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# Exact Calculations of Coherent Information for Toric Codes under
Decoherence: Identifying the Fundamental Error Threshold

Jong Yeon Lee<EMAIL_ADDRESS>Department of Physics, University of
California, Berkeley, California 94720, USA  Department of Physics, University
of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

###### Abstract

The toric code is a canonical example of a topological error-correcting code.
Two logical qubits stored within the toric code are robust against local
decoherence, ensuring that these qubits can be faithfully retrieved as long as
the error rate remains below a certain threshold. Recent studies have explored
such a threshold behavior as an intrinsic information-theoretic transition,
independent of the decoding protocol. These studies have shown that
information-theoretic metrics, calculated using the Renyi (replica)
approximation, demonstrate sharp transitions at a specific error rate.
However, an exact analytic expression that avoids using the replica trick has
not been shown, and the connection between the transition in information-
theoretic capacity and the random bond Ising model (RBIM) has only been
indirectly established. In this work, we present the first analytic expression
for the coherent information of a decohered toric code, thereby establishing a
rigorous connection between the fundamental error threshold and the
criticality of the RBIM.

_Introduction._ — In the realm of information transmission and utilization,
protecting data against errors stands as a paramount concern [1]. This issue
takes an even greater significance in the context of quantum information,
which is fragile and non-clonable [2]. Consequently, the study of robust
quantum memory and computation under the presence of a finite error rate has
emerged as both a practical and intriguing problem [3, 4, 5, 6, 7].

Particularly, the error threshold of the toric code, a stereotypical example
of a topological quantum error correction (QEC) code, has been extensively
studied [8, 9]. Dennis et al. [9] demonstrated that the error threshold of the
maximum entropy decoder under Pauli errors maps to the critical temperature of
a random bond Ising model (RBIM) along the Nishimori line [10, 11]. Following
this approach, decoding problems of various quantum codes under noises have
been associated with statistical models and their transition behaviors [12,
13, 14, 15]. However, the decoding error threshold depends on the specific
decoding algorithm in use, and the threshold obtained in this way can be
different from the fundamental one.

To understand the fundamental error threshold of the toric code, recent
studies have explored the information-theoretic properties of the toric code
subject to Pauli errors without specific consideration of the decoding process
[16, 17]. These studies have identified critical points in information-
theoretic measures by employing the replica method to establish an upper bound
for the error threshold. Furthermore, these works have indirectly suggested
that the extrapolation of these critical error rates to the
$n\,{\rightarrow}\,1$ limit would be consistent with the critical point of the
RBIM along the Nishimori line. Similar transition behavior induced by
decoherence has been studied via different approaches, such as separability
criterion [18, 19]. However, the exact behavior of the quantum information
retained within the decohered toric code has not been understood yet.

The main contribution of this work is to identify the fundamental error
threshold of the toric code subject to Pauli errors without the use of
approximation methods and to establish rigorous correspondence with the RBIM.
This is achieved by the analytic calculation of coherent information, a metric
quantifying the amount of decodable quantum information retained in the
system. Given that robust coherent information constitutes both a necessary
and sufficient condition for error correction [20, 21], the transition point
of coherent information establishes a fundamental upper bound for
decodability.

Figure 1: Coherent Information under Pauli-$Z$ and $X$ errors for (a) Two raw
physical qubits $I_{c}^{\textrm{raw}}$ and (b) Two logical qubits of the toric
code state $I_{c}^{\textrm{tc}}$ in the thermodynamic limit. The dashed line
in (b) is the contour of $I_{c}^{\textrm{raw}}=0$, which passes the point
$(0.1100,0.1100)$. This point is very close to the critical point of the
Nishimori line $(0.1094,0.1094)$ [22].

_Model._ — The toric code Hamiltonian is defined as

$\displaystyle H$ $\displaystyle=-\sum_{v}A_{v}-\sum_{p}B_{p}$ (1)

where $A_{v}:=\prod_{e\ni v}{Z}_{e}$ and $B_{p}:=\prod_{e\in p}X_{e}$. The
ground state is characterized by $A_{v}=B_{p}=1$. On the torus, the ground
state is 4-fold degenerate with two logical qubits. Logical qubits reside on
the space where the following effective Pauli operators act:
$\overline{\bm{X}}_{i}\,{:=}\,\prod_{e\in C_{i}}X_{e}$ and
$\overline{\bm{Z}}_{i}\,{:=}\,\prod_{e\in C^{\perp}_{\bar{i}}}Z_{e}$ where
$C_{i}^{\vphantom{\perp}}$ ($C_{i}^{\perp}$) is a (dual) cycle along the
$i$-th axis. Here, $C^{\perp}_{\bar{1}}\,{=}\,C^{\perp}_{2}$ and
$C^{\perp}_{\bar{2}}\,{=}\,C^{\perp}_{1}$ such that
$[\overline{\bm{Z}}_{i},\overline{\bm{X}}_{i}]=0$. While $C_{i}$ is defined
along the edges, $C_{i}^{\perp}$ is along the dual edges as in Fig. 2(a).

To further proceed, we maximally entangle two logical qubits of a toric code
with two reference qubits. In order to enforce the Bell-type maximal
entanglement, we require
$Z^{\textrm{r}}_{i}\overline{\bm{Z}}_{i}^{\vphantom{r}}\,{=}\,X_{i}^{\textrm{r}}\overline{\bm{X}}_{i}^{\vphantom{r}}\,{=}\,1$
for $i\,{=}\,1,2$, where $Z_{1,2}^{\textrm{r}}$ and $X_{1,2}^{\textrm{r}}$ are
Pauli operators acting on two reference qubits. Denoting the system of a toric
code as $Q$ and the reference as $R$, the full density matrix $\rho_{RQ}$ and
reduced density matrix $\rho_{Q}$ is given as

$\displaystyle\rho_{RQ}$
$\displaystyle:=4\prod_{i\in\\{1,2\\}}\quantity(\frac{1+Z_{i}\overline{\bm{Z}}_{i}}{2})\quantity(\frac{1+X_{i}\overline{\bm{X}}_{i}}{2})(\mathbb{I}_{R}\otimes\rho_{Q})$
$\displaystyle\rho_{Q}$
$\displaystyle:=\mathrm{tr}_{R}(\rho_{RQ})=\frac{1}{4}\prod_{v}\quantity(\frac{1+A_{v}}{2})\prod_{p}\quantity(\frac{1+B_{p}}{2}).$
(2)

The coherent information of the system $Q$ under a decoherence channel ${\cal
E}$ is defined as [20, 21]

$\displaystyle I_{c}(R,Q;{\cal E})$ $\displaystyle:=S({\cal
E}[\rho_{Q}])-S((\textrm{id}_{R}\otimes{\cal E})[\rho_{RQ}]).$ (3)

By maximally entangling logical qubits with a reference, this quantity
monitors the amount of identifiable information that persists despite
decoherence. When ${\cal E}$ is trivial, $I_{c}\,{=}\,2\log 2$ which shows
that the system and reference have entanglements of two Bell pairs. Under the
application of a non-trivial decoherence channel, the coherent information
monotonically decreases and the initial value sets its upper bound.

Throughout the paper, we consider decoherence channels for uncorrelated
Pauli-$X$ and $Z$ errors:

$\displaystyle{\cal E}^{z}_{e}:\rho$
$\displaystyle\rightarrow(1-p_{z})\rho+p_{z}Z^{\vphantom{\dagger}}_{e}\rho
Z^{\dagger}_{e},\quad{\cal E}^{z}=\prod_{e}{\cal E}^{z}_{e}$
$\displaystyle{\cal E}^{x}_{e}:\rho$
$\displaystyle\rightarrow(1-p_{x})\rho+p_{x}X^{\vphantom{\dagger}}_{e}\rho
X^{\dagger}_{e},\quad{\cal E}^{x}=\prod_{e}{\cal E}^{x}_{e}.$ (4)

Our goal is to diagonalize ${\cal E}[\rho_{Q}]$ and $(\textrm{id}\otimes{\cal
E})[\rho_{RQ}]$ to evaluate Eq. (3). One useful way to represent ${\cal
E}[\rho]$ is

$\displaystyle{\cal E}[\rho]=\sum_{{\bm{l}}_{x},{\bm{l}}_{z}}\prod_{i\in
x,z}(1-p_{i})_{\vphantom{i}}^{2N-|{\bm{l}}_{i}|}p_{i}^{|{\bm{l}}_{i}|}\cdot
X^{\vphantom{\dagger}}_{{\bm{l}}_{x}}Z^{\vphantom{\dagger}}_{{\bm{l}}_{z}}\rho
Z^{\dagger}_{{\bm{l}}_{z}}X^{\dagger}_{{\bm{l}}_{x}}$ (5)

where ${\bm{l}}_{i}\,{=}\,\\{l_{i,e}\\}$ with $l_{i,e}\,{\in}\,\\{0,1\\}$ is a
vector representing a string such that the presence of an edge $e$ in
${\bm{l}}_{i}$ is specified by $l_{i,e}\,{\neq}\,0$,
$|{\bm{l}}_{i}|\,{=}\,\sum_{i}l_{i,e}$,
$X_{\bm{l}}\,{:=}\,\prod_{e}X^{l_{e}}_{e}$,
$Z_{\bm{l}}\,{:=}\,\prod_{e}Z^{l_{e}}_{e}$, and $N$ is the number of vertices.
This convention allows us to encode the presence of edges within a string
through a straightforward numerical scheme.

Figure 2:  Conventions. (a) Cycles along the edges of the original (blue) and
dual (red) lattices $C_{i}$ and $C_{i}^{\perp}$ respectively. (b) A set of
blue plaquettes $\tilde{{\bm{m}}}$ in the dual lattice to denote $e$-anyons
and corresponding string (thick black lines) ${\bm{l}}_{\tilde{{\bm{m}}}}$ of
Pauli-$X$s. (c) A set of red plaquettes ${\bm{m}}$ in the original lattice to
denote $m$-anyons and corresponding string ${\bm{l}}_{\bm{m}}$ of Pauli-$Z$s.

_Diagonalization I_.— Consider a toric code state
$|\psi^{\textrm{tc}}_{0}\rangle$ satisfying
$\overline{\bm{X}}_{1,2}|\psi^{\textrm{tc}}_{0}\rangle\,{=}\,|\psi^{\textrm{tc}}_{0}\rangle$
and the density matrix $\rho_{0}\,{=}\,|\psi_{0}\rangle\langle\psi_{0}|$. The
key observation is that ${\cal E}[\rho_{0}]$ commutes with all local
stabilizers $\\{A_{v}\\}$ and $\\{B_{p}\\}$:

$\displaystyle A_{v}{\cal E}[\rho_{0}]={\cal E}[\rho_{0}]A_{v},\quad
B_{p}{\cal E}[\rho_{0}]={\cal E}[\rho_{0}]B_{p}\quad\forall v,p.$ (6)

Accordingly, we can diagonalize ${\cal E}[\rho_{0}]$ by the eigenvectors of
these stabilizers. However, since there are only $2(N\,{-}\,1)$ independent
stabilizers in the system of $2N$ qubits, the density matrix is diagonalized
into $2^{2(N-1)}$ blocks of $2^{2}\times 2^{2}$ logical subspace. To further
proceed, let us use eigenvalues of $\overline{\bm{X}}_{1,2}$ to index rows and
columns of this logical subspace. In this basis, non-trivial elements of a
decohered density matrix can be referred to by the following labels:

  * •

Eigenvalues of $B_{p}$s ${\bm{m}}\,{=}\,\\{m_{p}\\}$ ($m$-anyons).
${\bm{l}}_{\bm{m}}$ is a representative Pauli-$Z$ string living in the edges
of the dual lattice. $Z_{{\bm{l}}_{\bm{m}}}$ acting on
$|\psi_{0}^{\textrm{tc}}\rangle$ creates the $m$-anyon configuration
${\bm{m}}$, satisfying
$\partial_{\scriptscriptstyle{\perp}}{\bm{l}}_{\bm{m}}\,{\equiv}\,{\bm{m}}$,
where the subscript indicates the boundary operator is defined in the dual
lattice 111The boundary operator $\partial$
($\partial^{\scriptscriptstyle{\perp}}$) maps a set of values defined on
(dual) edges into a set of values defined on (dual) vertices;
$(\partial{\bm{l}})_{v}\,{:=}\,\sum_{v^{\prime}\ni v}l_{(v,v^{\prime})}$. Note
that the plaquettes of the original lattice corresponds the vertices of the
dual lattice. . See Fig. 2(b).

  * •

Eigenvalues of $A_{v}$s $\tilde{{\bm{m}}}\,{=}\,\\{m_{v}\\}$ ($e$-anyons).
${\bm{l}}_{\tilde{{\bm{m}}}}$ is a representative Pauli-$X$ string in the
original lattice. $X_{{\bm{l}}_{\tilde{\bm{m}}}}$ acting on
$|\psi_{0}^{\textrm{tc}}\rangle$ creates the $e$-anyon configuration
$\tilde{{\bm{m}}}$, satisfying
$\partial{\bm{l}}_{\tilde{\bm{m}}}\,{\equiv}\,{\tilde{\bm{m}}}$. See Fig.
2(c).

  * •

Eigenvalues of $(\overline{\bm{X}}_{1},\overline{\bm{X}}_{2})$ for the rows
and columns of the $2^{2}\times 2^{2}$ logical subspace respectively. Instead
of directly using eigenvalues, we use its logarithm
${\bm{a}}\,{=}\,(a_{1},a_{2})$ and
${\bm{a^{\prime}}}\,{=}\,(a^{\prime}_{1},a^{\prime}_{2})$ such that
$\bm{X}_{i}\,{=}\,e^{\mathrm{i}\pi a_{i}}$.

Accordingly, ${\cal E}[\rho_{0}]$ is spanned by the following basis:

$\displaystyle\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}}$
$\displaystyle:=X_{{\bm{l}}_{\tilde{{\bm{m}}}}}^{\vphantom{\dagger}}Z_{{\bm{l}}_{{\bm{m}}}}^{\vphantom{\dagger}}\overline{\bm{Z}}_{{\bm{a}}}^{\vphantom{\dagger}}|\psi_{0}\rangle\langle\psi_{0}|\overline{\bm{Z}}_{{\bm{a^{\prime}}}}^{\dagger}Z_{{\bm{l}}_{{\bm{m}}}}^{\dagger}X_{{\bm{l}}_{\tilde{{\bm{m}}}}}^{\dagger},$
$\displaystyle\overline{\bm{Z}}_{{\bm{a}}}$
$\displaystyle:=\overline{\bm{Z}}_{1}^{a_{1}}\overline{\bm{Z}}_{2}^{a_{2}}$
$\displaystyle{\cal E}[\rho_{0}]$
$\displaystyle=\sum_{{\bm{m}}{\tilde{\bm{m}}}{\bm{a}}{\bm{a^{\prime}}}}\mathrm{tr}({\cal
E}[\rho_{0}](\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}})^{\dagger})\cdot\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}}.$
(7)

since
$\mathrm{tr}(\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}}(\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}})^{\dagger})\,{=}\,1$.
The coefficient is evaluated by plugging Eq. (5) in Eq. (Exact Calculations of
Coherent Information for Toric Codes under Decoherence: Identifying the
Fundamental Error Threshold):

$\displaystyle\mathrm{tr}({\cal
E}[\rho_{0}](\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}})^{\dagger})=\sum_{{\bm{l}}_{x},{\bm{l}}_{z}}\Big{(}\prod_{i\in
x,z}(1-p_{i})_{\vphantom{i}}^{2N-|{\bm{l}}_{i}|}p_{i}^{|{\bm{l}}_{i}|}\Big{)}$
$\displaystyle\qquad\qquad\quad\times\absolutevalue{\langle\psi_{0}|Z_{{\bm{l}}_{z}}^{\dagger}X_{{\bm{l}}_{x}}^{\dagger}X_{{\bm{l}}_{\tilde{\bm{m}}}}^{\vphantom{\dagger}}Z_{{\bm{l}}_{{\bm{m}}}}^{\vphantom{\dagger}}\overline{\bm{Z}}_{{\bm{a}}}|\psi_{0}\rangle}^{2}\delta_{{\bm{a}},{\bm{a^{\prime}}}}.$
(8)

For the overlap not to vanish, two conditions must be satisfied: $(i)$
$\partial_{\scriptscriptstyle{\perp}}({\bm{l}}_{\bm{m}}\,{-}\,{\bm{l}}_{z})\,{\equiv}\,\bm{0}$
and
$\partial({\bm{l}}_{\tilde{\bm{m}}}\,{-}\,{\bm{l}}_{x})\,{\equiv}\,\bm{0}$.
$(ii)$ ${\bm{l}}_{z}\,{-}\,{\bm{l}}_{\bm{m}}$ is homologically equivalent to
the loop $\bm{L}^{\perp}_{\bm{a}}$ defined by the vector ${\bm{a}}$. If
$a_{i}\,{\neq}\,0$, $\bm{L}^{\perp}_{\bm{a}}$ has a non-trivial dual cycle
along $i$-th direction. Symbolically, this is represented as
$[{\bm{l}}_{z}\,{-}\,{\bm{l}}_{\bm{m}}]\,{\equiv}\,[\bm{L}^{\perp}_{\bm{a}}]\,{=}\,{\bm{a}}$
where the bracket notation indicates the homotopy class. Thus, we get

$\displaystyle\mathrm{tr}({\cal
E}[\rho_{0}](\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}})^{\dagger})$
$\displaystyle=\sum_{\begin{subarray}{c}\partial_{\perp}({\bm{l}}-{\bm{l}}_{\bm{m}})=0,\\\
[{\bm{l}}-{\bm{l}}_{\bm{m}}]={\bm{a}}\end{subarray}}(1-p_{z})^{2N-|{\bm{l}}|}p_{z}^{|{\bm{l}}|}\,\,\delta_{{\bm{a}},{\bm{a^{\prime}}}}$
$\displaystyle\times\sum_{\begin{subarray}{c}\partial({\bm{l}}^{\prime}-{\bm{l}}_{\tilde{\bm{m}}})=0\end{subarray}}(1-p_{x})^{2N-|{\bm{l}}^{\prime}|}p_{x}^{|{\bm{l}}^{\prime}|}.$
(9)

To facilitate the discussion, we generalize our notation. For a given string
${\bm{l}}$, we use the same symbol to denote a set of values defined on edges
${\bm{l}}\,{=}\,\\{l_{e}\\}$, where each $l_{e}\,{=}\,{-}1$ if
$e\,{\in}\,{\bm{l}}$ and $l_{e}\,{=}\,1$ if $e\,{\notin}\,{\bm{l}}$. This
convention allows us to encode the presence of links within a string through a
straightforward numerical scheme.

Now, how is ${\bm{l}}$ parameterized? Given $2^{N+1}$ constraints on $2^{2N}$
configurations, there are $2^{N-1}$ different configurations of ${\bm{l}}$
satisfying the constraints. One solution for the above condition is
${\bm{l}}\,{=}\,{\bm{l}}_{\bm{m}}\,{+}\,\bm{L}^{\perp}_{\bm{a}}$. Taking
${\bm{s}}^{{\bm{m}},{\bm{a}}}\,{:=}\,\exp(i\pi({\bm{l}}_{\bm{m}}\,{+}\,\bm{L}^{\perp}_{\bm{a}}))$,
all possible loops ${\bm{l}}$ satisfying
$\partial_{\scriptscriptstyle{\perp}}({\bm{l}}_{z}\,{-}\,{\bm{l}}_{\bm{m}})\,{=}\,0$
and $[{\bm{l}}_{z}\,{-}\,{\bm{l}}_{\bm{m}}]\,{\equiv}\,{\bm{a}}$ can be
parametrized by taking $e^{i\pi
l_{e}}\,{=}\,s^{{\bm{m}},{\bm{a}}}_{e}\sigma_{v}\sigma_{v^{\prime}}$ where
$\sigma_{v}\in\\{1,-1\\}$ is defined on vertices. Although there are $2^{N}$
different configurations of ${\bm{\sigma}}\,{=}\,\\{\sigma_{v}\\}$,
${\bm{\sigma}}$ and $-{\bm{\sigma}}$ give rise to the same ${\bm{l}}$, and
thus this parametrization gives exactly $2^{N-1}$ different configurations.
The first factor in Eq. (Exact Calculations of Coherent Information for Toric
Codes under Decoherence: Identifying the Fundamental Error Threshold) can be
rewritten as:

$\displaystyle\sum_{\begin{subarray}{c}\partial_{\scriptscriptstyle{\perp}}({\bm{l}}-{\bm{l}}_{\bm{m}})=0,\\\
[{\bm{l}}-{\bm{l}}_{\bm{m}}]={\bm{a}}\end{subarray}}(1-p_{z})^{2N-|{\bm{l}}|}p_{z}^{|{\bm{l}}|}=p_{z}^{N}(1-p_{z})^{N}\sum_{\begin{subarray}{c}\partial_{\scriptscriptstyle{\perp}}({\bm{l}}-{\bm{l}}_{\bm{m}})=0,\\\
[{\bm{l}}-{\bm{l}}_{\bm{m}}]={\bm{a}}\end{subarray}}e^{-\beta_{z}\sum_{e}l_{e}}$
$\displaystyle\quad\qquad=\frac{\sum_{{\bm{\sigma}}}e^{-\beta\sum_{e}s^{{\bm{m}},{\bm{a}}}_{e}\sigma_{v}\sigma_{v^{\prime}}}}{2(2\cosh\beta_{z})^{2N}}=\frac{Z[{\bm{s}}^{{\bm{m}},{\bm{a}}},\beta_{z}]}{2(2\cosh\beta_{z})^{2N}}$
(10)

where $1/(2\cosh\beta_{z})^{2}\,{=}\,p_{z}(1-p_{z})$ and
$Z[{\bm{s}}^{{\bm{m}},{\bm{a}}},\beta_{z}]$ is the partition function of the
RBIM with bond configuration ${\bm{s}}^{{\bm{m}},{\bm{a}}}$ at the inverse
temperature $\beta_{z}$ (see Appendix. A).

The second factor in Eq. (Exact Calculations of Coherent Information for Toric
Codes under Decoherence: Identifying the Fundamental Error Threshold) is more
subtle. First of all, the parametrization of ${\bm{l}}^{\prime}$ satisfying
$\partial({\bm{l}}_{\tilde{\bm{m}}}\,{-}\,{\bm{l}}^{\prime})\,{=}\,0$ as well
as the corresponding Ising model is defined in the dual lattice. Furthermore,
since there is no constraint on the homotopy class of
${\bm{l}}^{\prime}-{\bm{l}}_{\tilde{\bm{m}}}$, ${\bm{l}}^{\prime}$ is
parametrized in terms of both
$\tilde{{\bm{\sigma}}}\,{=}\,\\{\tilde{\sigma}_{v}\\}$ and
${\bm{b}}\,{=}\,(b_{1},b_{2})$ such that 222As in the case of a dual cycle,
the loop $\bm{L}_{\bm{b}}$ is defined by ${\bm{b}}$ such that
$\bm{L}_{\bm{b}}$ has a nontrivial cycle along $i$-th direction if
$b_{i}\,{\neq}\,0$.

$\displaystyle e^{i\pi
l^{\prime}_{e}}\,{=}\,s^{{\tilde{\bm{m}}},{\bm{b}}}\tilde{\sigma}_{v}\tilde{\sigma}_{v^{\prime}}\quad\textrm{with}\quad{\bm{s}}^{{\tilde{\bm{m}}},{\bm{b}}}\,{:=}\,e^{i\pi({\bm{l}}_{\tilde{\bm{m}}}\,{+}\,\bm{L}_{\bm{b}})},$
(11)

where $s^{{\tilde{\bm{m}}},{\bm{b}}}_{e}$ is defined on the edge of the dual
lattice. Accordingly, the second factor is expressed as:

$\displaystyle\sum_{\begin{subarray}{c}\partial({\bm{l}}^{\prime}-{\bm{l}}_{\tilde{\bm{m}}})=0\end{subarray}}(1-p_{x})^{2N-|{\bm{l}}^{\prime}|}p_{x}^{|{\bm{l}}^{\prime}|}$
$\displaystyle=\sum_{{\bm{b}}}\frac{Z[{\bm{s}}^{\tilde{{\bm{m}}},{\bm{b}}},\beta_{x}]}{2(2\cosh\beta_{x})^{2N}}.$
(12)

Therefore, we diagonalized ${\cal E}[\rho_{0}]$ with eigenvalue given in terms
of the RBIM partition functions.

Since
$\rho_{Q}=\frac{1}{4}\sum_{{\bm{a}}}\overline{\bm{Z}}_{{\bm{a}}}^{\vphantom{\dagger}}\rho_{0}\overline{\bm{Z}}_{{\bm{a}}}^{\dagger}$
${\cal E}[\rho_{Q}]$ is diagonalized as

$\displaystyle{\cal E}[\rho_{Q}]$
$\displaystyle=4\sum_{{\bm{m}},\tilde{{\bm{m}}},{\bm{a}}}\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a}}}\cdot\overline{p^{x}_{\tilde{\bm{m}}}}\cdot\overline{p^{z}_{\bm{m}\vphantom{\tilde{{\bm{m}}}}}}$
$\displaystyle p^{i}_{{\bm{m}},{\bm{a}}}$
$\displaystyle:=\frac{{Z[{\bm{s}}^{{\bm{m}},{\bm{a}}},\beta_{i}]}}{2(2\cosh\beta_{i})^{{2N}}},\quad\,\,\overline{p^{i}_{{\bm{m}}}}:=\sum_{\bm{a}}\frac{1}{4}p^{i}_{{\bm{m}},{\bm{a}}}.$
(13)

${\cal E}[\rho_{Q}]$ is properly normalized since
$\sum_{{\bm{m}},{\bm{a}}}p_{{\bm{m}},{\bm{a}}}^{i}\,{=}\,1$:

$\displaystyle\sum_{{\bm{m}},{\bm{a}}}\frac{{Z[{\bm{s}}^{{\bm{m}},{\bm{a}}},\beta]}}{2(2\cosh\beta)^{{2N}}}=\frac{1}{2^{N}}\sum_{\bm{s}}\frac{{Z[{\bm{s}},\beta]}}{(2\cosh\beta)^{{2N}}}=1.$
(14)

Therefore, the entanglement entropy is given as

$\displaystyle S(Q)=-2\log
2-\sum_{{\bm{m}}}\overline{p^{z}_{\bm{m}\vphantom{\tilde{{\bm{m}}}}}}\log\overline{p^{z}_{\bm{m}\vphantom{\tilde{{\bm{m}}}}}}-\sum_{\tilde{{\bm{m}}}}\overline{p^{x}_{\tilde{\bm{m}}}}\log\overline{p^{x}_{\tilde{\bm{m}}}}$
(15)

In the limit $\beta_{i}\,{\rightarrow}\,\infty$, the partition function of an
RBIM is dominated by the contribution with the trivial frustration pattern
with $\\{m_{p}\,{=}\,1\\}$ and $\\{a_{i}\,{=}\,1\\}$. This implies that
$\overline{p^{i}_{\bm{m}}}\,{\rightarrow}\,\delta_{{{\bm{m}}},{\bm{1}}}/4$.
Therefore, we correctly recover $S(Q)\,{=}\,2\log 2$ under the absence of
errors.

_Diagonalization II_.— To evaluate the coherent information, one has to
calculate $S({\cal E}[\rho_{RQ}])$ as well. In this combined system, the
density matrix is further labeled by the reference state with two qubits
labeled by eigenvalues $\bm{\alpha}\,{=}\,(\alpha_{1},\alpha_{2})$ of
$(X_{1},X_{2})$:

$\displaystyle\rho_{RQ}$
$\displaystyle=\sum_{\bm{\alpha},\bm{\alpha^{\prime}}}\frac{1}{4}|\bm{\alpha}\rangle\langle\bm{\alpha^{\prime}}|\otimes(\overline{\bm{Z}}_{\bm{\alpha}}^{\vphantom{\dagger}}\rho_{0}\overline{\bm{Z}}_{\bm{\alpha^{\prime}}}^{\dagger})$
(16)

Let us denote
$M_{\bm{\alpha},\bm{\alpha^{\prime}}}\,{:=}\,\overline{\bm{Z}}_{\bm{\alpha}}^{\vphantom{\dagger}}\rho_{0}\overline{\bm{Z}}_{\bm{\alpha^{\prime}}}^{\dagger}$.
In the basis $\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}}$,
each component of ${\cal E}[M_{\bm{\alpha},\bm{\alpha^{\prime}}}]$ is given as

$\displaystyle\mathrm{tr}\big{(}{\cal
E}[M_{\bm{\alpha},\bm{\alpha^{\prime}}}](\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}})^{\dagger}\big{)}=\sum_{{\bm{l}}_{x},{\bm{l}}_{z}}\Big{(}\prod_{i\in
x,z}(1-p_{i})_{\vphantom{i}}^{2N-|{\bm{l}}_{i}|}p_{i}^{|{\bm{l}}_{i}|}\Big{)}\times$
$\displaystyle\,\,\,\,\,\,\,\,\,\mathrm{tr}(X_{{\bm{l}}_{x}}^{\vphantom{\dagger}}Z_{{\bm{l}}_{z}}^{\vphantom{\dagger}}M_{\bm{\alpha},\bm{\alpha^{\prime}}}Z_{{\bm{l}}_{z}}^{\dagger}X_{{\bm{l}}_{x}}^{\dagger}X_{{\bm{l}}_{\tilde{{\bm{m}}}}}^{\vphantom{\dagger}}Z_{{\bm{l}}_{{\bm{m}}}}^{\vphantom{\dagger}}M_{{\bm{a^{\prime}}},{\bm{a}}}Z_{{\bm{l}}_{{\bm{m}}}}^{\dagger}X_{{\bm{l}}_{\tilde{{\bm{m}}}}}^{\dagger})$
(17)

For the trace not to vanish, two conditions are required:

$\displaystyle(i)$
$\displaystyle\quad\partial_{\scriptscriptstyle{\perp}}({\bm{l}}_{z}-{\bm{l}}_{\bm{m}})\equiv\bm{0},\quad\partial({\bm{l}}_{x}-{\bm{l}}_{\tilde{\bm{m}}})\equiv\bm{0}$
$\displaystyle(ii)$
$\displaystyle\quad[{\bm{l}}_{z}-{\bm{l}}_{\bm{m}}]\equiv{\bm{a^{\prime}}}-\bm{\alpha^{\prime}}\equiv{\bm{a}}-\bm{\alpha}.$
(18)

Contrary to the case in Eq. (Exact Calculations of Coherent Information for
Toric Codes under Decoherence: Identifying the Fundamental Error Threshold),
the homotopy class of $\bm{l}_{x}\,{-}\,\bm{l}_{\tilde{\bm{m}}}$ plays a
crucial role here. This distinction arises because the specific homotopy class
of it can lead to scenarios where the trace factor takes a negative value. As
we move Pauli-$X$ loops
$X_{{\bm{l}}_{x}}^{\dagger}X_{{\bm{l}}_{\tilde{{\bm{m}}}}}^{\vphantom{\dagger}}$
and
$X_{{\bm{l}}_{\tilde{{\bm{m}}}}}^{\dagger}X_{{\bm{l}}_{x}}^{\vphantom{\dagger}}$
to hit $M_{\bm{\alpha},\bm{\alpha^{\prime}}}$ to annihilate, we get

$\displaystyle
X_{{\bm{l}}_{x}}^{\dagger}X_{{\bm{l}}_{\tilde{{\bm{m}}}}}^{\vphantom{\dagger}}M_{\bm{\alpha},\bm{\alpha^{\prime}}}X_{{\bm{l}}_{\tilde{{\bm{m}}}}}^{\dagger}X_{{\bm{l}}_{x}}^{\vphantom{\dagger}}=(\xi_{\bm{\alpha}}^{[{\bm{l}}_{x}{-}{\bm{l}}_{\tilde{\bm{m}}}]})M_{\bm{\alpha},\bm{\alpha^{\prime}}}(\xi_{\bm{\alpha^{\prime}}}^{[{\bm{l}}_{x}{-}{\bm{l}}_{\tilde{\bm{m}}}]})^{-1}$
$\displaystyle\qquad\qquad\xi_{{\bm{a}}}^{[\bm{L}_{\bm{b}}]}:=e^{\mathrm{i}\pi({\bm{a}}\cdot{\bm{b}})},\quad{\bm{a}}\cdot{\bm{b}}=\sum_{i}a_{i}b_{i}.$
(19)

due to the non-trivial commutation relationship between logical operators
$\overline{\bm{Z}}_{\bm{\alpha}}$ and $\overline{\bm{X}}_{\bm{b}}$.
Accordingly, if we parametrize ${\bm{l}}_{x}$ as in Eq. (11), we get

$\displaystyle\mathrm{tr}\big{(}{\cal
E}[M_{\bm{\alpha},\bm{\alpha^{\prime}}}](\rho_{{\bm{m}}\tilde{{\bm{m}}}}^{{\bm{a}},{\bm{a^{\prime}}}})^{\dagger}\big{)}=p_{{\bm{m}},{\bm{a}}-{\bm{\alpha}}\vphantom{\tilde{{\bm{m}}}|\bm{\alpha}}}^{z}\,p_{\tilde{{\bm{m}}}|\bm{\alpha}-\bm{\alpha^{\prime}}}^{x}\cdot\delta_{{\bm{a}}-{\bm{\alpha}},{\bm{a^{\prime}}}-{\bm{\alpha^{\prime}}}}^{\vphantom{x}}$
$\displaystyle\quad\textrm{where}\quad
p_{\tilde{{\bm{m}}}|\bm{\alpha}}^{x}:=\sum_{\bm{b}}\xi_{\bm{\alpha}}^{[\bm{L}_{\bm{b}}]}p_{\tilde{{\bm{m}}},{\bm{b}}}^{x}.$
(20)

Therefore, the full density matrix is written as

$\displaystyle\rho_{RQ}=\frac{1}{4}\sum_{\bm{\alpha},\bm{\alpha^{\prime}}}\sum_{{\bm{a}}}p_{{\bm{m}},{\bm{a}}\vphantom{{\bm{\alpha^{\prime}}}}}^{z}\,p_{{\tilde{\bm{m}}}|{\bm{\alpha}}-{\bm{\alpha^{\prime}}}}^{x}|\bm{\alpha}\rangle\langle\bm{\alpha^{\prime}}|\otimes\rho^{{\bm{a}}+{\bm{\alpha}},{\bm{a}}+{\bm{\alpha^{\prime}}}}_{{\bm{m}},\tilde{{\bm{m}}}}.$
(21)

The density matrix is block-diagonal in $({\bm{m}},\tilde{{\bm{m}}})$, and
each $16\times 16$ dimensional block can be block diagonalized into four
smaller blocks because
$\rho^{{\bm{a}}+{\bm{\alpha}},{\bm{a}}+{\bm{\alpha^{\prime}}}}_{{\bm{m}},\tilde{{\bm{m}}}}$
and
$\rho^{{\bm{a^{\prime}}}+{\bm{\alpha}},{\bm{a^{\prime}}}+{\bm{\alpha^{\prime}}}}_{{\bm{m}},\tilde{{\bm{m}}}}$
are orthogonal if ${\bm{a}}\,{\neq}\,{\bm{a^{\prime}}}$. Therefore, our goal
is to diagonalize $4\times 4$ block denoted as
$B_{{\bm{m}},{\tilde{\bm{m}}}}^{\bm{a}}$, where

$\displaystyle B^{\bm{a}}_{{\bm{m}},{\tilde{\bm{m}}}}$
$\displaystyle=\frac{p_{{\bm{m}},{{\bm{a}}}}^{z}}{4}\sum_{\bm{\alpha},\bm{\alpha^{\prime}}}p_{\tilde{{\bm{m}}}|\bm{\alpha}-\bm{\alpha^{\prime}}}^{x}|\bm{\alpha}\rangle\langle\bm{\alpha^{\prime}}|\otimes\rho^{{\bm{a}}+{\bm{\alpha}},{\bm{a}}+{\bm{\alpha^{\prime}}}}_{{\bm{m}},{\tilde{\bm{m}}}}.$
(22)

Eigenvectors of $B^{\bm{a}}_{{\bm{m}},{\tilde{\bm{m}}}}$ can be directly
constructed as follows. Consider a tuple
$\bm{\beta}\,{=}\,(\beta_{1},\beta_{2})$ with $\beta_{i}\,{\in}\,\\{0,1\\}$.
We define the vector
$v^{\bm{\beta}}\,{:=}\,\sum_{{\bm{\alpha^{\prime}}}}\xi^{\bm{\beta}}_{\bm{\alpha^{\prime}}}(|{\bm{\alpha^{\prime}}}\rangle\otimes\overline{\bm{Z}}_{{\bm{a}}+{\bm{\alpha^{\prime}}}}^{\vphantom{\beta}}|\psi^{\textrm{tc}}_{0}\rangle)$.
It becomes the eigenvector with eigenvalue
$p_{{\bm{m}},{\bm{a}}\vphantom{\beta}}^{z}p_{{\tilde{\bm{m}}},\bm{\beta}}^{x}$:

$\displaystyle(B^{\bm{a}}_{{\bm{m}},{\tilde{\bm{m}}}}v^{\bm{\beta}})_{\bm{\alpha}}$
$\displaystyle=p_{{\bm{m}},{{\bm{a}}}}^{z}\cdot\frac{1}{4}\sum_{\bm{\alpha^{\prime}}}\xi^{\bm{\beta}}_{\bm{\alpha^{\prime}}}p_{\tilde{{\bm{m}}}|{\bm{\alpha}}-{\bm{\alpha^{\prime}}}}^{x}$
$\displaystyle=p_{{\bm{m}}\vphantom{\tilde{{\bm{m}}}},{{\bm{a}}}}^{z}\cdot\xi_{\bm{\alpha}}^{\bm{\beta}}\cdot
p_{\tilde{{\bm{m}}},{\bm{\beta}}}^{x}=p_{{\bm{m}}\vphantom{\tilde{{\bm{m}}}},{{\bm{a}}}}^{z}p_{\tilde{{\bm{m}}},{\bm{\beta}}}^{x}(v^{\bm{\beta}})_{\bm{\alpha}}$
(23)

as the summation is equivalent to performing a discrete inverse Fourier
transform, provided that Eq. (Exact Calculations of Coherent Information for
Toric Codes under Decoherence: Identifying the Fundamental Error Threshold) is
a two-dimensional discrete Fourier transformation with a periodicity of two
333We remark that this idea can be immediately generalized for
$\mathbb{Z}_{n}$ case.. Four eigenvalues of this small block labeld by
$({\bm{m}},{\tilde{\bm{m}}},{\bm{a}})$ is given as
$\\{p_{{\bm{m}}\vphantom{\tilde{{\bm{m}}}},{{\bm{a}}}}^{z}p_{\tilde{{\bm{m}}},{\bm{\beta}}}^{x}\\}_{\beta_{i}\,{\in}\,\\{0,1\\}}$.
This is exactly the product of two RBIM partition functions defined at
$\beta_{x}$ and $\beta_{z}$ with domain wall configurations ${\bm{a}}$ and
${\bm{b}}$, respectively.

The entanglement entropy of ${\cal E}[\rho_{RQ}]$ is evaluated as

$\displaystyle S({\cal E}[\rho_{RQ}])$
$\displaystyle=-\sum_{\bm{m},{\bm{a}}\vphantom{{\tilde{\bm{m}}},{\bm{b}}}}\sum_{{\tilde{\bm{m}}},{\bm{b}}}\,p_{{\bm{m}}\vphantom{\tilde{{\bm{m}}}},{{\bm{a}}}}^{z}\,p_{\tilde{{\bm{m}}},{{\bm{b}}}}^{x}\log\,p_{{\bm{m}}\vphantom{\tilde{{\bm{m}}}},{{\bm{a}}}}^{z}\,p_{\tilde{{\bm{m}}},{{\bm{b}}}}^{x}$
(24)

In the limit $\beta_{z,x}\,{\rightarrow}\,\infty$, both
$p_{{\bm{m}}\vphantom{\tilde{{\bm{m}}}},{{\bm{a}}}}^{z}$ and
$p_{\tilde{{\bm{m}}},{{\bm{b}}}}^{x}$ are nonzero only if
$({\bm{m}},{\bm{a}})$ and $(\tilde{{\bm{m}}},{\bm{b}})$ are trivial
configurations; in such a case, we obtain $S(R^{\prime}Q^{\prime})\,{=}\,0$ as
expected.

_Coherent Information_.— By plugging Eq. (15) and Eq. (24) into Eq. (3), the
coherent information of a decohered toric code is calculated as

$\displaystyle I_{c}^{{\textrm{tc}}}$ $\displaystyle=-2\log
2+\sum_{{{\bm{m}}},{\bm{a}},i}\quantity({p^{i}_{\bm{m},{\bm{a}}\vphantom{\tilde{{\bm{m}}}}}}\Big{[}\log{p^{i}_{\bm{m},{\bm{a}}\vphantom{\tilde{{\bm{m}}}}}}-\log\overline{p^{i}_{{\bm{m}}}}\Big{]})$
$\displaystyle=2\log
2+\sum_{{\bm{m}},{\bm{a}},i}p_{{\bm{m}},{\bm{a}}}^{i}\log(\frac{{Z[{\bm{s}}^{{\bm{m}},{\bm{a}}},\beta_{i}]}}{\sum_{{\bm{a^{\prime}}}}Z[{\bm{s}}^{{\bm{m}},{\bm{a^{\prime}}}},\beta_{i}]})$
(25)

where $i\,{\in}\,\\{x,z\\}$. Note that $({\bm{m}},{\bm{a}})$ completely
specifies the equivalence class of the RBIM, which corresponds to the bond
frustration pattern as elaborated in Appendix. A. With this analytic
expression, the information-theoretic capacity of the toric code under local
Pauli errors in the thermodynamic limit can be rigorously understood as the
following.

Let $\beta_{c}$ be the critical inverse temperature of the RBIM along the
Nishimori line [10, 11], where corresponding $p_{c}\,{=}\,0.1094$ [22]. To
facilitate the analysis, define $F_{{\bm{m}},{\bm{a}}}^{i}\,{:=}\,{-}\log
p^{i}_{{\bm{m}},{\bm{a}}}$, which is the free energy of a random bond Ising
model upto constant. The difference
$\Delta^{i}_{{\bm{m}},{\bm{a}}}\,{:=}\,F^{i}_{{\bm{m}},{\bm{a}}}\,{-}\,F^{i}_{{\bm{m}},\bm{0}}$
corresponds to the free energy cost of inserting a domain wall ${\bm{a}}$ from
the configuration $({\bm{m}},\bm{0})$ without any domain wall. The second term
in Eq. (Exact Calculations of Coherent Information for Toric Codes under
Decoherence: Identifying the Fundamental Error Threshold) decomposes into two
independent parts with the same functional form:
$I_{c}^{\textrm{tc}}\,{=}\,2\log 2\,{-}\,A_{x}\,{-}\,A_{z}$. With
$p^{i}_{{\bm{m}},{\bm{a}}}\,{=}\,p^{i}_{{\bm{m}},\bm{0}}e^{-\Delta^{i}_{{\bm{m}},{\bm{a}}}}$,
$A_{i}$ is given as

$\displaystyle
A_{i}=\sum_{{\bm{m}}}p_{{\bm{m}},\bm{0}}^{i}\sum_{\bm{a}}\quantity[e^{-\Delta^{i}_{{\bm{m}},{\bm{a}}}}\log\frac{\,\sum_{{\bm{a^{\prime}}}}e^{-\Delta^{i}_{{\bm{m}},{\bm{a^{\prime}}}}}}{e^{-\Delta^{i}_{{\bm{m}},{\bm{a}}}}}].$
(26)

Now, our goal is to understand the behavior of $A_{i}$ as a function of
$\beta_{i}$ in the thermodynamic limit.

$(1)$ $\beta_{i}\,{>}\,\beta_{c}$: In this case, the majority of the bond
configurations of the RBIM in the disorder ensemble are long-range ordered.
Accordingly, for typical configurations ${\bm{m}}$, the denominator inside
$\log(\cdots)$ in Eq. (26) is dominated by the term with
${\bm{a}}\,{=}\,\bm{0}$ (no domain wall), resulting in the vanishing value of
$\log(\cdots)$. Note that within the disorder ensemble of different bond
configurations, the fraction of paramagnetically ordered bond configurations
vanishes as the system size increases, as detailed in Appendix. B.
Furthermore, when a given configuration is long-range ordered, the cost of
domain wall insertion scales with system size,
$|\Delta^{x}_{{\bm{m}},{\bm{a}}}|\,{\geq}\,cL\delta_{{\bm{a}},\bm{0}}$ for
some non-zero constant $c$ and $L\,{=}\,\sqrt{N}$. With this premise, one can
establish that $\lim_{N\rightarrow\infty}A_{i}\,{=}\,0$. Therefore, if
$\beta_{x,z}\,{>}\,\beta_{c}$ ($p_{x,z}\,{<}\,p_{c}$),
$I_{c}^{\textrm{tc}}\rightarrow 2\log 2$ and two qubits of decodable quantum
information persist.

($2$) $\beta_{i}\,{<}\,\beta_{c}$: In this case, the majority of the bond
configurations are paramagnetic and
$\Delta^{z}_{{\bm{m}},{\bm{a}}}\,{\rightarrow}\,0$ in the thermodynamic limit
since domain walls are freely fluctuating in the paramagnetic phase 444This is
expected since disorder operators’ correlation functions decay exponentially
with their perimeter size in the disordered phase.. Accordingly,
$A_{i}\,{\rightarrow}\,2\log 2$, see Appendix. B. Without loss of generality,
if $\beta_{x}\,{<}\,\beta_{c}$ and $\beta_{z}\,{>}\,\beta_{c}$,
$A_{x}\,{=}\,2\log 2$ and $A_{z}\,{=}\,0$ and we get $I_{c}^{\textrm{tc}}=0$,
implying that there remain two bits of classical information that can be
restored, which corresponds to the eigenvalues of
$(\overline{\bm{X}}_{1},\overline{\bm{X}}_{2})$. In the doubled Hilbert space
formalism [27, 28], this corresponds to the phase where two copies of
$\mathbb{Z}_{2}$ topological order condense into a single $\mathbb{Z}_{2}$
topological order [17, 16]. On the other hand, if both
$\beta_{x,z}\,{<}\,\beta_{c}$, $I_{c}\approx-2\log 2$ and there remains
neither quantum nor classical information that can be decoded. This
corresponds to the trivial topological order in the doubled Hilbert space.

It is instructive to compare this expression with the Renyi-2 version of the
coherent information [16], where it is associated with the free energy of a
ferromagnetic Ising model defined at inverse temperature
$\tilde{\beta}_{i}\,{:=}\,-\frac{1}{2}\log(1-2p_{i})$ 555Note that at $p=0$,
$\tilde{\beta}=0$ while $\beta=\infty$.:

$\displaystyle{I^{{\textrm{tc}},(2)}_{c}}=\sum_{i={x,y}}\log\bigg(\sum_{{\bm{a}}}e^{-\Delta^{i}_{{\bm{a}}}}\bigg{missing})-2\log
2$ (27)

where $\Delta^{\alpha\beta}_{2,a}$ is the free energy of the domain wall in
the ferromagnetic Ising model. The Renyi-2 coherent information undergoes a
transition at $p_{c}^{\prime}\,{:=}\,0.178$ [17, 16], which is higher than the
threshold $p_{c}\,{:=}\,0.1094$ obtained in the current work.

To understand the advantage of storing information in the toric code logical
space, we can calculate the coherent information for two raw (physical) qubits
under the same decoherence channel as follows:

$\displaystyle I_{c}^{\textrm{raw}}$ $\displaystyle=2\Big{(}\log
2-\sum_{i\in\\{x,z\\}}H_{2}(p_{i})\Big{)}.$ (28)

where $H_{2}(p)\,{:=}\,\,{-}\,p\log p\,{-}\,(1-p)\log(1-p)$ is a binary
entropy function. We plot $I_{c}^{\textrm{raw}}$ and $I_{c}^{\textrm{tc}}$ as
a function of $(p_{x},p_{z})$ in Fig. 1(a,b). As we can observe, below the
error threshold $p_{x,z}\,{<}\,p_{c}$, $I_{c}^{\textrm{tc}}$ remains $2\log 2$
while $I_{c}^{\textrm{raw}}$ continuously decreases. However, if
$p_{i}\,{>}\,p_{c}$, $I_{c}^{\textrm{raw}}$ can be larger than
$I_{c}^{\textrm{tc}}$, and there is no advantage in storing information in the
form of toric code logical states. Along the line $p_{x}\,{=}\,p_{z}$, the
Nishimori critical point $p_{x,z}\,{=}\,0.1094$ is very close to the point
$p_{x,z}\,{=}\,0.1100$ where $I_{c}^{\textrm{raw}}$ vanishes.

_Relatitve Entropy._ — With a diagonalized decohered density matrix, it is
straightforward to evaluate a quantum relative entropy [30] between two
decohered toric code states, each of which is initialized at different logical
states. Consider two initial states $\rho_{0}$ and
$\rho_{1}\,{=}\,\overline{\bm{Z}}_{\bm{a}}\rho_{0}\overline{\bm{Z}}_{\bm{a}}$.
If $\rho_{n}^{\prime}\,{=}\,{\cal E}[\rho\vphantom{{}^{\prime}}_{n}]$, the
relative entropy between two decohered states is given as

$\displaystyle D(\rho_{0}^{\prime}\|\rho_{1}^{\prime})$
$\displaystyle:=\mathrm{tr}\big{(}\rho_{0}^{\prime}(\log\rho_{0}^{\prime}-\log\rho_{1}^{\prime})\big{)}$
$\displaystyle=\sum_{{\bm{m}},{\bm{a^{\prime}}}}p^{x}_{{\bm{m}},{\bm{a^{\prime}}}}(F^{x}_{{\bm{m}},{\bm{a}}{\bm{a^{\prime}}}}-F^{x}_{{\bm{m}},{\bm{a^{\prime}}}})=\langle\Delta
F_{\bm{{\bm{a}}}}\rangle$ (29)

which is the disorder-averaged free energy of a domain wall configuration
${\bm{a}}$ in the RBIM along the Nishimori line at $\beta_{x}$ (see Appendix.
B). If $\beta_{x}\,{>}\,\beta_{c}$ ($p_{x}\,{<}\,p_{c}$), the system is long-
range ordered on average, and $\langle\Delta F_{\bm{a}}\rangle\,{\sim}\,{\cal
O}(L)$. As the relative quantum entropy signifies the distinguishability of
two states, the linear scaling of $\langle\Delta F_{\bm{a}}\rangle$ implies
that two different logical states are well-distinguishable even after local
decoherence in the thermodynamic limit. If $\beta_{x}\,{<}\,\beta_{c}$,
$\langle\Delta F_{\bm{a}}\rangle\,{\sim}\,0$ in the thermodynamic limit and
two different logical states would be indistinguishable.

We note a critical distinction from coherent information: relative entropy
(disorder averaged free energy) would exhibit ${\cal O}(L)$ scaling even if a
constant fraction of the disorder ensemble were in the paramagnetic phase (see
Appendix. B), which would lead to coherent information being strictly smaller
than $2\log 2$ in the thermodynamic limit. Consequently, relative entropy
emerges as a less refined measure for quantifying decodable quantum
information.

_Conclusion_.— In this work, through the exact diagonalization of the
decohered system consisting of the toric code and reference, we obtained the
analytic expression for the coherent information of the toric code. This
analytic expression is directly tied to the free energy of the random bond
Ising model, from which we could derive the transition behavior of the
coherent information and the fundamental error threshold for toric code under
Pauli-$Z$ and $X$ errors. Our observation that exact diagonalization results
in statistical models sharing the same threshold as maximum entropy decoding
leads us to conjecture that the fundamental error threshold for various other
models, whose decoding thresholds come from mapping to the statistical models,
can be similarly understood by precisely calculating coherent information.
Furthermore, as the formalism developed in this work is directly generalized
to the $\mathbb{Z}_{n}$ degrees of freedom, studying the coherent information
in decohered $\mathbb{Z}_{n}$ topological phases by exact diagonalization
would be a promising future direction.

###### Acknowledgements.

We thank Matthew F.A. Fisher, Ehud Altman, Ruihua Fan, and Jeongwan Haah for
fruitful discussions and comments. The work is supported by the Simons
Investigator Award.

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##  Appendix A Review on Random Bond Isind Model

Consider a lattice with edges $(e)$ and vertices $(v)$. For a given bond
configuration ${\bm{b}}\,{=}\,\\{b_{e}\\}$ with $b_{e}\,{\in}\,\\{1,-1\\}$, a
random bond Ising model partition function at inverse temperature $\beta$ is
defined as

$\displaystyle
Z_{\textrm{RBIM}}[{\bm{b}},\beta]:=\sum_{\\{\sigma\\}}e^{-\beta\sum_{\langle
v,v^{\prime}\rangle}b_{v,v^{\prime}}\sigma_{v}\sigma_{v^{\prime}}}.$ (30)

One crucial property of the RBIM is that its partition function is invariant
under the gauge transformation of the bond configuration ${\bm{b}}$ defined as
follows. Consider a set of values defined on vertices
$\bm{t}\,{=}\,\\{t_{v}\\}$ with $t_{v}\,{\in}\\{1,-1\\}$. Define the
transformation ${\bm{b}}^{\prime}\,{=}\,{\cal G}[{\bm{b}},\bm{t}]$ such that
$b^{\prime}_{e}\,{=}\,t_{v}b_{e}t_{v^{\prime}}$ for $e\,{=}\,(v,v^{\prime})$.
Then for any $\bm{t}$ with ${\bm{b}}^{\prime}\,{=}\,{\cal
G}[{\bm{b}},\bm{t}]$,

$\displaystyle
Z_{\textrm{RBIM}}[{\bm{b}},\beta]=Z_{\textrm{RBIM}}[{\bm{b}}^{\prime},\beta].$
(31)

This is because the partition function for ${\bm{b}}^{\prime}$ is related to
${\bm{b}}$ via a change of variables for its spin degrees of freedom.
Therefore, the partition function only depends on the equivalence class of
${\bm{b}}$, which can be labeled by the gauge-invariant quantities
${\bm{m}}\,{=}\,\\{m_{p}\\}$ and ${\bm{a}}\,{=}\,(a_{1},a_{2})$ defined as

$\displaystyle m_{p}:=\prod_{e\in p}b_{e},\quad e^{i\pi a_{i}}:=\prod_{e\in
C_{i}}b_{e}$ (32)

where $C_{1}$ and $C_{2}$ are particular cycles along $x$ and $y$ directions,
respectively, see Fig. 2(a).

Let $P({\bm{b}})$ be the probability for a bond configuration to be
${\bm{b}}$. If each bond is independent and ferromagnetic with probability
$1-p$, then the probability distribution $P({\bm{b}})$ is expressed as

$\displaystyle
P({\bm{b}})=\prod_{\mathbf{e}}\sqrt{(1-p)^{1+b_{e}}p^{1-b_{e}}}=\frac{e^{\beta_{p}\sum_{e}b_{e}}}{(2\cosh\beta_{p})^{2N}},$
(33)

where $\beta_{p}=\tanh^{-1}(1-2p)$.

##  Appendix B Nishimori Line, Free Energy, and Variance across Disorder
Ensemble

Assuming $P({\bm{b}})$ in Eq. (33), the disorder averaged free energy is given
as

$\displaystyle\beta\overline{F}$ $\displaystyle:=-\sum_{\bm{b}}P({\bm{b}})\ln
Z_{\textrm{RBIM}}[{\bm{b}},\beta]$
$\displaystyle=-\sum_{\bm{b}}\frac{e^{\beta_{p}\sum_{e}b_{e}}}{(2\cosh\beta_{p})^{2N}}\ln
Z_{\textrm{RBIM}}[{\bm{b}},\beta]$
$\displaystyle=-\sum_{{\bm{b}}^{\prime}={\cal
G}[{\bm{b}},t]}\frac{e^{\beta_{p}\sum_{\langle
v,v^{\prime}\rangle}b^{\prime}_{v,v^{\prime}}t_{v}t_{v^{\prime}}}}{(2\cosh\beta_{p})^{2N}}\ln
Z_{\textrm{RBIM}}[{\bm{b}}^{\prime},\beta]$
$\displaystyle=-\sum_{\bm{t},{\bm{b}}^{\prime}}\frac{e^{\beta_{p}\sum_{\langle
v,v^{\prime}\rangle}b^{\prime}_{v,v^{\prime}}t_{v}t_{v^{\prime}}}}{2^{N}(2\cosh\beta_{p})^{2N}}\ln
Z_{\textrm{RBIM}}[{\bm{b}}^{\prime},\beta]$
$\displaystyle=-\sum_{{\bm{b}}^{\prime}}\frac{Z_{\textrm{RBIM}}[{\bm{b}}^{\prime},\beta_{p}]}{2^{N}(2\cosh\beta_{p})^{2N}}\ln
Z_{\textrm{RBIM}}[{\bm{b}}^{\prime},\beta]$ (34)

where the line above $F$ indicates it is disorder averaged. In the third line,
we used the change of variable from ${\bm{b}}$ to ${\bm{b}}^{\prime}$ by the
gauge transformation ${\cal G}[\cdot,\bm{t}]$. In the fourth line, we use the
fact that the summation over an auxiliary variable $\bm{t}$ introduces a
multiplicative factor $1/2^{N}$.

The Nishimori condition, as established in the literature [10, 11], identifies
a unique line in the phase diagram characterized by the equality
$\beta_{p}=\beta$, which gives $p=\frac{1-\tanh(\beta)}{2}$, directly linking
the disorder fraction in the system to the thermal energy scale governed by
$\beta$.

Previous numerical simulation has elucidated a critical phase transition point
at $\beta_{c}\,{=}\,1.048$ [22, 31] at which the system transitions from a
state exhibiting long-range ferromagnetic order to a paramagnetic phase. As
expected, it corresponds to a phase transition threshold at a lower critical
temperature (or higher $\beta$) compared to the conventional ferromagnetic
Ising model, which possesses a critical inverse temperature of
$\beta_{c}^{\textrm{FIM}}\,{=}\,0.371$.

In the context of a disorder-averaged system, the characterization of a long-
range ordered phase requires careful consideration. Specifically, the system
may present a scenario where a constant fraction (${<}\,1$) of the bond
configurations manifests long-range order, while the remaining configurations
are paramagnetic. In this case, the disorder-averaged order parameter would
suggest the presence of long-range order, even if the system is essentially
bifurcated between ferromagnetic and paramagnetic states. This observation
underscores the limitation of a naive order parameter in accurately reflecting
the system’s state under varying degrees of disorder.

To discern whether the majority of bond configurations truly exhibit long-
range order, the order parameter fluctuation across different bond
configurations should be examined. For a given bond configuration ${\bm{b}}$,
the order parameter is defined as

$\displaystyle\langle
O\rangle_{\bm{b}}:=\frac{1}{Z[{\bm{b}}_{i},\beta]}\sum_{\\{\sigma\\}}O(\\{\sigma\\})e^{-\beta\sum
b_{e}\sigma_{v}\sigma_{v^{\prime}}}.$ (35)

Assume that $O$ is normalized in such a way that $O\,{\in}\,[0,1]$; it takes a
finite value in the long-range ordered phase, and zero in the paramagnetic
phase. The variance across bond configurations is defined as

$\displaystyle\chi:=\overline{\langle
O\rangle^{2}_{\bm{b}}}^{\bm{b}}-\Big{(}\overline{\langle
O\rangle_{\bm{b}}}^{\bm{b}}\Big{)}^{2},$ (36)

where $\overline{(\,\cdot\,)}^{\bm{b}}\,{:=}\,\sum_{\bm{b}}P({\bm{b}})(\cdot)$
is disorder average. If $O$ is magnetization, $\chi$ is related to the
magnetic susceptibility. If $\chi\,{\sim}\,{\cal O}(1)$, then it implies that
the constant fraction of bond configurations is paramagnetic. On the other
hand, if $\chi\,{\sim}\,{\cal O}(1/N)$, then the paramagnetic fraction is
$1-{\cal O}(1/N)$, vanishing in the thermodynamic limit.

Figure S1: Random bond Ising model at $T=0$. The plot shows (a) order
parameter and (b) its variance across bond configurations as the function of
disorder probability $p$ at $T=0$ in the RBIM, adapted from [32]. The order
parameter $O$ is squared magnetization normalized to be within $[0,1]$. The
black dashed line is the critical point at $p_{c}\,{=}\,0.103$ [33].

In Fig. S1, the variance of the order parameter is shown along the $T=0$ line
of the RBIM phase diagram [33], where $\chi$ decays with the system size in
both the long-range ordered and paramagnetic phase. In this plot, the order
parameter $O$ is defined as the normalized squared magnetization [32]. This
numerically demonstrates that in the long-range ordered phase, the fraction of
paramagnetic configurations vanishes in the thermodynamic limit.

Therefore, when it comes to the evaluation of a disorder-averaged quantity
scaling sub-extensively with system size, one can assume that the majority of
bond configurations are in the same phase. This is crucial in the evaluation
of the coherent information of the decohered toric code state. In Eq. (26), we
encountered the following term:

$\displaystyle
f(\Delta^{x}_{{\bm{m}},{\bm{a}}}):=e^{-\Delta^{x}_{{\bm{m}},{\bm{a}}}}\log\frac{\,\sum_{{\bm{a^{\prime}}}}e^{-\Delta^{x}_{{\bm{m}},{\bm{a^{\prime}}}}}}{e^{-\Delta^{x}_{{\bm{m}},{\bm{a}}}}}>0,$
(37)

where $\Delta^{x}_{{\bm{m}},{\bm{a}}}$ is the difference between free energies
of the RBIM with bond equivalence classes $({\bm{m}},{\bm{a}})$ and
$({\bm{m}},\bm{0})$ at the inverse temperature $\beta_{x}$.

Our goal is to understand this quantity $f(\Delta^{x}_{{\bm{m}},{\bm{a}}})$ in
different regimes. First, assume that $\beta_{x}\,{>}\,\beta^{c}$. In this
case, for the typical long-range ordered configuration at the inverse
temperature $\beta_{x}$ along the Nishimori line, this free energy difference
increase with the system size such that
$|\Delta^{x}_{{\bm{m}},{\bm{a}}}|\,{>}\,cL$ for some non-zero constant $c$ if
${\bm{a}}\,{\neq}\,\bm{0}$ and $L\,{=}\,\sqrt{N}$ is the linear size of the
system. Note that $\Delta^{x}_{{\bm{m}},\bm{0}}\,{=}\,0$. Therefore, we get

$\displaystyle{\bm{a}}=0:$
$\displaystyle\,\,\,f(\Delta^{x}_{{\bm{m}},{\bm{a}}})\leq\log(1+3e^{-cL})<3e^{-cL}$
$\displaystyle{\bm{a}}\neq 0:$
$\displaystyle\,\,\,f(\Delta^{x}_{{\bm{m}},{\bm{a}}})\leq
cLe^{-cL}+e^{-cL}\log(1+3e^{-cL})$
$\displaystyle\,\,\qquad\qquad<(cL+3e^{-cL})e^{-cL},$ (38)

where we used that $xe^{-x}$ is monotonically decreasing for
$x\,{\in}\,[1,\infty]$. Accordingly, we obtain that

$\displaystyle\sum_{\textrm{LRO
}{\bm{m}},{\bm{a}}}p^{x}_{{\bm{m}},\bm{0}}\cdot
f(\Delta^{x}_{{\bm{m}},{\bm{a}}})$
$\displaystyle<3e^{-cL}(cL+4)\cdot\big{(}\sum_{\textrm{LRO
}{\bm{m}},{\bm{a}}}p^{x}_{{\bm{m}},\bm{0}}\big{)}$
$\displaystyle\xrightarrow{L\rightarrow\infty}0,$ (39)

since the sum of probabilities is upper-bounded by 1.

Similarly, consider bond configurations that are paramagnetic, whose fraction
$\sum_{\textrm{para }{\bm{m}}}p^{x}_{{\bm{m}},\bm{0}}\,{\sim}\,{\cal O}(1/N)$.
For these configurations, the free energy difference
$\Delta^{x}_{{\bm{m}},{\bm{a}}}$ is upper-bounded by constant, denoted as
$\Delta^{x}_{{\bm{m}},{\bm{a}}}\,{<}\,c^{\prime}$. In this case, one can show
that

$\displaystyle\sum_{{\bm{a}}}f(\Delta^{x}_{{\bm{m}},{\bm{a}}})\leq 2\log
2\quad\textrm{for para. ${\bm{m}}$}$
$\displaystyle\Rightarrow\quad\lim_{N\rightarrow\infty}\sum_{\textrm{para
}{\bm{m}},{\bm{a}}}p^{x}_{{\bm{m}},\bm{0}}\cdot
f(\Delta^{x}_{{\bm{m}},{\bm{a}}})=0.$ (40)

Therefore, we establish the limiting behavior in Eq. (26).

On the other hand, if there is a finite fraction $f\,{>}\,0$ of the bond
configurations with paramagnetic order with
$\Delta^{x}_{{\bm{m}},{\bm{a}}}\,{<}\,c^{\prime}$ at any system size, one can
show that

$\displaystyle\sum_{{\bm{a}}}f(\Delta^{x}_{{\bm{m}},{\bm{a}}})\geq
3c^{\prime}e^{-c^{\prime}}\quad\textrm{for para. ${\bm{m}}$}$
$\displaystyle\Rightarrow\,\,\,\sum_{\textrm{para
}{\bm{m}},{\bm{a}}}p^{x}_{{\bm{m}},\bm{0}}\cdot
f(\Delta^{x}_{{\bm{m}},{\bm{a}}})\geq 3fc^{\prime}e^{-c^{\prime}}>0.$ (41)

Accordingly, the coherent information will be strictly smaller than $2\log 2$
by an ${\cal O}(1)$ number.

A similar argument can be made in the paramagnetic phase at
$\beta_{x}\,{<}\,\beta^{c}$, where the majority of the bond configurations are
paramagnetic. In fact, in the paramagnetic configuration the free energy
difference is upper-bounded by a quantity $c^{\prime}$ which should vanish in
the thermodynamic limit, i.e.,

$\displaystyle\beta_{x}<\beta_{c}\quad\Rightarrow\quad\lim_{N\rightarrow\infty}c^{\prime}=0.$
(42)

Accordingly,
$\lim_{N\rightarrow\infty}\sum_{{\bm{a}}}f(\Delta^{x}_{{\bm{m}},{\bm{a}}})=-2\log
2$.