry $w$ (Eq.(31)). By replacing the
state at the top with a generic single qubit state $\ket{\psi}$, the same
tensor network encodes $\ket{\psi}$ into a codeword state of the quantum
repetition code.

We consider a setting in which each qubit experiences the same noise, as
modeled by a single qubit channel $\mathcal{N}$, resulting in the mixed state

$\rho_{L}:=\mathcal{N}^{\otimes
L}(\ket{\mathrm{GHZ}_{L}}\bra{\mathrm{GHZ}_{L}})\\\ $ (32)

We remark that the GHZ state is closely related to the quantum repetition
code, whose codespace is spanned by $\ket{0^{\otimes L}}$ and $\ket{1^{\otimes
L}}$. It is known that quantum information stored in a quantum repetition code
is robust against bit-flip noise ($X$ dephasing noise), but not phase-flip
noise ($Z$ dephasing noise). As we will see in this section, the robustness of
the GHZ state’s long-range entanglement has a parallel behavior when it is
subjected to these two types of noise. We postpone a detailed discussion of
the relation between mixed-state phases and quantum coding properties to
Sec.VII.

###  V.1 Bit-flip noise

We first consider dephasing each qubit in the $X$ direction:

$\mathcal{N}(\cdot)=\mathcal{N}^{X}_{p}(\cdot):=(1-p)(\cdot)+pX(\cdot)X,$ (33)

in which each qubit is flipped with probability $p$.

The resulting state is

$\rho^{X}_{p,L}=\frac{1}{2}\sum_{\textbf{s}}p^{|\textbf{s}|}(1-p)^{L-|\textbf{s}|}(\ket{\textbf{s}}\bra{\textbf{s}}+\ket{\textbf{s}}\bra{\bar{\textbf{s}}}+\ket{\bar{\textbf{s}}}\bra{\textbf{s}}+\ket{\bar{\textbf{s}}}\bra{\bar{\textbf{s}}})$
(34)

where $|\textbf{s}|:=\sum_{i}s_{i}$ is the number of $1$ in the bitstring s,
and $\bar{\textbf{s}}$ is the bitwise complement of s . Since
$\rho^{X}_{p}=\rho^{X}_{1-p}$, we only consider $p\in(0,0.5]$.

Inspired by decoders for the quantum repetition code, we use the $b$-qubit
majority-vote channel as the coarse-graining map. To define it, we first
introduce the unitary operator that re-parametrizes the bitstring:

$U\ket{\textbf{s}}:=\ket{\mathsf{maj}(\textbf{s})}\otimes\ket{\mathsf{diff}(\textbf{s})}$
(35)

where $\mathsf{maj}(\textbf{s})$ takes the majority vote of the bits within s:

$\mathsf{maj}(\textbf{s}):=\left\\{\begin{array}[]{ll}0&{\rm
if}~{}~{}|\textbf{s}|<b/2\\\ 1&{\rm if}~{}~{}|\textbf{s}|\geq
b/2\end{array}\right.$ (36)

and $\mathsf{diff}(\textbf{s})$ is a length $(b-1)$ bitstring that records
pairwise difference of s:

$\mathsf{diff}(\textbf{s})_{i}:=(s_{i+1}-s_{i})~{}\textrm{mod}~{}2~{}~{}~{}i=1,2,...,b-1$
(37)

Then the majority vote channel can be written as:

$\mathcal{E}_{b}(\cdot):=\text{tr}_{2}(U(\cdot)U^{\dagger})$ (38)

where $\text{tr}_{2}$ denotes tracing out the pairwise difference information,
regarded as unimportant short-distance degrees of freedom in the current
example.

We inspect the state’s RG flow under $\mathcal{E}_{b}$, the coarse-graining
map:

$\displaystyle\mathcal{E}_{b}^{\otimes{L/b}}(\rho^{X}_{p,L})$ (39)
$\displaystyle=$
$\displaystyle\mathcal{E}_{b}^{\otimes{L/b}}\circ(\mathcal{N}^{X}_{p})^{\otimes
L}(\ket{\mathrm{GHZ}_{L}}\bra{\mathrm{GHZ}_{L}})$ $\displaystyle=$
$\displaystyle\left(\mathcal{E}_{b}\circ(\mathcal{N}^{X}_{p})^{\otimes
b}\circ\mathcal{U}_{w_{b}}\right)^{\otimes
L/b}\left(\ket{\mathrm{GHZ}_{L/b}}\bra{\mathrm{GHZ}_{L/b}}\right)$

where $\mathcal{U}_{w_{b}}(\cdot):=w_{b}(\cdot)w_{b}^{\dagger}$ and we applied
the relation $\ket{\mathrm{GHZ}_{L}}=w_{b}^{\otimes
L/b}\ket{\mathrm{GHZ}_{L/b}}$. Thus after one iteration of coarse-graining,
the resulting state is a GHZ state with $1/b$ of the original size subject to
a ‘renormalized’ noise channel, which is still $X$-dephasing (see App.A.4 for
a derivation):

$\mathcal{E}_{b}\circ(\mathcal{N}_{p}^{X})^{\otimes
b}\circ\mathcal{U}_{w_{b}}=\mathcal{N}_{p^{\prime}}^{X},$ (40)

but with a renormalized noise strength
$p^{\prime}=\sum_{k=(b+1)/2}^{b}\binom{b}{k}p^{k}(1-p)^{b-k}$ .

Thus we obtain an exact description of the state’s RG flow:

$\rho^{(\ell)}=(\mathcal{N}^{X}_{p^{(\ell)}})^{\otimes
L^{(\ell)}}(\ket{\mathrm{GHZ}_{L^{(\ell)}}}\bra{\mathrm{GHZ}_{L^{(\ell)}}})$
(41)

where $L^{(\ell)}=Lb^{-\ell}$ is the renormalized system size at the $\ell$-th
iteration, and
$p^{(\ell+1)}=\sum_{k=(b+1)/2}^{b}\binom{b}{k}(p^{(\ell)})^{k}(1-p^{(\ell)})^{b-k}$.
It is straightforward to check that $p=0$ and $p=1/2$ are the two fixed points
of the RG transformation.

Around $p=1/2$, the iteration relation has the asymptotic behavior:

$(p^{\prime}-1/2)\simeq g(b)(p-1/2)$ (42)

where $g(b):=\sum_{k=(b+1)/2}^{b}4k\binom{b}{k}>1$. Thus this is an unstable
fixed point. Exactly at $p=1/2$, the fixed-point state is:

$\rho^{X}_{1/2,~{}L}=\frac{1}{2^{L}}\sum_{\textbf{s}\in\\{0,1\\}^{L}}\left(\ket{\textbf{s}}\bra{\textbf{s}}+\ket{\textbf{s}}\bra{\bar{\textbf{s}}}\right).$
(43)

The state is better understood in the eigenbasis of Pauli $X$ operators, i.e.
$\\{\ket{0^{X}}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1}),$
$\ket{1^{X}}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})\\}$:

$\rho^{X}_{1/2,~{}L}=\frac{1}{2^{L-1}}\sum_{\textbf{s}\in\\{0,1\\}^{L}}\ket{\textbf{s}^{X}}\bra{\textbf{s}^{X}}~{}\delta(|\textbf{s}|=0~{}\textrm{mod}~{}2)$
(44)

In this basis, the state is a uniform distribution of bitstrings with even
parity. Since it is diagonal in this basis, the state is a classical state
(recall the definition in Sec.II) and is not in the same phase as the GHZ
state 555In fact, the state $\rho^{X}_{1/2,L}$ is in the same phase as the
product state. One may prepare the state using LC by first preparing a one
dimensional cluster state of $2L$ spins and then tracing out qubits on all odd
sites..

On the other hand, $p=0$ is a stable fixed-point attracting $p\in[0,0.5)$.
Starting from any state in the interval, the RG process gradually removes
entropy within the state and brings it back to the noiseless state at $p=0$,
i.e. $\ket{\mathrm{GHZ}}$. We thus obtain the following LC transformation bi-
connection:

$\ket{\mathrm{GHZ}}~{}\xrightarrow{\mathcal{N}_{X}}~{}\rho^{X}_{p}~{}\xrightarrow{\rm
RG}~{}\ket{\mathrm{GHZ}}\quad\quad p\in[0,0.5)$ (45)

Thus we can conclude $\rho_{p}^{X}$ and $\ket{\mathrm{GHZ}}$ are in the same
phase. The analysis shows that the $X-$dephasing noise acts as an irrelevant
perturbation with respect to the $\ket{\mathrm{GHZ}}$ state and its long-range
entanglement.

Besides establishing the phase equivalence, the bi-connection in Eq.(45) also
yields information on the entanglement structure of the dephased state
$\rho_{p}^{X}$. Consider two sufficiently large subregions of the system,
referred to as $A$ and $B$, for which one can always choose the RG blocking
scheme such that coarse-graining channels never act jointly on $A$ and $B$.
Let $E_{A:B}(\cdot)$ be any quantum or classical correlation measure between
$A$ and $B$ that satisfies the data processing inequality. Then due to the bi-
connection we have:

$\displaystyle E_{A:B}(\ket{\mathrm{GHZ}})\geq E_{A:B}(\rho_{p}^{X})\geq
E_{A:B}(\ket{\mathrm{GHZ}})$ (46) $\displaystyle\Rightarrow{}$ $\displaystyle
E_{A:B}(\rho_{p}^{X})=E_{A:B}(\ket{\mathrm{GHZ}}).$

Some examples of correlation measures are quantum mutual information,
entanglement negativity, and entanglement of formation & distillation. All
quantities are easy to compute analytically for the GHZ state but are
difficult to obtain for the mixed-state $\rho_{p}^{X}$ by other means.

We point out that all conclusions in this section hold also for the dephasing
noise along other directions in the $X$-$Y$ plane. This can be most easily
seen by noticing that the expression Eq.(40) holds for any dephasing direction
in the $X$-$Y$ plane. As we will see in the next subsection, the $Z$-dephasing
acts very differently.

###  V.2 Phase-flip noise

Next we consider the GHZ state under another type of noise, namely the phase-
flip or Z-dephasing:

$\mathcal{N}^{Z}_{p}(\cdot):=(1-p)(\cdot)+pZ(\cdot)Z$ (47)

which leads to the density matrix

$\displaystyle\rho^{Z}_{p,L}=$ $\displaystyle\frac{1}{2}[\ket{0^{\otimes
L}}\bra{0^{\otimes L}}+\ket{1^{\otimes L}}\bra{1^{\otimes L}}+$ (48)
$\displaystyle(1-2p)^{L}(\ket{0^{\otimes L}}\bra{1^{\otimes
L}}+\ket{1^{\otimes L}}\bra{0^{\otimes L}})]$

In the thermodynamic limit, the off-diagonal term vanishes for any
$p\notin\\{0,1\\}$ and the state converges to a classical state
$\frac{1}{2}(\ket{0^{\otimes L}}\bra{0^{\otimes L}}+\ket{1^{\otimes
L}}\bra{1^{\otimes L}})$. This already indicates that the state is in a
different phase from the GHZ state.

To construct the RG for this mixed state, we still use the majority-vote
channel $\mathcal{E}_{b}$ (Eq.(38)) as the coarse-graining map. An important
difference of this case compared to the bit-flip case is that the majority
vote channel is now correlation-preserving with respect to $\rho^{Z}_{p,L}$.
To see this, we first verify the following relations:

$\displaystyle\mathcal{E}_{b}\circ\mathcal{U}_{w_{b}}$
$\displaystyle=\mathcal{I}$ (49) $\displaystyle(\mathcal{N}_{p}^{Z})^{\otimes
b}\circ\mathcal{U}_{w_{b}}$
$\displaystyle=\mathcal{U}_{w_{b}}\circ\mathcal{N}_{p^{\prime}}^{Z}$

where $\mathcal{I}$ is the identity channel and $p^{\prime}$ is given later in
Eq.(52). These equations imply that:

$\mathcal{U}_{w_{b}}\circ\mathcal{E}_{b}\left(\rho_{p,L}^{Z}\right)=\rho_{p,L}^{Z}$
(50)

where $\mathcal{U}_{w_{b}}\circ\mathcal{E}_{b}$ is applied to any block of $b$
sites. Thus $\mathcal{E}_{b}$ is reversible and correlation-preserving with
respect to $\rho_{p,L}^{Z}$.

Following a similar calculation as in the bit-flip noise case, we obtain that
the state after one step of RG maintains the same form:

$\mathcal{E}_{b}^{\otimes L/b}(\rho^{Z}_{p,L})=\rho^{Z}_{p^{\prime},L/b}$ (51)

but with a renormalized noise strength

$p^{\prime}=\frac{1}{2}(1-(1-2p)^{b}),$ (52)

The iteration relation has $p=0$ as an unstable fixed point, around which
$p^{\prime}\simeq bp$, and also $p=1/2$ as a stable fixed point, around which
$(p^{\prime}-1/2)=(p-1/2)^{b}$.

Since the RG is ideal, it leads to the following LC bi-connection:

$\rho^{Z}_{1/2}~{}\xrightarrow{{\rm
RG}^{-1}}~{}\rho^{Z}_{p}~{}\xrightarrow{\rm RG}~{}\rho^{Z}_{1/2}\quad\quad
p\in[0,0.5)$ (53)

and the analysis shows the noisy state is of the same phase as the classical
state $\frac{1}{2}(\ket{0^{\otimes L}}\bra{0^{\otimes L}}+\ket{1^{\otimes
L}}\bra{1^{\otimes L}})$. Therefore, for the GHZ state the phase flip noise is
relevant and destroys the long-range entanglement therein with an arbitrarily
small strength.

##  VI Thermal toric code state

In this section and the next, we discuss two mixed-states related to
$\mathbb{Z}_{2}$ topological order. In Sec.VI.1 we review key properties of
the toric code model and define the notations. In Sec.VI.2 we construct an
ideal RG to explicitly show that any finite temperature Gibbs state of the
toric code is in the trivial phase.

###  VI.1 Review of the toric code model

We consider a square lattice with periodic boundary conditions and qubits on
the links. Kitaev’s toric code model has the Hamiltonian

$H=-\sum_{\square\in P}A_{\square}-\sum_{+\in V}B_{+}$ (54)

where $A_{\square}=\prod_{i\in\square}X_{i}$ and $B_{+}=\prod_{i\in+}Z_{i}$.
$P,V$ represent plaquettes and vertices, respectively.

Since all terms in the Hamiltonian commute with each other, their common
eigenstates can be used to label the Hilbert space. But in order to construct
a complete basis, we need two more operators $\widetilde{X}_{1,2}=\prod_{i\in
S_{1,2}}X_{i}$, where $S_{1},S_{2}$ are the two homotopically inequivalent
non-contractable loops on the torus. Each $\widetilde{X}_{i}$ commutes with
$A_{\square}$s and $B_{+}$s, thus all of them together define a basis for the
Hilbert space:

$\displaystyle\ket{\textbf{m}=m_{1}...m_{|P|};\textbf{e}=e_{1}...e_{|V|};\textbf{l}=l_{1}l_{2}},\quad
m_{i},e_{i},l_{i}\in\\{0,1\\}$ (55)

satisfying

$\displaystyle A_{\square_{i}}\ket{\textbf{m};\textbf{e};\textbf{l}}$
$\displaystyle=(-1)^{m_{i}}\ket{\textbf{m};\textbf{e};\textbf{l}}$ (56)
$\displaystyle B_{+_{i}}\ket{\textbf{m};\textbf{e};\textbf{l}}$
$\displaystyle=(-1)^{e_{i}}\ket{\textbf{m};\textbf{e};\textbf{l}}$
$\displaystyle\widetilde{X}_{i}\ket{\textbf{m};\textbf{e};\textbf{l}}$
$\displaystyle=(-1)^{l_{i}}\ket{\textbf{m};\textbf{e};\textbf{l}}$

We call this the anyon number basis in contrast to the computational basis. If
$m_{i}=1$, there is a plaquette anyon (or $m$ anyon) at the corresponding
plaquette; while if $e_{i}=1$, there is a vertex anyon (or $e$ anyon) at the
corresponding vertex. The operator identities $\prod_{\square}A_{\square}=1$
and $\prod_{+}B_{+}=1$ enforce that the total number of either type of anyon
must be even:

$\pi(\textbf{m})=0\quad\pi(\textbf{e})=0,$ (57)

where the function $\pi(\cdot)$ evaluates the total parity of a bit string,
i.e. $\pi(\textbf{s}):=(\sum_{i}s_{i}~{}\textrm{mod}~{}2)$.

The Hamiltonian’s 4-dimensional ground state subspace $V$ is spanned by anyon-
free states:

$V:=\mathsf{span}\\{~{}\ket{\textbf{m}=\textbf{0};~{}\textbf{e}=\textbf{0};~{}\textbf{l}}:~{}~{}\textbf{l}\in\\{00,01,10,11\\}\\}$
(58)

States within this subspace are locally indistinguishable, i.e.
$\rho_{A}=\text{tr}_{\bar{A}}(\ket{\psi}\bra{\psi})$ is independent of
$\ket{\psi}\in V$ whenever $A$ is a topologically trivial region.

We define a mixed-state $\rho$ to be in the toric code phase if it is LC bi-
connected to states within $V$, namely:

$\rho_{a}~{}\xrightarrow{\mathcal{C}_{1}}~{}\rho~{}\xrightarrow{\mathcal{C}_{2}}~{}\rho_{b}$
(59)

for some states $\rho_{a},\rho_{b}$ within $V$, and some LC transformations
$\mathcal{C}_{1}$, $\mathcal{C}_{2}$.

###  VI.2 RG of the thermal toric code state

We consider the Gibbs state of the toric code model Eq.(54)
$\rho_{\beta}\propto\exp(-\beta H)$ at inverse temperature $\beta$. Ref.
Hastings (2011) showed that this state for finite $\beta$ is not long-range
entangled, and here we reproduce the conclusion by constructing an ideal
mixed-state RG under which the state flows to a trivial one.

We notice that the density matrix $\rho_{\beta}$ is diagonal in the anyon
number basis (Eq. (55)):

$\rho_{\beta}\ket{\textbf{m};\textbf{e};\textbf{l}}\propto\ket{\textbf{m};\textbf{e};\textbf{l}},$
(60)

and is thus a classical mixture of different anyon configurations, with
probabilities

$\displaystyle\mathrm{Pr}(\textbf{m},\textbf{e},\textbf{l}):=$
$\displaystyle\bra{\textbf{m};\textbf{e};\textbf{l}}\rho_{\beta}\ket{\textbf{m};\textbf{e};\textbf{l}}$
(61) $\displaystyle=$
$\displaystyle\mathrm{Pr}_{m}(\textbf{m})\mathrm{Pr}_{e}(\textbf{e})\mathrm{Pr}_{l}(\textbf{l})$

in which the three types of degrees of freedom are independent:

$\displaystyle\mathrm{Pr}_{m}(\textbf{m})$
$\displaystyle=C_{\beta}~{}\delta\left(\pi(\textbf{m})=0\right)\prod_{i}p_{\beta}^{m_{i}}(1-p_{\beta})^{1-m_{i}}$
(62) $\displaystyle\mathrm{Pr}_{e}(\textbf{e})$
$\displaystyle=C_{\beta}~{}\delta\left(\pi(\textbf{e})=0\right)\prod_{i}p_{\beta}^{e_{i}}(1-p_{\beta})^{1-e_{i}}$
$\displaystyle\mathrm{Pr}_{l}(\textbf{l})$ $\displaystyle=1/4$

with $p_{\beta}=\frac{e^{-\beta}}{e^{\beta}+e^{-\beta}}$ and $C_{\beta}$ a
normalization constant.

The key property is that $m$ anyons on each plaquette (and $e$ anyons on
vertices) are independently excited with probability $p_{\beta}$, up to a
global constraint that the total number of each anyon type is even. This
allows us to find an ideal RG, as we need only preserve the local anyon parity
$\pi(\textbf{m}_{B}),\pi(\textbf{e}_{B})$ of a block $B$ to maintain
correlations between the block and its complement.

We now describe how to coarse-grain to preserve this parity information.
Consider the following quantum channel acting on $12$ qubits in a $2\times 2$
block of plaquettes:

$\mathcal{E}^{X}(\cdot):=\sum_{\textbf{m}\in\\{0,1\\}^{\otimes
4}}U_{\textbf{m}}P_{\textbf{m}}(\cdot)P_{\textbf{m}}U_{\textbf{m}}^{\dagger}$
(63)

where $P_{\textbf{m}}$ is the projector to the subspace with anyon
configuration m, and the unitary operator $U_{\textbf{m}}$ is a product of
Pauli $Z$ matrices that brings $\ket{\textbf{m}=m_{1}m_{2}m_{3}m_{4}}$ to
$\ket{\pi(\textbf{m})000}$. For instance, if we assume plaquettes are labeled
as ${\begin{smallmatrix}1&2\\\ 3&4\end{smallmatrix}}$ and $\textbf{m}=0110$,
then $U_{\textbf{m}}$ can be $Z_{12}Z_{13}$, where $Z_{12(13)}$ is the Pauli-Z
matrix acting on the qubit separating 1 and 2 (1 and 3). We remark that
$U_{\textbf{m}}$ only acts on the inner four qubits.

In other words, $\mathcal{E}^{X}$ first measures the anyon configuration
within the block and then applies a unitary gate depending on the measurement
outcome that pushes all anyons to the top-left plaquette. Since $m$-anyon is
its own anti-particle, the top-left plaquette ends up with $\pi(\textbf{m})$
anyons while the other three end up with $0$. Importantly, neither step
disturbs the distribution of $e$-anyons.

$\mathcal{E}^{X}$ is a correlation-preserving map with respect to the state
$\rho_{\beta}$, and one can explicitly check that its action on $\rho_{\beta}$
can be reversed by the following channel:

$\mathcal{D}^{X}(\cdot):=\sum_{\textbf{m}}\mathrm{Pr}(\textbf{m}|\pi(\textbf{m}))U^{\dagger}_{\textbf{m}}P^{1}_{\pi(\textbf{m})}(\cdot)P^{1}_{\pi(\textbf{m})}U_{\textbf{m}}$
(64)

where $P^{1}_{x}$ is the projector to the subspace with $m_{1}=x$. The action
of $\mathcal{D}^{X}$ can be intuitively understood as follows: It first
measures the anyon occupancy of the site $1$, which we recall is the only site
that may host an anyon after the action of $\mathcal{E}^{X}$. Then based on
the measurement outcome (referred to as $x$), it randomly generates an anyon
configuration on the block according to the distribution
$\mathrm{Pr}(\textbf{m}|\pi(\textbf{m})=x)$.

Analogously, there is a channel for each $2\times 2$ block of vertices (i.e. a
block of plaquettes of the dual lattice) that coarse-grains $e$-anyons:

$\mathcal{E}^{Z}(\cdot):=\sum_{\textbf{e}\in\\{0,1\\}^{\otimes
4}}U_{\textbf{e}}P_{\textbf{e}}(\cdot)P_{\textbf{e}}U_{\textbf{e}}^{\dagger}$
(65)

where $P_{\textbf{e}}$ are projector for $e$-anyon configurations and
$U_{\textbf{e}}$ a product of $X$ operators that brings
$\ket{\textbf{e}=e_{1}e_{2}e_{3}e_{4}}$ to $\ket{\pi(\textbf{e})000}$.
$\mathcal{E}^{Z}$ only moves $e$-anyons and commutes with $\mathcal{E}^{X}$.

Figure 5:  RG scheme for the thermal toric code state– In all panels, a
plaquette (vertex) is shaded (dotted) if has a non-zero probability of holding
an $m$\- ($e$-) anyon, and physical qubits are associated with edges of the
lattice and drawn as circles. (left$\rightarrow$mid) $\mathcal{E}^{X}$ and
$\mathcal{E}^{Z}$ (see Eq.(63) and Eq.(65)) acts on each $2\times 2$ block of
plaquettes and vertices, respectively. The resulting state has anyons on one
of its sublattices’ plaquettes and vertices. (mid$\rightarrow$right) After
disentangling with the unitary $\mathcal{U}$ depicted in Eq.(66) and
discarding the decoupled qubits, the new state is still a toric code Gibbs
state, but with renormalized temperature $p^{\prime}$ (Eq.(68)) supported on a
coarse-grained lattice.

After applying $\mathcal{E}^{X(Z)}$ to each block of plaquettes (vertices),
the resulting state only has anyons on plaquettes and vertices corresponding
to a sublattice (see Fig.5, middle panel). To complete one iteration of the
RG, we need to discard some degrees of freedom and put the state on a coarse-
grained lattice. This step can be achieved by a series of local unitary
operators called elementary moves introduced in Dennis  _et al._ (2002);
Aguado and Vidal (2008).

This step is most easily described graphically:

(66)

At each step, multiple controlled-not gates are applied, represented with
arrows from the control qubit to target qubit. These gates decouple qubits
into a product state which can then be removed, and the remaining qubits form
a toric code state with anyons on a coarser lattice. Operations shown in each
panel are applied in parallel to all the $2\times 2$ blocks on the lattice.

In summary, one iteration of the RG consists of:

$\mathcal{C}=\mathcal{U}\circ\left(\bigotimes_{B\in\mathcal{B}^{\prime}}\mathcal{E}_{B}^{Z}\right)\circ\left(\bigotimes_{B\in\mathcal{B}}\mathcal{E}_{B}^{X}\right)$
(67)

where $\mathcal{B}$ contains $2\times 2$ blocks of plaquettes and
$\mathcal{B}^{\prime}$ contains $2\times 2$ blocks of vertices. $\mathcal{U}$
stands for the disentangling operations in Eq.(66).

After one step of RG, a plaquette (vertex) contains an anyon if and only the
four plaquettes (vertices) it was coarse-grained from contain an odd number of
anyons. The renormalized state is still a thermal toric code state, but with a
renormalized probability (or renormalized temperature):

$\displaystyle
p^{\prime}_{\beta^{\prime}}=4p_{\beta}(1-p_{\beta})^{3}+4p^{3}_{\beta}(1-p_{\beta})$
(68) $\displaystyle\Leftrightarrow$
$\displaystyle\tanh{\beta^{\prime}}=\tanh^{4}{\beta}$

We thus conclude that any finite temperature state $\rho_{\beta<\infty}$ flows
to the infinite temperature one $\rho_{\beta=0}$ under the RG.

Furthermore, since all channels in the RG are correlation-preserving with
respect to their inputs, the RG is ideal and can be reversed. Thus there is
the following bi-connection:

$\rho_{\beta=0}~{}\xrightarrow{{\rm
RG}^{-1}}~{}\rho_{\beta}~{}\xrightarrow{\rm
RG}~{}\rho_{\beta=0}\quad\quad\beta<\infty$ (69)

Since the infinite temperature state $\rho_{\beta=0}\propto\mathbb{I}$ is in
the trivial phase, we conclude that $\rho_{\beta}$ is also in the trivial
phase.

##  VII Noisy toric code state

Now we consider the mixed-state obtained by applying noise to a pure toric
code state. All notations in the section follow those introduced in Sec.VI.1.

The toric code model is naturally a quantum memory that stores quantum
information in its ground state subspace $V$. In Sec.VII.1, we discuss the
relation between the preservation of logical information and the preservation
of the phase of matter. We prove that if an LC transformation $\mathcal{C}$
does not bring a pure toric code state out of its phase, then $\mathcal{C}$
must preserve any quantum information stored in $V$.

In Sec.VII.2 and VII.3, we describe two ways to show that the dephased toric
code state is in the toric code phase when dephasing strength is small. More
specifically, we show that there exist LC transformations that bring the
dephased state back to a pure toric code state. In Sec.VII.2, the LC
transformation is motivated by the Harrington decoder of the toric code and
takes the form of an RG transformation. In Sec.VII.3, the LC transformation is
obtained by spatially truncating the minimum weight perfect matching (MWPM)
decoder.

###  VII.1 Logical information and long-range entanglement

The toric code model, as its name suggests, is naturally a quantum error
correcting code whose codespace is the ground state subspace $V$ (Eq.(58)). In
this context, an important question is whether a noise channel $\mathcal{N}$
destroys logical information stored in a quantum memory. Mathematically, this
is equivalent to asking whether there exists a recovery channel $\mathcal{R}$
such that:

$\mathcal{R}\circ\mathcal{N}(\ket{\psi}\bra{\psi})=\ket{\psi}\bra{\psi}\ \ \
\forall\ket{\psi}\in V.$ (70)

In quantum error correction, $\mathcal{R}$ is often realized by a decoder,
which maps any input state into an output supported within $V$ 666Since we are
only concerned about in-principle recoverability of logical information, we
assume all operations within $\mathcal{R}$ are noiseless.. If such
$\mathcal{R}$ exists, we say the logical information is preserved by
$\mathcal{N}$. Otherwise, we say the logical information is destroyed.

To relate the phase of the mixed state, as defined by two-way LC connection,
to preservation of logical information, we will need to first prove the
following theorem.

######  Theorem 2.

Let $V$ be the code subspace of a toric code defined on a torus. Suppose
$\mathcal{C}$ is a local channel transformation satisfying
$\mathsf{supp}~{}\mathcal{C}(\ket{\psi}\bra{\psi})\subseteq V\ \
\forall\ket{\psi}\in V$, then $\mathcal{C}$’s action when restricted to $V$ is
a unitary channel.

To gain some intuition for why locality of the channel is essential in the
theorem, consider the following channel:

$\mathcal{N}(\rho):=\frac{1}{2}\rho+\frac{1}{2}\widetilde{X}_{1}\rho\widetilde{X}_{1}$
(71)

where $\widetilde{X}_{1}=\Pi_{i\in S_{1}}X_{i}$ is the logical $X$ operator of
the first encoded qubit (see Sec.VI.1). $\mathcal{N}$ is not an LC
transformation: $\mathcal{N}(\ket{0}^{\otimes L}\bra{0}^{\otimes
L})=\frac{1}{2}(\ket{0}^{\otimes L}\bra{0}^{\otimes L}+\ket{1}^{\otimes
L}\bra{1}^{\otimes L})$, a non-trivial mixed-state with long-range
correlations. Furthermore, $\mathcal{N}$ preserves $V$, but its action within
$V$ is dephasing the first logical qubit, which is not a unitary action.

We now prove Thm.2.

###### Proof.

$\mathcal{C}$ can be dilated into an LU circuit $U$ that acts jointly on the
physical qubits (referred to as $P$) and the ancilla qubits (referred to as
$A$). Consider $U$’s action on a codeword state:

$\ket{\psi;\textbf{0}}:=\ket{\psi}_{P}\ket{\textbf{0}}_{A}~{}\xrightarrow{U}~{}\ket{\phi}_{PA}$
(72)

where $\ket{\psi}$ is any code word state in $V$. For later convenience we
define the expanded codespace $V_{0}$, which is the subspace of
$\mathcal{H}_{PA}$ spanned by
$\\{\ket{\psi}_{P}\ket{\textbf{0}}_{A}\\}_{\ket{\psi}\in V}$. We use $V_{0}$
to refer to both the subspace and the code defined by it. $V_{0}$ is still a
stabilizer code, whose stabilizers are those of $V$ combined with $\\{Z_{i}:\
i\in A\\}$.

Figure 6: Geometry of operators $L_{1}$, $L_{2}$, and $K$.

Let $L_{1}$ and $L_{2}$ be two Pauli logical operators of the toric code that
act in the same way in the code subspace $V$ (see Fig.6). Thus $L_{1}L_{2}$ is
a stabilizer of the toric code. Since $\mathcal{C}$ preserves the code
subspace, we have:

$\bra{\phi}L_{1}L_{2}\ket{\phi}=\text{tr}(C(\ket{\psi}\bra{\psi})L_{1}L_{2})=1$
(73)

This leads to:

$\bra{\psi;\textbf{0}}L_{1}^{U}L_{2}^{U}\ket{\psi;\textbf{0}}=1$ (74)

where $L_{i}^{U}:=U^{\dagger}L_{i}U$ has support on both $P$ and $A$, and is
not necessarily a Pauli operator. Recalling that $L_{1}^{U}L_{2}^{U}$ is a
unitary operator and the above expression holds for any
$\ket{\psi;\textbf{0}}$, we conclude that $L_{1}^{U}L_{2}^{U}$ acts as logical
identity in the extended codespace $V_{0}$.

To proceed, we assume the spatial separation between $L_{1}$ and $L_{2}$ to be
much larger than the range of $U$, so that $L_{1}^{U}$ and $L_{2}^{U}$ are
also well-separated.

We claim that both $L_{1}^{U}$ and $L_{2}^{U}$ are logical operators of
$V_{0}$. Otherwise, there needs to be a codeword state $\ket{a}\in V_{0}$ such
that $L_{1}^{U}\ket{a}\notin V_{0}$. This implies at least one stabilizer $S$
of $V_{0}$ is violated by the state:
$\bra{a}(L_{1}^{U})^{\dagger}SL_{1}^{U}\ket{a}\neq 1$, and $S$ must have
spatial overlap with $L_{1}^{U}$. Further, since $L_{2}^{U}$ is far from both
$L_{1}^{U}$ and $S$,

$\bra{a}(L_{1}^{U}L_{2}^{U})^{\dagger}SL_{1}^{U}L_{2}^{U}\ket{a}\neq 1$ (75)

But this cannot be true because $L_{1}^{U}L_{2}^{U}\ket{a}=\ket{a}$ and $S$ is
a stabilizer.

The same reasoning applies to any Pauli logical operator, referred to as $K$,
whose spatial support is perpendicular to $L=L_{1}$ (see Fig.6). By varying
$K$ and $L$, their product $R=K\cdot L$ can represent all of the $15$
inequivalent Pauli logical operators of the toric code. We fix such a set:
$\mathcal{P}=\\{R_{1},R_{2},...,R_{15}\\}$. The image of $\mathcal{P}$ under
$R\rightarrow R^{U}$ is a set of 15 logical operators of $V_{0}$, as we just
proved. Furthermore, since the map $R\rightarrow R^{U}$ preserves all the
multiplication and commutation relations, we know $\mathcal{P}^{U}$ must act
as a set of 15 inequivalent Pauli logical operators on $V_{0}$, up to a basis
rotation.

We consider the part of $R^{U}_{i}$ (or $R_{i}$) when restricted to the
codespace $V_{0}$ (or $V$):

$\displaystyle\Pi_{V_{0}}R^{U}_{i}$
$\displaystyle=\widetilde{R^{U}_{i}}\otimes\ket{\textbf{0}}\bra{\textbf{0}}$
(76) $\displaystyle\Pi_{V}R_{i}$ $\displaystyle=\widetilde{R_{i}}$

where $\widetilde{R^{U}_{i}}$ and $\widetilde{R_{i}}$ are operators acting
within $V$ only. $\Pi_{V}$ is the projector to the subspace $V$ and
$\Pi_{V_{0}}=\Pi_{V}\otimes\ket{\textbf{0}}\bra{\textbf{0}}$. As explained,
both $\\{\widetilde{R_{i}}\\}$ and $\\{\widetilde{R^{U}_{i}}\\}$ realize the
algebra of Pauli operators in the logical space.

We have:

$\displaystyle\mathcal{C}(\widetilde{R^{U}_{i}})$
$\displaystyle=\text{tr}_{A}(U(\widetilde{R^{U}_{i}}\otimes\ket{\textbf{0}}\bra{\textbf{0}})U^{\dagger})$
(77) $\displaystyle=\text{tr}_{A}(U(R_{i}^{U}\Pi_{V_{0}})U^{\dagger})$
$\displaystyle=\text{tr}_{A}(R_{i}U\Pi_{V_{0}}U^{\dagger})$
$\displaystyle={R_{i}}\mathcal{C}(\Pi_{V})$
$\displaystyle=\widetilde{R_{i}}\mathcal{C}(\Pi_{V})$

The second to last equality holds because $R_{i}$ is supported on $P$ only,
while the last one holds because
$\mathsf{supp}~{}\mathcal{C}(\Pi_{V})\subseteq V$ by assumption.

On the r.h.s. of the second equality above $R_{i}^{U}$ and $\Pi_{V_{0}}$
commute. Thus if we change their order then the same derivation gives:

$\mathcal{C}(\widetilde{R^{U}_{i}})=\mathcal{C}(\Pi_{V})\widetilde{R_{i}}$
(78)

Since the relation holds for any $i\in\\{1,...,15\\}$ and
$\mathsf{supp}~{}\mathcal{C}(\Pi_{V})\subseteq V$, we know
$\mathcal{C}(\Pi_{V})\propto\Pi_{V}$. Further, since $\mathcal{C}$ is trace-
preserving, we have $\mathcal{C}(\Pi_{V})=\Pi_{V}$. Thus:

$\mathcal{C}(\widetilde{R^{U}_{i}})=\widetilde{R_{i}}$ (79)

This implies that when restricted to $V$, $\mathcal{C}(\cdot)$ is a
$*$-isomorphism and must be a unitary channel. ∎

We use the theorem to explore the relation between the phase of the noisy
toric code state and the preservation of quantum information stored.

Consider a toric code’s codeword state $\ket{\psi}\in V$. Suppose
$\mathcal{N}_{\psi}$ is an LC transformation that preserves the toric code
phase. By definition, there exists another LC transformation
$\mathcal{D}_{\psi}$ such that
$\mathsf{supp}~{}\mathcal{D}_{\psi}\circ\mathcal{N}_{\psi}(\ket{\psi})\subseteq
V$ 777We emphasize that the condition should not be
$\mathcal{D}_{\psi}\circ\mathcal{N}_{\psi}(\ket{\psi})=\ket{\psi}$, according
to our definition of the toric code phase in Sec.VI.1. We first point out that
the pair $(\mathcal{N}_{\psi},\mathcal{D}_{\psi})$ satisfies

$\mathcal{D}_{\psi}\circ\mathcal{N}_{\psi}(\ket{\psi^{\prime}}\bra{\psi^{\prime}})\subseteq
V~{}~{}~{}\forall\ket{\psi^{\prime}}\in V,$ (80)

which we prove in App.A.5. Since the choice of
$(\mathcal{N}_{\psi},\mathcal{D}_{\psi})$ does not depend on the codeword
state $\ket{\psi}$, we drop the $\psi$ subscript henceforth.

The channel $\mathcal{D}\circ\mathcal{N}$ thus satisfies the condition in
Thm.2, according to which we have

$\mathcal{D}\circ\mathcal{N}(\ket{\psi}\bra{\psi})=U\ket{\psi}\bra{\psi}U^{\dagger}~{}~{}~{}\forall\ket{\psi}\in
V$ (81)

for some logical unitary operator $U$.

We thus conclude that if the noise $\mathcal{N}$ preserves the toric code
phase, then it also preserves the logical information stored. In particular,
the recovery map can be chosen as $R(\cdot)=U^{\dagger}\mathcal{D}(\cdot)U$.

We consider a more detailed scenario where the channel
$\mathcal{N}=\mathcal{N}_{p}$ has a strength parameter $p$. When the noise is
very strong, both the toric code phase and the logical information stored
should be destroyed. Thus one can define two critical noise strengths: $p_{\rm
t.c.}$, beyond which the noisy state is no longer in the toric code phase; and
$p_{\rm coding}$, beyond which the stored logical information is no longer
recoverable. The previous analysis shows that

$p_{\rm t.c.}\leq p_{\rm coding}$ (82)

Namely, the loss of logical information must occur after transitioning out of
the toric code phase.

If there is a gap between $p_{\rm t.c.}$ and $p_{\rm coding}$, then the noisy
state $\mathcal{N}_{p}(\ket{\psi}\bra{\psi})$ for $p\in(p_{\rm t.c.},p_{\rm
coding})$ is not in the toric code phase but still contains logical
information. In this case, the corresponding recovery map $\mathcal{R}$ that
recovers logical information must be non-LC.

###  VII.2 RG of the dephased toric code state

We illustrate these general results in a specific example, for which we
construct explicit RG channels. We consider a toric code ground state
$\ket{\rm t.c.}\in V$ subject to bit-flip noise with strength $p$ ( Eq.(33)):

$\rho_{p}:=(\mathcal{N}^{X}_{p})^{\otimes L}\left(\ket{\rm t.c.}\bra{\rm
t.c.}\right)$ (83)

We ask whether the state is in the same phase as $\ket{\rm t.c.}$, when $p$ is
small.

(a)

(b)  (c)

Figure 7:  RG of the dephased toric code state– (a) In each RG iteration,
$\mathcal{G}$ is applied in parallel to all the odd blocks (blue), then
$\mathcal{E}^{X}$ is applied to all the even blocks (green). Finally
disentangling unitaries (see Eq.(66), not drawn in the figure) are applied to
reduce the lattice size by half. (b) RG flow of the anyon density
$q_{p}^{(\ell)}$. (c) Iteration relation of $q_{p}^{(\ell)}$ when approaching
$0$, for various choices of $p$.

It is convenient to work in the anyon number basis Eq.(55), and since all
states in this example are in the $e$-anyon-free subspace, we omit the e
labeling henceforth. An $X$ operator acting on a qubit on an edge will create
a pair of anyons in the two plaquettes adjacent to the edge. But if two anyons
meet in the same plaquette, they annihilate. Thus if we fix the set of edges
acted on by $X$, then anyons appear on faces adjacent to an odd number of $X$s
(see Fig.8).

Figure 8: A sample anyon configuration for noisy toric code– A dashed line on
an edge denotes the corresponding qubit is flipped by an $X$ operator. Anyons
are created in plaquettes (shaded) where an odd number of dashed lines meet.

The noisy state is a classical mixture of anyon configurations which differ
significantly from those of the Gibbs state. When $p$ is sufficiently small,
the typical size of an error cluster is much smaller than the typical distance
between clusters. The errors create anyons at the boundary of each cluster.

This picture suggests that by locally identifying clusters and pairing up
anyons therein, one can remove all errors if $p$ is sufficiently small. This
intuition underlies the design of several decoding algorithms for the toric
code Duclos-Cianci and Poulin (2010); Bravyi and Haah (2013); Breuckmann  _et
al._ (2016); Harrington (2004), which aim to pair up anyons such that the
quantum information stored in the code remains intact. As we show now, these
decoders can be modified into RG schemes to reveal mixed-state phases of the
noisy toric-code states.

We construct a simplified version of the Harrington decoder for the toric code
Harrington (2004); Breuckmann  _et al._ (2016) to demonstrate that $\rho_{p}$
and $\ket{\rm t.c.}$ are in the same phase when $p$ is small. We first
partition the lattice into even blocks $\mathcal{B}_{\rm even}$ and odd blocks
$\mathcal{B}_{\rm odd}$ (see Fig.7). Odd blocks are obtained by translating
even blocks by one lattice spacing in both spatial directions. The two types
of blocks will play different roles: coarse-graining will occur on even blocks
and anyons will be paired up within odd blocks, regarded as boundary regions
of even blocks.

Each step of the RG is composed of three layers of local channels:

$\mathcal{C}=\mathcal{U}\circ\left(\bigotimes_{B\in\mathcal{B}_{\rm
even}}\mathcal{E}_{B}^{X}\right)\circ\left(\bigotimes_{B\in\mathcal{B}_{\rm
odd}}\mathcal{G}_{B}\right)$ (84)

where the final step $\mathcal{U}$ is the disentangling operation depicted in
Eq.(66), and $\mathcal{E}^{X}$ is the coarse-graining channel defined in
Eq.(63).

The main difference between this RG scheme and the one for the thermal toric
code state (see Sec.VI.2) is the introduction of $\mathcal{G}$s.
$\mathcal{G}_{B}$ annihilates all anyons within $B$ only if there are an even
number of them; otherwise it leaves anyons within $B$ unmodified:

$\mathcal{G}_{B}:=\sum_{\textbf{m}\in\\{0,1\\}^{\otimes
4}}\widetilde{U}_{\textbf{m}}P_{\textbf{m}}(\cdot)P_{\textbf{m}}\widetilde{U}_{\textbf{m}}^{\dagger}$
(85)

where $\widetilde{U}_{\textbf{m}}$ equals $U_{\textbf{m}}$ (Eq.(63)) if
$\pi(\textbf{m})=0$, and is $\mathbb{I}$ when $\pi(\textbf{m})=1$. The
heuristic reason for introducing $\mathcal{G}_{B}$ is to pair up anyon
clusters across the boundaries of the even blocks before the coarse-graining
step. If such anyons were not paired, the coarse-graining on even blocks
potentially prolongs them into clusters with a larger size, thus hindering
effective anyon removal.

After each RG step $\mathcal{C}$, the new state is still an ensemble of
different anyon configurations, albeit one that is not analytically tractable.
Thus, we numerically compute how the RG steps affect the anyon density:

$q_{p}^{(\ell)}=\frac{1}{|P^{(\ell)}|}\sum_{\square\in
P^{(\ell)}}\text{tr}\left(\rho_{p}^{(\ell)}\frac{1-A_{\square}}{2}\right)$
(86)

where $P^{(\ell)}$ is the set of plaquettes on the renormalized lattice and
$\rho_{p}^{(\ell)}$ is the renormalized state after $\ell$ iterations. $q=0$
implies the state is in the ground state subspace $V$ of the toric code
Hamiltonian Eq.(54).

We use Monte Carlo method to study the flow of $q^{(\ell)}_{p}$ under RG. The
simulation (Fig.7 (b)) shows that there is a sharp transition of
$q_{p}^{(\ell)}$ at $p_{c}\approx 0.041$:

$\lim_{l\rightarrow\infty}q_{p}^{(\ell)}=\left\\{\begin{array}[]{lr}0&p<p_{c}\\\
O(1)&p>p_{c}\end{array}\right.$ (87)

When $p<p_{c}$ the RG successfully annihilates all anyons and the fixed-point
state is in the ground state subspace of $V$; while when $p>p_{c}$, the fixed-
point state has finite anyon density.

Furthermore, when the anyon density $q^{(\ell)}$ approaches $0$, it transforms
under each RG step as (see Fig.7 (c)):

$q^{(\ell+1)}\simeq(q^{(\ell)})^{\gamma}$ (88)

for $\gamma>1$. This behavior guarantees that a small number of iterations is
sufficient for the convergence to the toric code ground state subspace (see
the appendix App.B.4 for a details).

We thus obtain the LC bi-connection:

$\ket{\rm t.c.}~{}\xrightarrow{{\rm noise}}~{}\rho_{p}~{}\xrightarrow{\rm
RG}~{}\ket{\rm t.c.^{\prime}}\quad\quad p<p_{c}$ (89)

where $\ket{\rm t.c.^{\prime}}$ is another toric code state. This shows that
$\rho_{p}$ is in the same phase as the pure toric code state, when $p<p_{c}$
and therefore $X$-dephasing noise is an irrelevant perturbation to the
topologically ordered phase.

We emphasize that this analysis does not show that $\rho_{p}$ with $p>p_{c}$
is in a different phase, because no bi-connection has been identified with
this decoder. In fact, in the next section, we will construct another local
channel that establishes that the phase boundary of the toric code phase
extends to a much higher $p_{c}$.

###  VII.3 Truncated minimal weight perfect matching channel

The seminal work Dennis  _et al._ (2002) showed that the dephased toric code
state (Eq.(83)) retains its logical information up to a critical point $p_{\rm
coding}\approx 0.108$, by relating the coding phase transition to the
ferromagnetic-paramagnetic transition in the random bond Ising model. A
recovery channel called the maximal likelihood decoder Dennis  _et al._
(2002); Bravyi  _et al._ (2014) decodes the logical information for any
$p<p_{\rm coding}$, but the channel is not an LC transformation.

The minimal weight perfect matching (MWPM) decoder is another decoder
introduced in Dennis  _et al._ (2002). It has a decoding threshold $p_{\rm
MWPM}\approx 0.103$ very close to $p_{\rm coding}$ Wang  _et al._ (2003). The
MWPM decoder, as a quantum channel, is also not an LC transformation. In the
rest of this section, we show that it is possible to approximate the MPWM
decoder’s action arbitrarily well with an LC transformation whenever $p<p_{\rm
MWPM}$. Consequently, we show that any dephased toric code state with
$p<p_{\rm MWPM}$ is in the toric code phase.

The core component of the MWPM decoder (henceforth referred to as
$\mathcal{C}^{\rm MWPM}$) is a classical algorithm that solves the MWPM
problem, namely looking for an anyons pairing scheme that minimizes the total
length of the strings connecting pairs. Afterward, the decoder annihilates
each anyon pair by acting with the string of $X$ operators connecting the
pair.

(a)

(b)

Figure 9: Truncated minimum weight perfect matching channel– (a) For a given
block $B$, the corresponding channel $\mathcal{E}_{B,a}$ acts on both $B$ and
a buffer region $F$ of a width $a$. (b) $\mu(a,p)$ is the probability that the
truncated and global MWPM algorithms produce the same anyon pairings. It is
plotted against $p$ for various choices of $a$.

We now devise a way to truncate the $\mathcal{C}^{\rm MWPM}$ into an LC
transformation. We first partition plaquettes into disjoint blocks, each with
a size $b\times b$. For each block $B$, we apply a local channel
$\mathcal{E}_{B,a}$ which acts jointly on $B$ and a buffer region $F$ of width
$a$ surrounding $B$ (see Fig.9). The local channel first solves the MWPM of
anyons within the truncated region $B\cup F$, with the additional requirement
that each anyon can either pair with another anyon or with the outer boundary
of $F$ (the dashed line in Fig.9). Then given the pairing scheme suggested by
the MWPM solution, the channel only accepts a subset of it, namely pairs with
at least one anyon within $B$. The truncated MWPM (tMWPM) channel applies the
above channel to every block:

$\mathcal{C}^{\rm tMWPM}_{a}:=\prod_{B\in\mathcal{B}}\mathcal{E}_{B,a}$ (90)

Note that different $\mathcal{E}_{B,a}$ can have overlapping domains. But
since each $\mathcal{E}_{B,a}$ acts only on a patch of $(b+2a)^{2}$ qubits, we
can always rearrange $\\{\mathcal{E}_{B,a}\\}$ into an $O((b+2a)^{2})$-layer
circuit so that each layer is composed of channels with non-overlapping
domains. After the rearrangement, it is apparent that $\mathcal{E}^{\rm
tMWPM}_{a}$ is a range-$O((b+2a)^{4})$ LC transformation (because both the
depth and the range of gate is $O((b+2a)^{2})$).

The assumption behind the design of the tMWPM channel is the existence of a
correlation length $\xi(p)$, such that when $a\gg\xi(p)$, solving MWPM on
$B\cup F$ only and solving MWPM on the whole system produce the same pairing
for anyons in $B$. If the assumption holds for every block $B$, then the
$\mathcal{C}_{a}^{\rm tMWPM}$ pairs all anyons in the same way as
$\mathcal{C}^{\rm MWPM}$:

$\mathcal{C}^{\rm tMWPM}_{a}(\rho_{p})\approx\mathcal{C}^{\rm
MWPM}(\rho_{p})=\ket{\rm t.c.}\quad a\gg\xi(p)$ (91)

We provide a rough estimate of how large $a$ needs to be for the ‘$\approx$’
above to hold (agreement between local and global MWPM with probability
$1-\epsilon$). Given the correlation length assumption, the probability that
for a single block $B$ the global and the truncated MWPM agrees should be
$(1-e^{-a/\xi(p)})$. Assuming these probabilities for different blocks are
independent (which should hold for far apart blocks), then we need

$(1-e^{-a/\xi(p)})^{L^{2}/b^{2}}>1-\epsilon$ (92)

which occurs when

$a=\xi(p)O\left(\log\frac{L^{2}}{\epsilon}\right).$ (93)

$a$ diverges whenever $\xi(p)$ does, and this is expected to happen when
$p\rightarrow p_{\rm MWPM}$.

To numerically support the assumption that there exists a correlation length
for MWPM, we sample anyon configurations in the $X$-dephased toric code state
and solve the MWPM first for all the anyons, and then only for anyons within
$B\cup F$ (Fig.9(a)). Then we compute the probability $\mu(a,p)$ that the two
solutions are identical on $B$. We let both the system size $L$ and the
diameter of $B$ be proportional to $a$, the width of the buffer $F$, so that
the system has only one length scale $a$.

The simulation result is shown in Fig.9(b) and suggests there is a critical
point $p_{\rm tMWPM}$ in the interval $(0.10,0.11)$, presumably consistent
with $p_{\rm MWPM}$ in the thermodynamic limit. Below $p_{\rm tMWPM}$, we
observe that $\lim_{a\rightarrow\infty}\mu(a,p)=1$. This indicates that the
MWPM solution within $B$ is independent of anyons that are more than $O(\xi)$
away from $B$, for some correlation length $\xi$ which diverges at $p_{\rm
tMWPM}$. tMWPM thus serves as a local channel which, along with the noise
channel, establishes the two-way connection demonstrating the toric code phase
up to $p_{\rm tMWPM}\approx 0.1$. Above $p_{\rm tMWPM}$,
$\lim_{a\rightarrow\infty}\mu(a,p)=0$, implying non-locality in the MPWM
solution.

We point out that the simulation method above provides a way to detect the
toric code phase using the anyon distribution data. One can fix a region $B$,
implement MWPM on $B\cup F$, and gradually increase the buffer width $a$. If
the MWPM solution when restricted to $B$ becomes stationary after $a$ is
larger than some $a^{*}=O(1)$ with high probability, then the original mixed
state is in the toric code phase because the tMWPM channel with $a\gtrsim
a^{*}\log L$ can transform it into a pure toric code state. The method can
potentially be used to detect mixed-state topological order in experiments.

##  VIII Discussion and outlook

Our work provides two routes (RG and local versions of decoders) for
constructing local channels connecting two mixed states to prove they are in
the same phase. We formulated a real-space RG scheme for mixed states and
proposed the correlation-preserving property as a guiding criterion for
finding coarse-graining maps; this property is necessary and sufficient for
the map’s action to be reversible (Thm.1). We applied this formalism to
identify the phases of several classes of mixed states obtained by perturbing
a long-range entangled pure state with noise or finite temperature, and in
particular we constructed an exact RG flow of the finite temperature 2D toric
code state to infinite temperature.

For toric code subject to decoherence, we also established a relation between
the mixed state phase of the toric code and the integrity of logical
information. In Thm.2, we proved that if local noise preserves the long-range
entanglement of the toric code (and the resulting mixed state remains within
the same phase as toric code), it must also preserve logical information
encoded in the initial pure state. We conjecture that the converse statement
is also true, namely, if local noise destroys the long-range entanglement of
toric code, it must also destroy any encoded logical information. Even though
the theorem and subsequent discussion focused on the toric code state, the
main proof idea generalizes to many other topological codes and their
corresponding phases.

  * •

After formalizing the definition of mixed-state phase, one natural question to
ask is whether there is a nontrivial phase that has no pure state nor
classical state in this phase. A promising candidate is the ZX-dephased toric
code state recently studied in Wang  _et al._ (2023). Since the state (when
noise is strong) loses logical information Wang  _et al._ (2023), it is
provably not in the toric code phase according to our Thm.2. Thus if the state
is not in the trivial phase, it is an example of intrinsic mixed-state
topological order. Another class of potential examples are decohered critical
ground states Zou  _et al._ (2023); Lee  _et al._ (2023), because such states
naturally sit between a long-range entangled pure state and a long-range
correlated classical state.

  * •

One related question is whether one can find a computable quantity to detect
nontrivial mixed-state phases. Topological entanglement negativity has been
successfully used as a probe of mixed-state topological order Lu  _et al._
(2020); Fan  _et al._ (2023), but its robustness under LC transformations
needs to be studied further.

  * •

The decoherence-induced toric code transition can also be understood as a
separability transition of the mixed-state Chen and Grover (2023), and it
would be valuable to relate this perspective with the mixed state phase and
local channel perspective.

  * •

Pure state RG methods like DMRG serve as powerful computational methods for
analyzing many-body systems. It is thus important to develop a numerical
implementation of our mixed-state RG scheme. To facilitate simulations, one
needs to first find an efficient representation of the mixed state (e.g. using
tensor networks), then update it iteratively using exact or approximately
correlation-preserving maps, obtained by solving the optimization problem
Eq.(13). We leave this problem for future exploration.

  * •

As presented in Sec.VII.3, tMWPM also serves as a practical probe of the mixed
state toric code phase using anyon measurements. However, in experiments
imperfect measurements lead to a finite density of ‘fake’ anyons as well as
unprobed anyons. To address this, one needs to consider a specific model of
measurement errors and perform more than one round of measurements. Another
potential direction is generalizing tMWPM to other topologically ordered
mixed-state phases in two or higher dimensions.

###### Acknowledgements.

We thank Arpit Dua, Tyler Ellison, Matthew P.A. Fisher, Tarun Grover, Ethan
Lake, Yaodong Li, Zi-Wen Liu, Tsung-Cheng Lu, Ruochen Ma, and Michael Vasmer
for helpful discussions and feedback. We also thank Roger G. Melko and Digital
Research Alliance of Canada for computational resources. SS acknowledges the
KITP graduate fellow program, during which part of this work was completed.
This work was supported by the Perimeter Institute for Theoretical Physics
(PI) and the Natural Sciences and Engineering Research Council of Canada
(NSERC). Research at PI 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. This research was also supported in part by the National Science
Foundation under Grants No. NSF PHY-1748958 and PHY-2309135.

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##  Appendix A Various short proofs

###  A.1 Relation between pure- and mixed- state phase equivalence

We sketch a proof of the following statement: On a given lattice, two pure
states $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ are of the same mixed-state phase
if and only if there exists an invertible state $\ket{a}$ on the same lattice
such that $\ket{\psi_{1}}$ and $\ket{\psi_{2}}\otimes\ket{a}$ are of the same
pure state phase. A many-body ground state $\ket{a}$ is called an invertible
state if there exists another state $\ket{\widetilde{a}}$ such that
$\ket{a}\otimes\ket{\widetilde{a}}$ can be LU transformed into a product
state.

Since a LU transformation is also a LC transformation, pure-state phase
equivalence trivially implies mixed-state phase equivalence.

Now we show the other direction. Assume that there exists a pair of LC
transformations $\mathcal{C}_{1,2}$ such that
$\mathcal{C}_{1}(\rho_{1})\approx\rho_{2}$ and
$\mathcal{C}_{2}(\rho_{2})\approx\rho_{1}$. We use $U_{1},U_{2}$ to denote the
unitary circuit within the definition of $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$.

We observe that: Since $C_{1}(\rho_{1})\approx\rho_{2}$ and $\rho_{2}$ is a
pure state, the state right before the tracing-out operation must factorize
as:

$U(\ket{\psi_{1}}\otimes\ket{\textbf{0}})\approx\ket{\psi_{2}}\otimes\ket{a_{1}}$
(94)

where $\ket{a_{1}}$ is the state that supports on qubits to be traced out and
is defined on the same lattice. Recalling the definition of pure-state phases,
we conclude the (pure-state) phase equivalence: $\ket{\psi_{1}}\underset{{\rm
pure}}{\sim}\ket{\psi_{2}}\otimes\ket{a_{1}}$. Following a similar argument,
we can also obtain $\ket{\psi_{2}}\underset{{\rm
pure}}{\sim}\ket{\psi_{1}}\otimes\ket{a_{2}}$. We thus have:

$\ket{\psi_{1}}\underset{{\rm
pure}}{\sim}\ket{\psi_{2}}\otimes\ket{a_{1}}\underset{{\rm
pure}}{\sim}\ket{\psi_{1}}\otimes\ket{a_{2}}\otimes\ket{a_{1}}$ (95)

As a result, the state $\ket{a_{1}}\otimes\ket{a_{2}}$ is in the trivial
phase, and $\ket{a_{1}}$ is either in the trivial phase or an invertible
state. Note that the proof assumes there is no ‘catalyst’ effect in phase
equivalence relation: if $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ cannot be LU
connected to each other, then neither does the pair
$\ket{\psi_{1}}\otimes\ket{a}$ and $\ket{\psi_{2}}\otimes\ket{a}$, for any
state $\ket{a}$.

###  A.2 Derivation of Eq.(12)

Let us assume $\mathcal{E}_{A\rightarrow A^{\prime}}$ approximately preserves
the correlation of a bi-partite state $\rho_{AB}$:

$I_{A:B}(\rho)-I_{A^{\prime}:B}(\mathcal{E}_{A\rightarrow
A^{\prime}}(\rho))=\epsilon,$ (96)

After defining $W$ and $\sigma_{A^{\prime}EB}$ the same way as in the proof of
Thm.1, the condition above is equivalent to:

$I_{B:E|A^{\prime}}(\sigma_{A^{\prime}EB})=\epsilon.$ (97)

Using the result in Fawzi and Renner (2015), there exists a reconstruction map
$\mathcal{T}_{A^{\prime}\rightarrow A^{\prime}E}$ to approximately reconstruct
$\sigma_{A^{\prime}EB}$ from $\sigma_{A^{\prime}B}$, with the approximation
error bounded as:

$\epsilon\geq-2\log_{2}F(\sigma_{A^{\prime}EB},\mathcal{T}_{A^{\prime}\rightarrow
A^{\prime}E}(\sigma_{A^{\prime}B})).$ (98)

We still define the recovery map $\mathcal{D}_{A^{\prime}\rightarrow A}$ using
Eq.(9). We have:

$\displaystyle F(\rho_{AB},\mathcal{D}\circ\mathcal{E}(\rho_{AB}))$ (99)
$\displaystyle=$ $\displaystyle
F\left(\text{tr}_{R}\left(U_{W}^{\dagger}U_{W}\left((\rho_{AB})\otimes\ket{0}_{R}\bra{0}\right)U^{\dagger}_{W}U_{W}\right),\text{tr}_{R}\left(U_{W}^{\dagger}\mathcal{T}(\sigma_{A^{\prime}B})U_{W}\right)\right)$
$\displaystyle\geq$ $\displaystyle
F\left(U_{W}\left((\rho_{AB})\otimes\ket{0}_{R}\bra{0}\right)U^{\dagger}_{W},\mathcal{T}(\sigma_{A^{\prime}B})\right)$
$\displaystyle=$ $\displaystyle
F\left(\sigma_{A^{\prime}EB},\mathcal{T}(\sigma_{A^{\prime}B})\right),$

where the inequality is due to the monotonicity of $F$ under quantum channels.
Combining the two expressions above, we arrive at the approximate
recoverability we want:

$\epsilon\geq-\log_{2}F(\rho_{AB},\mathcal{D}\circ\mathcal{E}(\rho_{AB}))$
(100)

###  A.3 Derivation of Eq.(23)

We assume that the two operators $o_{1}^{(\ell)}=o_{1}$ and
$o_{2}^{(\ell)}=o_{2}$ have non-overlapping lightcones $\mathcal{L}(o_{1})$
and $\mathcal{L}(o_{2})$ in the circuit representation of
$\mathcal{F}^{\dagger}$, as illustrated below:

To prove Eq.(23), we decompose $\mathcal{F}^{\dagger}$ into $O(1)$ number of
layers (as is drawn in the figure above). Each individual layer is of the form
$\mathcal{E}^{\dagger}_{1}\otimes\mathcal{E}^{\dagger}_{2}\otimes...\otimes\mathcal{E}^{\dagger}_{K}$,
where each $\mathcal{E}_{I}^{\dagger}$ is a dual channel that acts on a block
of sites referred to as $B_{I}$ such that $B_{I}\cap B_{J}=\varnothing$
whenever $I\neq J$. We inspect $o_{1}o_{2}$’s transformation under a single
layer. Assuming $o_{1}$, $o_{2}$ are supported on $B_{I}$ and $B_{J}$
respectively, we have

$\mathcal{E}^{\dagger}_{1}\otimes\mathcal{E}^{\dagger}_{2}\otimes...\otimes\mathcal{E}^{\dagger}_{K}(o_{1}o_{2})=\mathcal{E}_{I}^{\dagger}(o_{1})\mathcal{E}_{J}^{\dagger}(o_{2})$
(101)

because dual channels are unital. By doing induction over all the layers in
$\mathcal{F}^{\dagger}$, we arrive at our conclusion Eq.(23).

###  A.4 Derivation of Eq.(40)

We prove the channel equality by checking its action on $\ket{0}\bra{0}$,
$\ket{0}\bra{1}$ and $\ket{1}\bra{1}$, which form a basis of linear operators
for a single qubit.

We first act on the isometry and the noise channel:

$\displaystyle(\mathcal{N}_{p}^{X})^{\otimes
b}\circ\mathcal{U}_{w_{b}}(\ket{0}\bra{0})$
$\displaystyle=\sum_{\textbf{s}\in\\{0,1\\}^{b}}p^{|\textbf{s}|}(1-p)^{b-|\textbf{s}|}\ket{\textbf{s}}\bra{\textbf{s}}=\sigma_{00}$
(102) $\displaystyle(\mathcal{N}_{p}^{X})^{\otimes
b}\circ\mathcal{U}_{w_{b}}(\ket{0}\bra{1})$
$\displaystyle=\sum_{\textbf{s}\in\\{0,1\\}^{b}}p^{|\textbf{s}|}(1-p)^{b-|\textbf{s}|}\ket{\textbf{s}}\bra{\bar{\textbf{s}}}=\sigma_{01}$
$\displaystyle(\mathcal{N}_{p}^{X})^{\otimes
b}\circ\mathcal{U}_{w_{b}}(\ket{1}\bra{1})$
$\displaystyle=\sum_{\textbf{s}\in\\{0,1\\}^{b}}p^{|\textbf{s}|}(1-p)^{b-|\textbf{s}|}\ket{\bar{\textbf{s}}}\bra{\bar{\textbf{s}}}=\sigma_{11}$

Then we act the majority vote channel. Its action on $\sigma_{00}$ is:

$\displaystyle\mathcal{E}(\sigma_{00})$
$\displaystyle=\sum_{\textbf{s}}p^{|\textbf{s}|}(1-p)^{b-|\textbf{s}|}\ket{\mathsf{maj}(\textbf{s})}\bra{\mathsf{maj}(\textbf{s})}\otimes\text{tr}(\ket{\mathsf{diff}(\textbf{s})}\bra{\mathsf{diff}(\textbf{s})})$
(103)
$\displaystyle=\left(\sum_{\textbf{s}:|\textbf{s}|<b/2}p^{|\textbf{s}|}(1-p)^{b-|\textbf{s}|}\right)\ket{0}\bra{0}+\left(\sum_{\textbf{s}:|\textbf{s}|>b/2}p^{|\textbf{s}|}(1-p)^{b-|\textbf{s}|}\right)\ket{1}\bra{1}$
$\displaystyle=(1-p^{\prime})\ket{0}\bra{0}+p^{\prime}\ket{1}\bra{1}$
$\displaystyle=\mathcal{N}^{X}_{p^{\prime}}(\ket{0}\bra{0}),$

where

$p^{\prime}=\sum_{s:|s|>b/2}p^{|s|}(1-p)^{b-|s|}=\sum_{k=(b+1)/2}^{b}\binom{b}{k}p^{k}(1-p)^{b-k}$
(104)

Following a very similar calculation, we have:

$\displaystyle\mathcal{E}(\sigma_{01})$
$\displaystyle=(1-p^{\prime})\ket{0}\bra{1}+p^{\prime}\ket{1}\bra{0}$
$\displaystyle=\mathcal{N}^{X}_{p^{\prime}}(\ket{0}\bra{1})$ (105)
$\displaystyle\mathcal{E}(\sigma_{11})$
$\displaystyle=(1-p^{\prime})\ket{1}\bra{1}+p^{\prime}\ket{0}\bra{0}$
$\displaystyle=\mathcal{N}^{X}_{p^{\prime}}(\ket{1}\bra{1})$

Then we obtain Eq.(40).

###  A.5 Derivation of Eq.(80)

The statement is equivalent to stating that
$\mathcal{D}_{\psi}\circ\mathcal{N}_{\psi}(\ket{\phi})$ is anyon free, namely:

$\text{tr}(\mathcal{D}_{\psi}\circ\mathcal{N}_{\psi}(\ket{\phi})O)=1\quad\forall
O\in\left\\{A_{\square},B_{+}\right\\},~{}\ket{\phi}\in V$ (106)

which is further equivalent to:

$\text{tr}(\ket{\phi}\bra{\phi}\mathcal{N}^{\dagger}_{\psi}\circ\mathcal{D}^{\dagger}_{\psi}(O))=1$
(107)

We use $A$ to denote the spatial support of
$\mathcal{N}^{\dagger}_{\psi}\circ\mathcal{D}^{\dagger}_{\psi}(O)$. $A$ is a
topologically trivial region because both $\mathcal{N}$ and $\mathcal{D}$ are
LC transformations. Thus the reduced density matrix of $\ket{\psi}$ and
$\ket{\phi}$ over the region $A$ must be the same, and we can conclude the
above equation holds for arbitrary $\ket{\phi}\in V$.

##  Appendix B Convergence of real-space RGs

In Sec.III.4 it is stated that if a mixed state’s RG flow
$\\{\rho^{(0)},\rho^{(1)},...\\}$ satisfies the following conditions for large
enough $\ell$:

$\displaystyle\text{Eq.\eqref{eq:
convergence_condition1}:~{}~{}~{}}F(\rho^{(\ell)},\rho^{(\infty)})\simeq\exp(-\alpha\theta^{(\ell)}L^{(\ell)})$
$\displaystyle\text{Eq.\eqref{eq:
convergence_condition2}:~{}~{}~{}}\theta^{(\ell+1)}\lesssim(\theta^{(\ell)})^{\gamma}~{}~{}~{}{\rm
when}~{}~{}~{}\theta^{(\ell)}\rightarrow 0_{+},$

then choosing

$\ell^{*}=O(\log\log(L/\epsilon))$ (108)

guarantees $F(\rho^{(\ell^{*})},\rho^{(\infty)})>1-\epsilon$ for a small
$\epsilon$. $L^{(\ell)}=L/b^{\ell}$ is the renormalized system size.

In this appendix we first prove the statement, then show the validness of
conditions Eqs.(25),(26) in several scenarios, including all examples we
studied in the main text.

Suppose $\ell_{0}=O(1)$ simultaneously satisfies the following two conditions:
Eq.(26) holds when $\ell>\ell_{0}$ and that
$\theta^{(\ell_{0})}:=\theta_{0}<1$. Iterating the condition $(\ell-\ell_{0})$
times, we get:

$\theta^{(\ell)}<\left(\theta_{0}\right)^{\gamma^{\ell-\ell_{0}}}.$ (109)

We need to find how large $\ell$ needs to be, in order to satisfy:

$\exp(-\alpha\theta^{(\ell)}L^{(\ell)})>1-\epsilon,$ (110)

which is implied by (note that $L^{(\ell)}<L$):

$\theta^{(\ell)}<\epsilon/L,$ (111)

which is further implied by:

$\ell>\ell_{0}+\log_{\gamma}\log_{\theta_{0}^{-1}}(L/\epsilon)=O(\log\log(L/\epsilon))$
(112)

This completes the proof.

In the rest of this appendix section, we show several scenarios where the
condition Eq.(25)(26) is satisfied. For the case of matrix product state with
a tree tensor network RG, the analysis is done thoroughly in a recent work
Malz  _et al._ (2023).

###  B.1 Classical statistical mechanics models

Let us consider a classical statistical mechanics model with the Hamiltonian:

$H_{g}=H_{0}-gH^{\prime}$ (113)

We assume that under some given RG process, $H_{0}$ is the RG fixed point
while $H^{\prime}$ is an irrelevant perturbation with respect to $H_{0}$.
Further, we assume both $H_{0}$ and $H^{\prime}$ are summations of spatially
local terms each involving $O(1)$ number of sites. Further, each spin only
appears $O(1)$ number of terms, and terms are uniformly bounded.

Let $\rho_{g}\propto\exp(-H_{g})$ be the Gibbs state of $H_{g}$. We are
interested in how fast it approaches the fixed-point $\rho_{0}$. We use the
fidelity as a measure of closeness between the two states:

$F(\rho_{g},\rho_{0})=|\sqrt{\rho_{g}}\sqrt{\rho_{0}}|_{1}=\text{tr}(e^{-H_{g}/2}e^{-H_{0}/2})/\sqrt{Z_{g}Z_{0}}=Z_{g/2}/\sqrt{Z_{g}Z_{0}}$
(114)

where $Z_{g}:=\text{tr}(\rho_{g})$ is the partition function. We note that
since $\rho_{g}$ is a classical state, the fidelity function $F$ coincides
with the Bhattacharyya coefficient, a measure of similarity for classical
distributions.

We have:

$Z_{g}=\text{tr}\left(e^{-H_{0}}\sum_{n}\frac{g^{n}}{n!}(H^{\prime})^{n}\right)=Z_{0}\sum_{n}\frac{g^{n}}{n!}\langle(H^{\prime})^{n}\rangle_{0}$
(115)

Without loss of generality, we assume each term in $H^{\prime}$ is positive,
so that: $0\leq\langle(H^{\prime})^{n}\rangle_{0}\leq|H^{\prime}|^{n}$. Thus
we have:

$Z_{0}\leq Z_{g}\leq Z_{0}e^{g|H^{\prime}|}$ (116)

Combining with Eq.(114), we get:

$F(\rho_{g},\rho_{0})\geq e^{-g|H^{\prime}|}$ (117)

Since $|H^{\prime}|$ is a summation of spatially local terms whose norms are
uniformly bounded, $|H^{\prime}|$ must be upper-bounded by $\alpha L$ for some
$\alpha=O(1)$ and large $L$. This leads to:

$F(\rho_{g},\rho_{0})\geq e^{-g\alpha L}$ (118)

Thus Eq.(25) is satisfied. The Eq.(26) is satisfied because $g$ is an
irrelevant coupling for a non-critical fixed point.

###  B.2 Noisy GHZ state in Sec.V

(a) Bit flip noise

The fidelity function is:

$\displaystyle F(\rho^{X}_{p,L},~{}\rho^{X}_{0,L})$
$\displaystyle=\bra{\mathrm{GHZ}}\rho^{X}_{p,L}\ket{\mathrm{GHZ}}$ (119)
$\displaystyle=\frac{1}{2}\sum_{\textbf{s}\in\\{0,1\\}^{L}}p^{|\textbf{s}|}(1-p)^{L-|\textbf{s}|}\bra{\mathrm{GHZ}}(\ket{\textbf{s}}\bra{\textbf{s}}+\ket{\bar{\textbf{s}}}\bra{\textbf{s}}+\ket{\textbf{s}}\bra{\bar{\textbf{s}}}+\ket{\bar{\textbf{s}}}\bra{\bar{\textbf{s}}})\ket{\mathrm{GHZ}}$
$\displaystyle=(1-p)^{L}+p^{L}$ $\displaystyle\approx e^{-pL}$

The approximation follows from $p\ll 1$. The iteration relation of $p$
satisfies:

$p^{\prime}=\sum_{k=(b+1)/2}^{b}\binom{b}{k}p^{k}(1-p)^{b-k}\leq
2^{b-1}p^{(b+1)/2}\leq p^{b/2}$ (120)

the last inequality holds if $p<2^{-2b+2}$, which can be always achieved after
an $O(1)$ number of RG iterations starting from any $p\in(0,0.5)$.

(b) Phase-flip noise

The fidelity function is:

$\displaystyle F(\rho^{Z}_{p,L},~{}\rho^{Z}_{1/2,L})$
$\displaystyle=\text{tr}\sqrt{\sqrt{\rho^{Z}_{1/2,L}}~{}\rho^{Z}_{p,L}~{}\sqrt{\rho^{Z}_{1/2,L}}}$
(121) $\displaystyle=\frac{1}{2}+\frac{1}{2}\sqrt{1-(1-2p)^{2L}}$

We notice that the fidelity function actually goes to $1$ when
$L\rightarrow\infty$. This can be treated as a special case of the condition
Eq.(25) for $\alpha=0$.

The iteration relation for $p$ around the stable fixed-point $p=0.5$ is:

$(p^{\prime}-1/2)=(p-1/2)^{b}$ (122)

###  B.3 Thermal toric code state in Sec.VI.2

We consider the fidelity between the finite temperature state and the infinite
temperature one. Note that the latter is proportional to identity. Thus,

$\displaystyle F(\rho_{\beta},~{}\rho_{0})$
$\displaystyle=2^{-\frac{L^{2}}{2}}\text{tr}\sqrt{\rho_{\beta}}$ (123)
$\displaystyle=2^{-\frac{L^{2}}{2}}\sum_{\textbf{m},\textbf{e},\textbf{l}}\sqrt{\mathrm{Pr}_{m}(\textbf{m})\mathrm{Pr}_{e}(\textbf{e})\mathrm{Pr}_{l}(\textbf{l})}$
$\displaystyle=2^{-\frac{L^{2}}{2}+1}\left(\sum_{\textbf{m}}\sqrt{\mathrm{Pr}_{m}(\textbf{m})}\right)^{2}$
$\displaystyle\geq
2^{-\frac{L^{2}}{2}+1}\left(\sum_{\textbf{m}:\pi(\textbf{m})=0}p_{\beta}^{|\textbf{m}|/{2}}(1-p_{\beta})^{(L^{2}/2-|\textbf{m}|)/2}\right)^{2}$
$\displaystyle=2^{-\frac{L^{2}}{2}+1}(1-p_{\beta})^{L^{2}/2}\left(\sum_{\textbf{m}:\pi(\textbf{m})=0}e^{-\beta|\textbf{m}|}\right)^{2}$
$\displaystyle\geq 2^{-\frac{L^{2}}{2}+1}2^{-L^{2}/2}\left(2^{L/2-1}e^{-\beta
L^{2}/2}\right)^{2}$ $\displaystyle=2^{-\beta L^{2}-1}$

The iteration relation of the inverse temperature $\beta$ is given by

$\beta^{\prime}=\tanh^{-1}\tanh^{4}\beta\approx\beta^{4}$ (124)

at small $\beta$, which satisfies the condition.

###  B.4 Noisy toric code state in Sec.VII

The renormalized state’s overlap with the ground state subspace $V$ is:

$F^{(\ell)}:=\text{tr}\left(\rho^{(\ell)}\Pi_{V}\right)$ (125)

where $\Pi_{V}$ is the projection to $V$.

Since $\rho$ is always diagonal in the anyon number basis throughout the RG
process, the quantity is the probability of having zero anyon after $l$ steps
of RG. The quantity is bounded by the anyon density $q_{p}^{(\ell)}$:

$q_{p}^{(\ell)}=\sum_{a=1}^{L^{2}}\mathrm{Pr}(|m|=a)\frac{a}{L^{2}}\geq\frac{1}{L^{2}}\sum_{a=1}^{L^{2}}\mathrm{Pr}(|m|=a)=\frac{1}{L^{2}}(1-F^{(\ell)})$
(126)

leading to:

$F^{(\ell)}\geq 1-L^{2}q_{p}^{(\ell)}\simeq e^{-L^{2}q_{p}^{(\ell)}}.$ (127)

The last approximate equal sign holds only when $L^{2}q_{p}^{(\ell)}$ is
small.

Furthermore, according to the numerical results Fig.7(c), the decay of anyon
density follows:

$q_{p}^{(\ell+1)}<(q_{p}^{(\ell)})^{\gamma}$ (128)

for some $\gamma>1$.

##  Appendix C RG of a mixed $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT state

In this appendix, we consider the RG of a mixed SPT (symmetry protected
topological) state. The problem was first considered in de Groot  _et al._
(2022), where the authors use the string order parameter as a definition for
the mixed state SPT. They show that, when a pure SPT state is subject to
noise, the string order parameter (i.e. the SPT phase) is preserved if and
only if the noise is strongly symmetric, meaning that all its Kraus operators
commute with the symmetry operator.

The above definition (via string order parameters) of mixed-state SPT actually
agrees with the LC transformation based definition. To show this, we put
forward a symmetric RG transformation that brings the noisy state back to a
clean one. The RG is the same as the one proposed in Lake  _et al._ (2022).
Although in Lake  _et al._ (2022) the circuit is designed for recognizing
pure-state phases, we point out that it can be readily applied on a mixed-
state SPT states.

Here we demonstrate the principle with a simple (1+1)D SPT state, namely the
$Z_{2}\times Z_{2}$ SPT. Consider a 1D lattice spin chain where in the bulk
each site contains 2 qubits, labeled as $A$ and $B$. The pure SPT wavefunction
can be written as:

$\ket{\psi}=\bigotimes_{i=-\infty}^{+\infty}\ket{\mathrm{EPR}_{i_{B},(i+1)_{A}}},$
(129)

where each
$\ket{\mathrm{EPR}_{i,j}}:=\frac{1}{\sqrt{2}}(\ket{0_{i}0_{j}}+{\ket{1_{i}1_{j}}})$.
Note that the above way of defining the $Z_{2}\times Z_{2}$ SPT is related to
the cluster state by rotating the two spins within a unit cell with a CNOT
gate. It is straightforward to verify that the state has a $Z_{2}\times Z_{2}$
symmetry generated by 2 generators:

$U^{X}=\prod_{i}X_{i_{A}}X_{i_{B}},\quad U^{Z}=\prod_{i}Z_{i_{A}}Z_{i_{B}}.$
(130)

Two generators define two string order parameters:

$S^{X}_{ij}=X_{i_{B}}\left(\prod_{k=i+1}^{j-1}X_{k_{A}}X_{k_{B}}\right)X_{j_{A}},\quad
S^{Z}_{ij}=Z_{i_{B}}\left(\prod_{k=i+1}^{j-1}Z_{k_{A}}Z_{k_{B}}\right)Z_{j_{A}}$
(131)

One signature of the SPT order in $\ket{\psi}$ is that expectation value of
string order parameters does not decay with $|i-j|$.

We consider the $XX$ dephasing channel:

$\mathcal{N}^{XX}_{p}(\cdot)=(1-p)(\cdot)+pX_{A}X_{B}(\cdot)X_{A}X_{B},$ (132)

whose action can be realized by the pair of spins within each site being
flipped simultaneously with probability $p$. The channel is strongly symmetric
under $U_{X}$ and $U_{Z}$, because its Kraus operators commute with symmetry
operators.

The noisy SPT state of our interest is obtained by applying $\mathcal{N}_{p}$
uniformly upon $\psi$:

$\sigma_{p,L}=\mathcal{N}_{p}^{\otimes L}(\ket{\psi}\bra{\psi}).$ (133)

It will be useful to unravel $\sigma_{p}$ into a classical mixture of pure SPT
states decorated by domain walls:

$\displaystyle\sigma_{p}$
$\displaystyle=\mathbb{E}[\ket{\psi_{\textbf{s}}}\bra{\psi_{\textbf{s}}}]=\sum_{\textbf{s}\in\\{0,1\\}^{L}}P(\textbf{s})\ket{\psi_{\textbf{s}}}\bra{\psi_{\textbf{s}}}$
(134) $\displaystyle P(\textbf{s})$
$\displaystyle:=p^{|\textbf{s}|}(1-p)^{|\textbf{s}|}$
$\displaystyle\ket{\psi_{\textbf{s}}}$
$\displaystyle:=\bigotimes_{i}X^{\textbf{s}_{i}}_{i_{A}}X^{\textbf{s}_{i+1}}_{(i+1)_{B}}\ket{\mathrm{EPR}_{i_{B},(i+1)_{A}}}$

Now we focus on a block of $b$ sites within the state and devise the channel
that coarse-grains the block. The part of the $\ket{\psi_{\textbf{s}}}$ within
the block can be drawn as (for $b=3$):

where each hollowed circle is either an identity gate if $s=0$ or an $X$ gate
if $s=1$. Each $s_{i}$ is an independent Bernoulli random variable with chance
$p$.

The renormalized site is formed by qubits $\\{1_{A},b_{B}\\}$, while the
$(b-2)$ entangled pairs supporting on the remaining qubits
$\\{1_{B},2_{A},2_{B},...,b_{A}\\}$ can help us infer and correct errors
$s_{1}$ and $s_{b}$ before they get traced out. More concretely, the coarse-
graining channel $\mathcal{E}$’s action is the following:

  1. 1.

Measure each Bell pairs within the block in the $ZZ$ basis, whose outcome
records the domain-walls between $s_{i}$, i.e.,
$\\{t_{1}=s_{1}-s_{2},t_{2}=s_{3}-s_{2},...,t_{b-1}=s_{b-1}-s_{b}\\}$. The
outcomes decide all the $s_{i}$ up to a global flip, i.e. once we assume a
value for the first error $s_{1}=\hat{s}_{1}$, the remaining $s$ are also
uniquely determined by: $\hat{s}_{k}=(\sum_{j=1}^{k-1}t_{j})-\hat{s}_{1}$.

  2. 2.

Assume

$\hat{s}_{1}=\mathrm{argmax}_{s_{1}\in\\{0,1\\}}\mathrm{Pr}(s_{1}|t_{1},...,t_{b-1})$
(135)

to be the actual error, and correct all sites within the block. Namely,
applying $X_{i_{A}}X_{i_{B}}$ if $\hat{s}_{i}=1$. After this step, all Bell
pairs within the block are noiseless and decoupled from $\\{1_{B},b_{A}\\}$.

  3. 3.

Trace out $\\{1_{B},2_{A},2_{B},...,b_{A}\\}$. Then we obtain the renormalized
site with $\\{1_{B},b_{A}\\}$,

It is clear that channel $\mathcal{E}$ is strongly symmetric under
$Z_{2}\times Z_{2}$ since each individual step is. The renormalized site gets
a $XX$ error if and only if $\hat{s}_{1}\neq s_{1}$, whose probability we
denote as $p^{\prime}$. Thus we find that the normalized state is still a
symmetrically dephased SPT state, but with a normalized noise strength
$p^{\prime}$:

$\mathcal{E}^{\otimes\frac{L}{b}}(\sigma^{\mathrm{SPT}}_{p,L})=\sigma^{\mathrm{SPT}}_{p^{\prime},L/b}$
(136)

The explicit form of $p^{\prime}$ is:

$p^{\prime}=\sum_{k=(b+1)/2}^{b}\binom{b}{k}p^{k}(1-p)^{b-k}$ (137)

Thus we obtain a similar RG flow as in the $X$-dephased GHZ state studied in
Sec.V: symmetrically decohered SPT state flows back to a pure SPT state when
$0\leq p<0.5$.