# Approximate quantum error correcting codes from conformal field theory

Shengqi Sang  Perimeter Institute for Theoretical Physics, Waterloo, Ontario
N2L 2Y5, Canada  University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Timothy H. Hsieh  Perimeter Institute for Theoretical Physics, Waterloo,
Ontario N2L 2Y5, Canada  Yijian Zou<EMAIL_ADDRESS>Perimeter
Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

###### Abstract

The low-energy subspace of a conformal field theory (CFT) can serve as a
quantum error correcting code, with important consequences in holography and
quantum gravity. We consider generic 1+1D CFT codes under extensive local
dephasing channels and analyze their error correctability in the thermodynamic
limit. We show that (i) there is a finite decoding threshold if and only if
the minimal nonzero scaling dimension in the fusion algebra generated by the
jump operator of the channel is larger than $1/2$ and (ii) the number of
protected logical qubits $k\geq\Omega(\log\log n)$, where $n$ is the number of
physical qubits. As an application, we show that the one-dimensional quantum
critical Ising model has a finite threshold for certain types of dephasing
noise. Our general results also imply that a CFT code with continuous symmetry
saturates a bound on the recovery fidelity for covariant codes.

Introduction.– Quantum information is fragile and can be lost when subject to
decoherence. In order to robustly store and manipulate quantum information
against noise, one embeds the logical qubits into a larger set of physical
qubits, forming quantum error correcting codes (QECC) [1, 2]. QECCs provide
promising routes to fault-tolerant quantum computing [3, 4] and have been
recently realized on quantum simulators [5]. On the theoretical side, code
properties of QECCs provide insight into patterns of long-range quantum many-
body entanglement, connecting quantum information with condensed matter
physics [6, 7] and quantum gravity [8, 9]. More recently, studies have
suggested a close relationship between mixed-state topological phases and the
ability of the QECCs to correct certain errors [10, 11]. It is thus important,
both from a practical and a theoretical point of view, to understand which
physical systems can serve as QECCs and which errors are correctable given the
system or code.

Most previous studies focus on stabilizer codes [2]. The stabilizer formalism
offers a natural way for decoding, where in some cases the optimal decoder can
be found [6]. However, the notion of QECC goes far beyond stabilizer codes. In
order to define a QECC, one may specify a $D$-dimensional code subspace
spanned by a set of mutually orthogonal codeword states,
$\\{|\phi_{\alpha}\rangle,~{}\alpha=1,2,\cdots,D\\}$, where $D=2^{k}$ to
encode $k$ logical qubits. In the seminal work [12], the codeword states are
chosen as the eigenstates of a local Hamiltonian. The code properties can then
be related to physical properties of the system in an energy window. Another
example that goes beyond stabilizers is AdS/CFT correspondence, where the code
subspace is the CFT low-energy subspace corresponding to bulk graviton
excitations. For this code, the code properties reveal important aspects of
quantum gravity [13]. Going beyond stabilizer codes poses significant
challenges to determine error correctability as there are usually no explicit
decoders. Furthermore, existing approximate error correctability [14, 15, 16]
criteria are numerically hard to compute.

In this work, we show that generic CFTs furnish approximate quantum error
correcting codes [17, 18], where the code subspace is given by the low-energy
subspace of the CFT. As opposed to the holographic correspondence, here the
CFT can be realized by a simple spin chain at criticality, such as the
transverse field Ising model. We analyze the decodability of CFT codes under a
finite-time evolution of a translation-invariant Lindbladian, such as uniform
dephasing noise. Our main result is that the errors can be corrected at a
finite threshold if and only if $\Delta_{\min}>1/2$, where $\Delta_{\min}$ is
the minimal nonzero scaling dimension of CFT operators in fusion algebra
generated by the Lindbladian jump operator. A natural corollary is that a CFT
code can correct uniform depolarization noise at a finite threshold if and
only if the algebra generated by the jump operator only contains CFT operators
with scaling dimension larger than $1/2$. In addition, we show that for any
CFT code the number of protected logical qubits $k\geq\Omega(\log\log n)$,
where $n$ is the number of physical qubits. As an example, we analyze the
Ising CFT code realized by the low-energy subspace of the critical transverse-
field Ising model (TFIM) in one dimension. We show that for dephasing noise,
the code does not have a finite threshold for $X$ dephasing but can correct
$Y$ and $Z$ dephasing up to maximal strength.

Our work is in part inspired by recent developments in decoherence-induced
mixed-state phases [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40]. While previous approaches mainly focus on
finite number of replicas of the system (for computing Renyi entanglement
quantities), our work directly addresses the replica limit, which is provably
related to the decoding transition. As a byproduct, this work also features an
example where the replica limit has fundamentally different physics from other
integer Renyi indices. In contrast to previous works on QECCs in gapless
systems [41, 42, 12, 14], our results involve the decoding threshold for
extensive noise rather than code distance. Furthermore, our results are
applicable to generic CFTs as opposed to specific models.

CFT code– Let us start with properly defining the CFT code. Given a critical
quantum spin chain with $n$ spins and Hamiltonian $H$, the low-energy physics
is described by a CFT. Each low-energy eigenstate $|\phi_{\alpha}\rangle$
corresponds to a scaling operator $\phi_{\alpha}$ in the CFT due to the state-
operator correspondence. The energy of the state is given by
$E_{\alpha}=\frac{2\pi}{n}(\Delta_{\alpha}-c/12)$, where $\Delta_{\alpha}$ is
the scaling dimension and $c$ is the central charge. As $n\rightarrow\infty$,
there are infinite number of low-energy eigenstates whose energies are
degenerate. The encoding circuit can be chosen as a MERA tensor network [43,
44, 45, 46, 42], where the logical qubits are located at top layers.

In order to discuss the error-correcting properties, we specify the error
model to be a finite-time evolution $e^{t\mathcal{L}}$ of a local Lindbladian
$\mathcal{L}$, where $\mathcal{L}(\rho)=\sum_{i}\left[L_{i}\rho
L^{\dagger}_{i}-\frac{1}{2}\\{L^{\dagger}_{i}L_{i},\rho\\}\right]$ and $L_{i}$
are local jump operators. More concretely, we consider two types of noise,
dephasing and flagged dephasing. For both error models, the noise channel
$\mathcal{N}=\otimes_{j}\mathcal{N}^{[j]}$ is a product of local channels
$\mathcal{N}^{[j]}$ acting on a single spin $j$. A dephasing channel of
strength $p$ is defined by

$\mathcal{N}^{[j]}_{p,\alpha}(\rho)=\left(1-\frac{p}{2}\right)\rho+\frac{p}{2}\sigma^{[j]}_{\alpha}\rho\sigma^{[j]}_{\alpha},$
(1)

where $\sigma^{[j]}_{\alpha}$ is the Pauli operator acting on site $j$ and
$\alpha=x,y,z$. In terms of Lindbladian, this corresponds to $n$ jump
operators $L_{i}=\sigma^{[i]}_{\alpha}/\sqrt{2}$ and evolution time
$t=-\log(1-p)$. A flagged dephasing channel (also known as heralded noise in
quantum optics [47]) of strength $p$ is defined by

$\mathcal{N}^{[j]}_{p,\alpha;\mathrm{F}}(\rho)=\left(1-p\right)\rho\otimes|0_{F}\rangle\langle
0_{F}|+p\mathcal{N}^{[j]}_{1,\alpha}(\rho)\otimes|1_{F}\rangle\langle 1_{F}|,$
(2)

where we have introduced a “flag” qubit $F$ at each site $j$. The flagged
dephasing means that each site has a probability $p$ to undergo a complete
dephasing, but we know which sites are subject to error by measuring the flag
qubit. The flagged noise has $2n$ jump operators,
$L_{2i-1}=I\otimes\sigma^{[i]}_{+}/\sqrt{2},L_{2i}=\sigma^{[i]}_{\alpha}\otimes\sigma^{[i]}_{+}/\sqrt{2}$.

We say the noise is correctable if there exists a decoding channel
$\mathcal{D}$ at each size $n$ such that
$\lim_{n\rightarrow\infty}F(\mathcal{D}\circ\mathcal{N}(\rho),\rho)=1$ for any
$\rho$ in $\mathcal{H}_{\mathrm{code}}$, where $F$ is the Uhlmann fidelity.
The uniform dephasing is harder to correct than the flagged dephasing, as one
can discard the flag to obtain the uniformly dephased state,
$\textrm{Tr}_{\mathrm{F}}\mathcal{N}_{p,\alpha;\mathrm{F}}(\rho)=\mathcal{N}_{p,\alpha}(\rho)$.

Error correcting condition.– In order to characterize the recoverability of a
QEC, Ref. [48] introduces the entanglement fidelity for a decoding channel
$\mathcal{D}$,
$F_{e}=\langle\psi_{RQ}|\mathcal{D}\circ\mathcal{N}(|\psi_{RQ}\rangle\langle\psi_{RQ}|)|\psi_{RQ}\rangle$,
where

$|\psi_{RQ}\rangle=\frac{1}{\sqrt{D}}\sum_{\alpha=1}^{D}|\alpha\rangle_{R}|\phi_{\alpha}\rangle_{Q}$
(3)

is the maximal-entangled state between the reference $R$ and the code subspace
of the physical qubits $Q$. The fidelity between any logical state and the
recovered state is lower-bounded by $F_{e}$. Let
$\rho_{RQ}=\mathcal{N}(|\psi_{RQ}\rangle\langle\psi_{RQ}|)$ (see Fig. 1), one
can further define the coherent information

$I_{c}=S_{Q}-S_{RQ},$ (4)

where $S_{A}:=-\textrm{Tr}(\rho_{A}\log\rho_{A})$. It has been shown that if
the coherent information is close to its maximal value, that is, $I_{c}=\log
D-\epsilon$, then there exists a decoding channel $\mathcal{D}$ which achieves
entanglement fidelity $F_{e}>1-2\sqrt{\epsilon}$ 111There is also an upper
bound of $F_{e}$ by $1-F_{e}>O(\epsilon)$ due to monotonicity of coherent
information and Fannes inequality.. A QEC can perfectly correct the error if
and only if $I_{c}=\log D$. If $\epsilon>0$ then the code is an approximate
QEC.

Figure 1: (Left) The coherent information of the CFT code is defined by
$I_{c}=S_{Q}-S_{RQ}$ for the state $\rho_{RQ}$ in the figure, where
$W_{\mathrm{encode}}$ is an encoding isometry which outputs states in the CFT
low-energy subspace and $\mathcal{N}$ denote noise channels acting on all
physical sites. (Right) Coherent information of correctable (blue) and
uncorrectable (red) noise for the CFT code. The correctability condition is
$\Delta_{\min}>1/2$, where $\Delta_{\min}$ is the smallest scaling dimension
in the fusion algebra of the noise channel jump operators.

A CFT code can approximately correct any noise that only acts on one lattice
site. The reason is that the jump operator $L_{i}$ on site $i$ connects
different codeword states with an amplitude [50, 51]

$\langle\phi_{\beta}|L_{i}|\phi_{\alpha}\rangle=\left(\frac{2\pi}{n}\right)^{\Delta}C_{\alpha\beta
L},$ (5)

where $\Delta$ is the scaling dimension of the operator $L_{i}$ and
$C_{\alpha\beta L}$ is the operator product expansion (OPE) coefficient that
is independent of $n$. This violation of Knill-Laflamme condition implies that
the best entanglement infidelity decreases to zero polynomially in $n$ [14].

Below we consider the CFT code under noise that uniformly acting on all sites,
which is much harder to treat than the single-site noise. The coherent
information $I_{c}(p,n)$ depends on the noise rate and the system size. The
noise is correctable at error rate $p$ if and only if

$\lim_{n\rightarrow\infty}I_{c}(p,n)=\log
D~{}(\mathrm{correctable~{}condition}).$ (6)

Given a noise model, such as flagged dephasing Eq. (2) or unflagged dephasing
Eq. (1), the code is said to have a threshold $p_{c}$ if all noise channels
with $p<p_{c}$ are correctable.

Perturbative expansion of coherent information– In order to derive our main
result, we make use of a perturbative expansion of coherent information
together with the scaling hypothesis. The derivation is sketched below.

The crucial assumpution is the scaling hypothesis, that is, the coherent
information has a scaling collapse of the form

$I_{c}(p,n)=f(pn^{\nu})$ (7)

as $n\rightarrow\infty$ and $p\rightarrow 0$, where $\nu$ is a constant that
is analogous to the critical exponent. The same scaling form, in particular,
the same exponent $\nu$, is assumed to hold for any scaling of $p$ with
respect to $n$ as long as $p\rightarrow 0$. The underlying physical intuition
is to treat the noisy channel as a perturbation, which flows towards or flows
away from the no-noise fixed point under the renormalization group flow.
Furthermore, the scaling form is also confirmed numerically, as we demonstrate
later. The scaling function $f$ is monotonically decreasing and satisfies
$f(0)=\log D$ and $f(\infty)<\log D$. For dephasing noise one can easily show
$f(\infty)=0$. Therefore the sign of $\nu$ determines the error correctability
in the thermodynamic limit. If $\nu>0$, then $I_{c}(p,n)\rightarrow 0$ in the
thermodynamic limit for arbitrary $p>0$, thus $p_{c}=0$. If $\nu<0$, then the
correctability condition Eq. (6) is satisfied for small enough $p$ and thus
the error has a finite threshold.

Next, we perform a perturbation theory on the coherent information assuming
that $p=o(1/n)$. The quantum channel can be approximated by

$\mathcal{N}_{p}(\rho)=\rho+\frac{p}{2}\mathcal{L}(\rho)+O(p^{2}),$ (8)

where $\mathcal{L}$ is the Lindbladian. The state
$\rho_{RQ}(p)=\mathcal{N}(|\psi_{RQ}\rangle\langle\psi_{RQ}|)$ can be expanded
to first order in $p$. Expanding $S_{Q}$ and $S_{QR}$ to the first order (see
details in appendix), we obtain

$I_{c}(p,n)=\log D+b(n)p\log p+O(p),$ (9)

where

$b(n)=\frac{1}{D^{2}}\sum_{i=1}^{n}[D\textrm{Tr}(L_{i}PL^{\dagger}_{i}P)-\textrm{Tr}(L_{i}P)\textrm{Tr}(L^{\dagger}_{i}P)]$
(10)

and $P=\sum_{\alpha}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|$ is the
projector onto the code subspace. The noise is correctable only if
$\lim_{n\rightarrow\infty}b(n)=0$. It is worth noting that $b(n)\geq 0$ and
the equality holds if and only if $PL_{i}P\propto P$, which is part of the
Knill-Laflamme conditions for a single-qubit error. Higher-order perturbations
can reproduce Knill-Laflamme conditions with multi-qubit errors.

Finally, using Eq. (5) we obtain $b(n)\propto n^{1-2\Delta}$, where $\Delta$
is the scaling dimension of the jump operator. The only way that $b(n)\propto
n^{1-2\Delta}$ can be compatible with the scaling form
$I_{c}(p,n)=f(pn^{\nu})$ is that

$\nu=1-2\Delta.$

Crucially, the scaling hypothesis allows us to infer the exponent $\nu$ for
$p=O(1)$ from finite-order of perturbation theory performed at $p=o(1/n)$.
Thus the error correctability condition $\nu>0$ holds at finite $p$ if and
only if $\Delta>1/2$.

Expanding to higher orders of perturbation theory, we find that correctability
requires that all the OPE between $L_{i}$ and itself only contains operators
with scaling dimension larger than $1/2$, excluding the identity operator. If
the lowest operator with scaling dimension $\Delta_{\min}$ appears at fusion
order $r$, then $\nu=(1-2\Delta_{\min})/r$. We provide a calculation to the
second order and an argument for higher orders in the appendix.

In order to verify our result, we numerically compute coherent information for
both unflagged (Eq. (1)) and flagged (Eq. (2)) dephasing noise. We remark that
two noise models should have the same $\nu$ according to our argument, because
they contain the same jump operators acting on the physical system. For the
unflagged case, we are restricted to small system sizes due to exponentially
growing simulation complexity. For the flagged case, we can access much larger
system sizes by sampling over measurement trajectories, which is a trick first
explored in Ref. [52] and detailed in appendix.

Figure 2: Coherent information of the Ising CFT code under flagged and flagged
dephasing with $p\leq 0.2$

Example: Ising CFT code.– As an example, let us consider the Ising CFT
realized by the transverse field Ising model
$H=-\sum_{i}(\sigma^{[i]}_{x}\sigma^{[i]}_{x}+g\sigma^{[i]}_{z})$ at
$g=g_{c}=1$. It has three primary operators labelled by $I,\sigma,\varepsilon$
with scaling dimensions
$\Delta_{I}=0,\Delta_{\sigma}=1/8,\Delta_{\varepsilon}=1$ [53]. The code
subspace can be minimally chosen to be spanned by the ground state $|I\rangle$
and the second excited state $|\varepsilon\rangle$. The three single-site
Pauli operators $\sigma_{x},\sigma_{y},\sigma_{z}$ corresponds to
$\sigma,\partial\sigma,\varepsilon$ in the CFT [51]. We can read off the
exponent $\nu$ for the $X$, and $Z$ dephasing
$\nu_{x}=1-2\Delta_{\sigma}=0.75$ and $\nu_{z}=1-2\Delta_{\varepsilon}=-1$.
For $Y$ dephasing, we can use second order perturbation theory to show that
$\nu_{y}=(1-2\Delta_{\varepsilon})/2=-0.5$. The underlying reason is that the
OPE $\partial\sigma\times\partial\sigma$ contains an operator $\varepsilon$
with $\Delta_{\varepsilon}<\Delta_{\partial\sigma}$, and further fusion does
not produce a lower one than $\varepsilon$. Thus our theory predicts that the
$Y$ and $Z$ dephasing have a finite threshold and $X$ dephasing has zero
threshold.

We compute the coherent information for the unflagged dephasing up to $n=16$
222We use MPO to store the purified state $|\psi\rangle_{RQE}$ and coarse-
grain $Q$ and $E$ separately. The effective physical dimension must increase
exponentially with subsystem sizes, so we have limited system size $n\leq 16$.
and flagged dephasing up to $n=128$ 333For the sampling algorithm we use 56000
samples. with $p\leq 0.2$, see Fig. 2. The scaling collapses fit best with
$\nu_{x}\approx 0.65,\nu_{y}\approx-0.5,\nu_{z}\approx-1.0$, which are close
to the theoretical values.

In order to obtain the threshold, we compute $I_{c}$ for flagged noise at
large $p$ and find that the threshold $p=1$ for both $Y$ and $Z$ dephasing. In
Fig. 3 we show the extrapolation of $I_{c}$ to the thermodynamic limit and
show that $I_{c}=\log 2$ up to $p\leq 0.8$. We also perform a scaling collapse
near $p=1$ to confirm that it is indeed an unstable RG fixed point, which
indicates that $p=1$ is the threshold. For unflagged dephasing, the threshold
can be estimated by the crossing point of the curves $I_{c}(p)$ of different
$n$. Our small-size simulation suggests that the threshold is also
$p_{y}=p_{z}=1$, see appendix for details.

Figure 3: Coherent information of the Ising CFT code under $Z$ (left) and $Y$
(right) flagged dephasing with $0.2\leq p\leq 0.8$

We see that the correctable and uncorrectable dephasing of the Ising CFT code
is identical to the repetition code. It is not unexpected since the repetition
code can be realized by the spontaneous symmetry broken phase $0<g<1$ of TFIM.
However, as opposed to the repitition code where the number of protected
logical qubits is $k=O(1)$, the CFT code can protect more logical information
as we increase $n$, as we demonstrate below.

Dimension of logical subspace.– The low-energy subspace is infinite-
dimensional as $n\rightarrow\infty$, as each primary operator has infinitely
many descendants. It is then natural to ask how large the code subspace can be
chosen to maintain a finite threshold for correctable dephasing noise with
$\Delta>1/2$. Firstly, as $b(n)\propto n^{1-2\Delta}$ in Eq. (10) for any
finite-dimensional code subspace, we can store at least constant number of
qubits. Below we further argue that the dimension of code subspace can scale
at least as $\mathrm{polylog}(n)$, thus $k\geq\Omega(\log\log n)$. Let us
choose the code subspace to be spanned by a primary state $|\phi\rangle$ and
its global descendants $|\partial^{m}\bar{\partial}^{\bar{m}}\phi\rangle$,
where $m$ and $\bar{m}$ are natural numbers smaller than a cutoff $M$. Then
the projector to the code subspace is

$P=\sum_{m,\bar{m}=0}^{M-1}|\partial^{m}\bar{\partial}^{\bar{m}}\phi\rangle\langle\partial^{m}\bar{\partial}^{\bar{m}}\phi|$
(11)

As we show in the appendix, if $M=\mathrm{polylog}(n)$, then
$\lim_{n\rightarrow\infty}b(n)\rightarrow 0$ given any $\Delta_{K}>1/2$. The
total dimension of code subspace $2^{k}=M^{2}$, which means
$k\propto\log(M)=O(\log\log n)$. Thus, we establish that the number of logical
qubits increases with the number of physical qubits. In principle one can
encode more logical qubits by including Virasoro descendants in the code
subspace. By including all descendants below scaling dimension
$M=\mathrm{polylog}(n)$, one may reach $k=O(\mathrm{polylog}(n))$ by the Cardy
formula [50]. We leave it to future work to investigate its error protection
properties.

Covariant CFT code.– Let us consider a particular case of the CFT code where
the CFT has a global $U(1)$ symmetry. Then there exists a conserved charge
density which has scaling dimension $\Delta=1$. The dephasing noise on the
conserved charge is a correctable error since $\nu=1-2\Delta=-1$. The coherent
information gives $1-F_{e}>\sqrt{\delta I_{c}}\propto\sqrt{p}n^{-1/2}$ at
small $p$. The same conclusion holds for arbitrary dimensions. On the other
hand, it has been shown in Refs. [56, 57] that $1-F_{e}\leq O(n^{-1/2})$ for
any covariant code with $U(1)$ symmetry. Thus, the CFT code with $U(1)$
symmetry saturates the covariant code bound.

Discussion.– Our result can be readily applied to depolarization noise, where
all Pauli errors can happen. The CFT code can correct depolarization noise at
a finite threshold if and only if all scaling dimensions $\Delta>1/2$. Such a
CFT does not exist for Virasoro minimal models and all $c=1$ free boson
models. CFTs that satisfy this condition have $c>1$ and may only be realized
beyond spin-$1/2$ chains and $2-$local interactions [58]. One such example is
the monster CFT with the lowest primary $\Delta=4$, which has been conjectured
to be dual to a bulk quantum gravity theory [59]. A holographic CFT also
satisfies $\Delta>1/2$ in the large-$N$ limit, as the CFT spectrum is assumed
to be sparse. One interesting feature of such a CFT code is that it may have a
finite threshold in one spatial dimension, which cannot be achieved by the
ground state subspace of any gapped local Hamiltonian.

In terms of characterization of mixed-state phases, one of our technical
contributions is the perturbative calculation of coherent information without
applying replica trick, while prior works mostly focus on finite replica
calculations. In fact, the Renyi coherent information may give incorrect
predictions of the code properties. For example, it gives a wrong decoding
threshold for the toric code [20]. This work has given another example which
shows that working with integer Renyi index predicts wrong error-
correctability of infinitesimally small noise strength. For the Ising CFT
code, uniform $Z$ dephasing can be corrected at a finite threshold but the
Renyi coherent information drops as long as $p\neq 0$, see appendix for
details.

Our work opens up several future directions. Firstly, even though we have
shown that some errors can be corrected for CFT code, it is still nontrivial
to construct an explicit decoder, an important ingredient of QECC. Secondly,
we can consider logical operations for CFT code and see if it can be performed
fault-tolerantly. Thirdly, it is interesting to consider AdS/CFT with boundary
dephasing, which may inspire the solution of the two questions above.

Acknowledgements.– We thank Charles Cao, Yu-Hsueh Chen, Tarun Grover, Yangrui
Hu, Tianci Zhou, Yaodong Li, David Lin, Peter Lu, Ruochen Ma, Daniel Parker,
Bowen Shi, Guifre Vidal, Sisi Zhou and Zheng Zhou for helpful discussions.
Y.Z. in particular thanks Patrick Hayden for his insight that leads this work.
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 supported in part by grant NSF PHY-2309135 to
the Kavli Institute for Theoretical Physics (KITP).

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## Flagged dephasing and unflagged dephasing

###  .1 Coherent information

In order to compute the coherent information, let us start with an unravelling
of the flagged dephasing. At each site there is a probability $1-p$ that we
don’t measure the state, and probability $p$ that we measure the state in the
$\sigma_{\alpha}$ basis, with a Born probability
$q_{\pm}=\mathrm{tr}(\rho_{RQ}\frac{I\pm\sigma_{\alpha}}{2})$ to project onto
the $|\pm\alpha\rangle$ state. A crucial observation is the output states of
the above three cases are mutually orthogonal. This is because, firstly, if
the measured sites are different, then the flags are orthogonal, and secondly,
if the measurement outcomes are different, then the measured qubits are
orthogonal. Thus,
$\rho_{RQF}=\mathcal{N}_{p,\alpha;F}(|\psi_{RQ}\rangle\langle\psi_{RQ}|)$ can
be unravelled as

$\rho_{RQF}=\sum_{\mathbf{m}}p_{\mathbf{m}}|\psi^{\mathbf{m}}_{RQ}\rangle\langle\psi^{\mathbf{m}}_{RQ}|\otimes|\mathbf{m}\rangle\langle\mathbf{m}|.$
(12)

Note that here $|\mathbf{m}\rangle$ records not only which sites are measured
but also the outcome state in the measured sites. More precisely, denote
$m=\pm$ as measurement outcome and $m=0$ as no measurement, and
$P^{[j]}_{\pm}=\frac{I\pm\sigma^{[j]}_{\alpha}}{2}$, $P^{[j]}_{0}=I$ then

$\displaystyle|\psi^{\mathbf{m}}_{RQ}\rangle=\sqrt{p_{\mathbf{m}}}\left(\prod_{j}P^{[j]}_{\mathbf{m}_{j}}\right)|\psi_{RQ}\rangle.$
(13)

It follows that

$\displaystyle S(\rho_{RQF})$ $\displaystyle=$
$\displaystyle-\sum_{\mathbf{m}}p_{\mathbf{m}}\log p_{\mathbf{m}}$ (14)
$\displaystyle S(\rho_{QF})$ $\displaystyle=$
$\displaystyle-\sum_{\mathbf{m}}p_{\mathbf{m}}\log
p_{\mathbf{m}}+\sum_{\mathbf{m}}p_{\mathbf{m}}S_{Q}(|\psi^{\mathbf{m}}_{RQ}\rangle).$
(15)

Note that $S_{Q}=S_{R}$ for a pure state $|\psi^{\mathbf{m}}_{RQ}\rangle$. The
coherent information $I_{c}=S(\rho_{QF})-S(\rho_{QRF})$ is then

$I_{c}=\mathbb{E}_{\mathbf{m}}S_{R}(|\psi^{\mathbf{m}}_{RQ}\rangle),$ (16)

where $\mathbb{E}_{\mathbf{m}}$ is the average with respect to the joint
probability distribution $p_{\mathbf{m}}$ of the flags and measurement
outcomes.

###  .2 Threshold of flagged dephasing

We can calculate the coherent information of the Ising CFT code under flagged
dephasing using MPS techniques. We first use DMRG to diagonalize the low-
energy eigenstates in the code subspace. We then construct the maximal
entangled state $\psi_{RQ}$ as an MPS by concatenating the MPS of the
eigenstates. We compress the MPS to lower bond dimension by allowing a small
amount $10^{-8}$ of truncation error. We then perform measurement on $Q$ and
obtain a sample $|\psi^{\mathbf{m}}_{RQ}\rangle$ of post-measurement states as
an MPS, from which we can read off the entanglement entropy $S_{R}$. Finally
we average over $S_{R}$ to obtain the coherent information. The results for
the flagged dephasing up to $n=128$ and $p\leq 0.8$ are shown in the main
text. Here we provide additional data for the flagged $Y$ dephasing and the
data near $p\approx 1$.

Figure 4: Flagged $Y$ (left) and $Z$ (right) dephasing coherent information of
the Ising CFT code near $p=1$.

For both $Y$ and $Z$ flagged dephasing, we can perform a scaling collapse near
$p=1$ to obtain a scaling form $I_{c}((1-p)L^{\nu^{\prime}})$ with
$\nu^{\prime}>0$. This indicates that $p=1$ is an unstable RG fixed point
which flows to $p=0$. Thus, for both $Y$ and $Z$ flagged dephasing, the
decoding threshold is $p_{c}=1$.

###  .3 Threshold of unflagged dephasing

We provide additional evidence tha that the threshold for unflagged $Y$ and
$Z$ dephasing for the Ising CFT code is also $p_{c}=1$. We plot $I_{c}(p)$
with system sizes $7\leq n\leq 11$ and find that the curves only intersect at
$p=0$ and $p=1$. This suggests that the threshold is $p_{c}=1$ but the
evidence is not conclusive due to very small system sizes.

Figure 5: $Y$ (left) and $Z$ (right) dephasing coherent information of the
Ising CFT code. The only intersections of the curves are at $p=0$ and $p=1$,
indicating a threshold $p_{c}=1$ in the thermodynamic limit.

## Perturbation of coherent information

In this section we consider the change of coherent information under an
infinitesimal evolution with the Lindbladian.

###  .4 First order perturbation

Under the first-order perturbation, we can approximate

$\rho_{RQ}(p)=|\psi\rangle\langle\psi|+p\sum_{i=1}^{n}\left(L_{i}|\psi\rangle\langle\psi|L^{\dagger}_{i}-\frac{1}{2}L_{i}^{\dagger}L_{i}|\psi\rangle\langle\psi|-\frac{1}{2}|\psi\rangle\langle\psi|L_{i}^{\dagger}L_{i}\right)+O(p^{2}),$
(17)

where

$|\psi\rangle=\frac{1}{\sqrt{D}}\sum_{\alpha=1}^{D}|\alpha\rangle_{R}|\phi_{\alpha}\rangle_{Q}.$
(18)

We will look at the eigenvalues of the density matrix $\rho_{RQ}(p)$. It is
clear that there are at most $n+1$ nonzero eigenvalues, which are
$\lambda_{0}=1-p\nu_{0}$ and $\lambda_{i}=p\nu_{i}$ to the first order in $p$.
Since the first eigenvalue is non-degenerate, we can use first-order
perturbation theory on the eigenvalue to obtain

$-\nu_{0}=\sum_{i}\left(|\langle\psi|L_{i}|\psi\rangle|^{2}-\langle\psi|L^{\dagger}_{i}L_{i}|\psi\rangle\right)$
(19)

Now we can compute the entropy $S_{RQ}$ to be

$\displaystyle S_{RQ}$ $\displaystyle=$
$\displaystyle-\sum_{i}\lambda_{i}\log\lambda_{i}$ (20) $\displaystyle=$
$\displaystyle-(1-p\nu_{0})\log(1-p\nu_{0})-\sum_{i}p\nu_{i}\log(p\nu_{i})$
(21) $\displaystyle=$ $\displaystyle-\left(\sum_{i}\nu_{i}\right)p\log
p+\left(\nu_{0}-\sum_{i}\nu_{i}\log\nu_{i}\right)p+O(p^{2}).$ (22)
$\displaystyle=$ $\displaystyle\nu_{0}p\log
p+\left(\nu_{0}-\sum_{i}\nu_{i}\log\nu_{i}\right)p+O(p^{2})$ (23)

where we have used the fact that $\textrm{Tr}\rho_{RQ}(p)=1$ in the last step.
Now we turn to the state $\rho_{Q}(p)=\textrm{Tr}_{R}\rho_{RQ}(p)$ and
consider its eigenvalues. To the first order in $p$,

$\rho_{Q}(p)=\sum_{\alpha=1}^{D}\frac{1}{D}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|+\frac{p}{D}\sum_{\alpha=1}^{D}\sum_{i=1}^{n}\left(L_{i}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|L^{\dagger}_{i}-\frac{1}{2}L_{i}^{\dagger}L_{i}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|-\frac{1}{2}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|L_{i}^{\dagger}L_{i}\right)+O(p^{2}).$
(24)

The first $D$ eigenvalues are then $(1/D-p\alpha_{1},1/D-p\alpha_{2},\cdots
1/D-p\alpha_{D})$ and the rest are proportional to $p$ with at most $Dn$ of
them, $(p\beta_{1},\cdots p\beta_{Dn})$. Using the degenerate perturbation
theory on the subspace spanned by the first $D$ eigenvalues, we can show that
$-\alpha_{k}$’s are the eigenvalues of a $D\times D$ matrix $V$ whose
components are

$\displaystyle V_{kk}$ $\displaystyle=$
$\displaystyle\frac{1}{D}\left(\sum_{i,\alpha}\langle\phi_{k}|L_{i}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|L^{\dagger}_{i}|\phi_{k}\rangle-\sum_{i}\langle\phi_{k}|L^{\dagger}_{i}L_{i}|\phi_{k}\rangle\right)$
(25) $\displaystyle V_{kl}$ $\displaystyle=$
$\displaystyle\frac{1}{D}\left(\sum_{i,\alpha}\langle\phi_{k}|L_{i}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|L^{\dagger}_{i}|\phi_{l}\rangle-\sum_{i}\mathrm{Re}\langle\phi_{k}|L^{\dagger}_{i}L_{i}|\phi_{l}\rangle\right)~{}~{}~{}~{}~{}(k\neq
l)$ (26)

In terms of the eigenvalues of $\rho_{Q}$, we can express

$S_{Q}=\log D+\left(\sum_{k=1}^{D}\alpha_{k}\right)p\log p+\left((1-\log
D)\left(\sum_{k=1}^{D}\alpha_{k}\right)-\sum_{i}\beta_{i}\log\beta_{i}\right)p+O(p^{2})$
(27)

Since the sum of eigenvalues equals the trace,

$\sum_{k=1}^{D}\alpha_{k}=-\sum_{k}V_{kk}.$ (28)

Combining Eqs. (18), (19), (23), (25), (27), (28) we obtain

$I_{c}=S_{Q}-S_{QR}=\log D+bp\log p+O(p)$ (29)

where

$b=\frac{1}{D^{2}}\sum_{i,\alpha,\beta}(D|\langle\phi_{\beta}|L_{i}|\phi_{\alpha}\rangle|^{2}-\langle\phi_{\alpha}|L_{i}|\phi_{\alpha}\rangle\langle\phi_{\beta}|L^{\dagger}_{i}|\phi_{\beta}\rangle).$
(30)

More compactly we can write

$b=\frac{1}{D^{2}}\sum_{i=1}^{n}[D\textrm{Tr}(L_{i}PL^{\dagger}_{i}P)-\textrm{Tr}(L_{i}P)\textrm{Tr}(L^{\dagger}_{i}P)]$
(31)

where

$P=\sum_{\alpha=1}^{D}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|$ (32)

is the projector onto code subspace.

We have shown in the main text that $b\propto n^{1-2\Delta}$, where $\Delta$
is the scaling dimension of the jump operator. If $\Delta<1/2$. then the
leading correction term $bp\log p$ dominates Eq. (29) at small $p$ and gives
$I_{c}<\log D$, thus the noise is uncorrectable. If $\Delta>1/2$, then the
leading correction term $bp\log p\rightarrow 0$ as $n\rightarrow\infty$. We
further argue below that the subleading correction $O(p)$ term in Eq. (29)
gives the same exponent $1-2\Delta$ as $n\rightarrow\infty$. Using the scaling
hypothesis $I_{c}=I_{c}(pn^{\nu})$ thus fixes $\nu=1-2\Delta$ and gives a
finite threshold for the noise.

###  .5 Second-order perturbation

Before we do an explicit calculation, We will first intuitively argue how
higher-order terms could potentially dominate the change of $I_{c}$ and how it
affects the error correctability condition. Let us consider a block spin
transformation of two spins $i$ and $i+1$ which are blocked into a four-
dimensional local Hilbert space. Then for each site there is $O(p)$
probability where one error $L_{i}$ or $L_{i+1}$ happens and there is
$O(p^{2})$ probability where two errors $L_{i}L_{i+1}$ happen. Now
$L_{i}L_{i+1}$ is a local operator and has its own scaling dimension
$\Delta^{(2)}$. For a lattice realization of CFT, if $L_{i}$ corresponds to a
scaling operator $O_{\Delta}$ with scaling dimension $\Delta$, then
$L_{i}L_{i+1}$ corresponds to the lowest operator (except identity) in the
operator product expansion $O_{\Delta}\times O_{\Delta}\rightarrow
I+:O^{2}:+\cdots$. The normal-ordered operator $:O^{2}:$ has its own scaling
dimension $\Delta^{(2)}$. If $\Delta^{(2)}<\Delta$, then the change of
coherent information is dominated by this error for large system sizes,
$\delta I_{c}\propto p^{2}n^{1-2\Delta^{(2)}}\log p$. This then indicates that
$\nu=(1-2\Delta^{(2)})/2$. More generally, within the self-OPE of the jump
operator, if the lowest operator with scaling dimension $\Delta^{(r)}$ appears
at the $r$-th order of fusion, then $\nu=(1-2\Delta^{(r)})/r$. In order to
require the error to be correctable, we need all $\Delta^{(r)}>1/2$. This
could happen if all of the CFT primary operators have scaling dimensions
larger than $1/2$, or there is a symmetry of the jump operator $L$ that
forbids it from fusing into primary operators with scaling dimension smaller
than $1/2$.

For an explicit example, consider the $Z$ operator the TFIM model which
corresponds to the $I+\varepsilon$ operator in the CFT. Since we have the
fusion rule $I\times\varepsilon=\varepsilon$ and
$\varepsilon\times\varepsilon=I$, we will never encounter the $\sigma$
operator in the OPE, thus the lowest operator
$\Delta^{(r)}=\Delta_{\varepsilon}=1$ occurs at $r=1$. Thus
$\nu=1-2\Delta_{\varepsilon}=-1$. A more sophisticated example happens for the
$Y$ dephasing, which corresponds to $Y=\partial_{t}\sigma$ in the Ising CFT.
Two $Y$’s could fuse
$\partial_{t}\sigma\times\partial_{t}\sigma=I+\varepsilon$ and three $Y$’s can
only fuse into itself,
$\partial_{t}\sigma\times\partial_{t}\sigma\times\partial_{t}\sigma=\partial_{t}\sigma$.
Note that fusion into $\sigma$ is not possible due to time reversal symmetry.
Thus, the lowest operator in the fusion appears at $r=2$, which is again the
$\varepsilon$ operator. As a result $\nu=(1-2\Delta_{\varepsilon})/2=-0.5$.

Now we perform a second order perturbation theory on the coherent information
$I_{c}=S_{Q}-S_{QR}$ and see how the intuition above works. To start with, we
can expand

$I_{c}=\log D+bp\log p+b^{(2)}p^{2}\log p+\mathrm{regular}$ (33)

and we will compute $b^{(2)}$ as a function of jump operators and system size.
Let us consider the perturbed density matrix

$\rho_{QR}(p)=\rho_{QR}(0)+p\mathcal{L}(\rho_{QR}(0))+\frac{p^{2}}{2}\mathcal{L}(\mathcal{L}(\rho_{QR}(0)))+O(p^{3})$
(34)

where $\rho_{QR}(0)=|\psi\rangle\langle\psi|$. The eigenvalues of
$\rho_{QR}(p)$ can be parameterized as $1-p\nu_{0}+p^{2}\nu^{(2)}_{0}$ and
$p\nu_{i}+p^{2}\nu^{(2)}_{i}$. To this order, we can expand

$S_{QR}=\nu_{0}p\log p-\nu^{(2)}_{0}p^{2}\log p+\mathrm{regular.}$ (35)

where $\mathrm{regular}$ means terms analytic in $p$. We will see that
$\nu^{(2)}_{0}\propto n^{1-2\Delta^{(2)}}$ using perturbation theory. There
are two contributions, one coming from the second-order perturbation with the
$p\mathcal{L}(\rho_{QR}(0))$ term and one coming from the first-order
perturbation with the $\frac{p^{2}}{2}\mathcal{L}(\mathcal{L}(\rho_{QR}(0)))$
term. Note that $\rho_{QR}(0)$ has one eigenvalue $1$ and the rest are $0$.
Denote for shorthand
$\rho:=\rho_{QR}(0),\delta\rho:=\mathcal{L}(\rho_{QR}(0))$ and
$\delta^{2}\rho:=\mathcal{L}(\mathcal{L}(\rho_{QR}(0)))$ we have

$\nu^{(2)}_{0}=\textrm{Tr}(\rho\delta\rho(1-\rho)\delta\rho)+\frac{1}{2}\textrm{Tr}(\rho\delta^{2}\rho).$
(36)

For dephasing noise, we have $L^{\dagger}_{i}L_{i}=I$. This simplifies the
expression but does not affect the conclusion. For this case we have

$\delta\rho=\sum_{i}\frac{L_{i}\rho
L^{\dagger}_{i}-\rho}{2},~{}~{}\delta^{2}\rho=\sum_{i,j}\frac{L_{i}L_{j}\rho
L^{\dagger}_{j}L^{\dagger}_{i}-\rho}{4}$ (37)

Now we can substitute the above equation to $\nu^{(2)}_{0}$ to obtain

$\nu^{(2)}_{0}=\frac{1}{8}\sum_{i,j}\left(2\langle L_{i}\rangle\langle
L^{\dagger}_{i}L_{j}\rangle\langle L^{\dagger}_{j}\rangle-2\langle
L_{i}\rangle\langle L^{\dagger}_{i}\rangle\langle L_{j}\rangle\langle
L^{\dagger}_{j}\rangle+\langle L_{i}L_{j}\rangle\langle
L^{\dagger}_{i}L^{\dagger}_{j}\rangle-1\right)$ (38)

where $\langle\cdot\rangle$ is taken with respect to $|\psi\rangle_{QR}$, that
is, $\langle\cdot\rangle=\frac{1}{D}\mathrm{Tr}(\cdot P)$.

Now we turn to $\rho_{Q}(p)$, which takes the same form as Eq. (34) but with
$QR$ substituted as $Q$. The initial state is $\rho_{Q}(0)=\frac{1}{D}P$ where
$P$ is the projector onto the code subspace. Thus
$\rho_{Q}(0)^{2}=\frac{1}{D}\rho_{Q}(0)$. The first $D$ eigenvalues of
$\rho_{Q}(p)$ can be expanded as $1/D-p\alpha_{k}+p^{2}\alpha^{(2)}_{k}$,
where $k=1,2,\cdots,D$. The entropy $S_{Q}$ can be expanded as

$S_{Q}=\log D+\left(\sum_{k=1}^{D}\alpha_{k}\right)p\log
p-\left(\sum_{k=1}^{D}\alpha^{(2)}_{k}\right)p^{2}\log p+\mathrm{regular}$
(39)

We can use second order perturbation theory to compute the coefficient in
front of $p^{2}\log p$.

$\sum_{k=1}^{D}\alpha^{(2)}_{k}=D\mathrm{Tr}(P\delta P(1-P)\delta
P)+\frac{1}{2}\mathrm{Tr}(P\delta^{2}P),$ (40)

where the factor of $D$ comes from the eigenvalue $1/D$ in the unperturbed
$\rho_{Q}$ and we have denoted

$\delta
P=\frac{1}{D}\sum_{i}\frac{L_{i}PL^{\dagger}_{i}-P}{2},~{}~{}\delta^{2}P=\frac{1}{D}\sum_{i,j}\frac{L_{i}L_{j}PL^{\dagger}_{j}L^{\dagger}_{i}-P}{4}.$
(41)

Substituting Eq. (41) into Eq. (40), we obtain

$\sum_{k=1}^{D}\alpha^{(2)}_{k}=\frac{1}{8D}\sum_{i,j}\left(2\mathrm{Tr}(PL_{i}PL^{\dagger}_{i}L_{j}PL^{\dagger}_{j})-2\mathrm{Tr}(PL_{i}PL^{\dagger}_{i}PL_{j}PL^{\dagger}_{j})+\mathrm{Tr}(PL_{i}L_{j}PL^{\dagger}_{i}L^{\dagger}_{j})-D\right)$
(42)

Now we can identify the coefficient $b^{(2)}$ in Eq. (33) as

$b^{(2)}=\sum_{k=1}^{D}\alpha^{(2)}_{k}-\nu^{(2)}_{0}:=b^{(2)}_{1}+b^{(2)}_{2}+b^{(2)}_{3},$
(43)

where we separate the sum into three terms,

$\displaystyle b^{(2)}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4D^{4}}\sum_{i,j}\left(D^{3}\mathrm{Tr}(PL_{i}PL^{\dagger}_{i}PL_{j}PL^{\dagger}_{j})-\mathrm{Tr}(PL_{i})\mathrm{Tr}(PL^{\dagger}_{i})\mathrm{Tr}(PL_{j})\mathrm{Tr}(PL^{\dagger}_{j})\right)$
(44) $\displaystyle b^{(2)}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4D^{3}}\sum_{i,j}\left(D^{2}\mathrm{Tr}(PL_{i}PL^{\dagger}_{i}L_{j}PL^{\dagger}_{j})-\mathrm{Tr}(PL_{i})\mathrm{Tr}(PL^{\dagger}_{i}L_{j})\mathrm{Tr}(PL^{\dagger}_{j})\right)$
(45) $\displaystyle b^{(2)}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{8D^{2}}\sum_{i,j}\left(D\mathrm{Tr}(PL_{i}L_{j}PL^{\dagger}_{i}L^{\dagger}_{j})-\mathrm{Tr}(PL_{i}L_{j})\mathrm{Tr}(PL^{\dagger}_{i}L^{\dagger}_{j})\right)$
(46)

Again the result makes sense since $b^{(2)}=0$ if and only if the Knill-
Laflamme condition is satisfied to the second order,
$PL^{\dagger}_{i}L_{j}P\propto P$ and $PL_{i}L_{j}P\propto P$. The first
condition says that any codeword state with one error can be distinguished
from any other codeword state with one error. The second condition says that
any codeword state with two errors is distinguishable from any other codeword
states with no error.

Now we analyze the scaling of each term for the CFT code. For the first term
$b^{(2)}_{1}$, it scales as $O(n^{2-4\Delta})$. Combining with $p^{2}$ gives
$(pn^{1-2\Delta})^{2}$, thus this term does not change the exponent from the
first-order perturbation. For the second term $b^{(2)}_{2}$, we can separate
the sum into $i=j$ and $i\neq j$ pieces. The $i=j$ piece is identical to Eq.
(31) up to a constant factor. For the $i\neq j$ piece, we first analyze the
scaling of the term

$\sum_{i\neq
j}\mathrm{Tr}(PL^{\dagger}_{i}L_{j})=n\sum_{\alpha=1}^{D}\sum_{i=1}^{n-1}\langle\phi_{\alpha}|L^{\dagger}_{0}L_{i}|\phi_{\alpha}\rangle,$
(47)

where we have made use of translation invariance. Given a lattice realization
of the CFT, we can do OPE on the lattice with
$O^{\dagger}(i)O(j)\sim|i-j|^{-2\Delta+\Delta_{O^{\dagger}O}}:O^{\dagger}O:((i+j)/2)+\cdots$,
where we recall that $\Delta$ denotes the scaling dimension of $O$. Now the
sum over $i$ can be estimated based on whether
$\Delta_{O^{\dagger}O}-2\Delta>-1$. If $\Delta_{O^{\dagger}O}-2\Delta>-1$,
then the sum diverges as $n\rightarrow\infty$, then

$\sum_{i\neq j}\mathrm{Tr}(PL^{\dagger}_{i}L_{j})\propto
n^{2-2\Delta}~{}~{}(\Delta_{O^{\dagger}O}<2\Delta-1)$ (48)

otherwise we have a converge sum in $n$ which gives rise to

$\sum_{i\neq j}\mathrm{Tr}(PL^{\dagger}_{i}L_{j})\propto
n^{1-\Delta_{O^{\dagger}O}}~{}~{}(\Delta_{O^{\dagger}O}>2\Delta-1)$ (49)

In either case, we consistently have

$\sum_{i\neq j}\mathrm{Tr}(PL^{\dagger}_{i}L_{j})\leq O(n^{2-2\Delta})$ (50)

Taking into account the other factor $\mathrm{Tr}(PL_{i})\propto n^{-\Delta}$,
we conclude that the second term $b^{(2)}_{2}\leq O(n^{2-4\Delta})$ cannot
give a larger exponent than the first-order perturbation. The first term in
$b^{(2)}_{2}$ can be argued analogously.

Finally, we consider $b^{(2)}_{3}$. Following a similar derivation as above,
we use the OPE
$O(i)O(j)\sim|i-j|^{-2\Delta+\Delta^{(2)}}:O^{2}:((i+j)/2)+\cdots$. We can
obtain

$b^{(2)}_{3}\propto\begin{cases}n^{2-4\Delta}&\text{if
}2\Delta^{(2)}>1+4\Delta\\\ n^{1-2\Delta^{(2)}}&\text{if
}2\Delta^{(2)}<1+4\Delta\end{cases}$ (51)

If $\Delta^{(2)}<\Delta$, then clearly the second condition is satisfied and
$b^{(2)}_{3}\propto n^{1-2\Delta^{(2)}}$. This term has larger exponent
compared to the first-order perturbation scaling $n^{1-2\Delta}$. Thus if $n$
is large we obtain $\delta I_{c}\propto p^{2}n^{1-2\Delta^{(2)}}\log p$ and
$\nu=(1-2\Delta^{(2)})/2$.

###  .6 Higher order perturbation

Our main claim is that the noise is correctable at a finite threshold if and
only if $\Delta_{\min}>1/2$, where $\Delta_{\min}$ is the minimum of nonzero
scaling dimensions in the fusion algebra of the jump operators. So far we have
shown it within first and second order perturbation theory. We have argued for
the “only if” direction to all orders using the RG argument in the last
section. For the “if” direction one has to work out the full perturbation
theory to all orders, but it will be technically very difficult. Instead,
based on the second-order perturbation result, we give an argument in this
section that $\Delta_{\min}>1/2$ implies $\nu<0$ to all orders, which in turn
implies that the noise is correctable.

The singular term in the coherent information can be expanded in all orders in
$p$,

$I_{c}=\log D+\sum_{r=1}^{\infty}b^{(r)}p^{r}\log p+\mathrm{regular}.$ (52)

Each of the coefficient $b^{(r)}\propto n^{1-2\Delta^{(r)}}$. The exponent
$\nu$ is determined by the smallest $\Delta^{(r)}$ in the expansion by
$\nu=(1-2\Delta^{(r)})/r$. We will argue that $\Delta^{(r)}$ is the lowest
scaling dimension of the first $r$-th order OPE of the jump operator below.

As we have shown, to the first-order $r=1$ and second order $r=2$, $b^{(r)}$
reproduces part of the Knill-Laflamme conditions, see Eq. (31) and Eqs.
(44)-(46). This feature should persist to all orders, because the Knill-
Laflamme condition is equivalent to $I_{c}=\log D$, which follows from the
fact that both are equivalent to exact decodability. Furthermore, to the
$r$-th order, there are $2r$ jump operator insertions between the $P$ operator
inside the trace. This observation allows us to write down the form of
$b^{(r)}$ without an explicit calculation. Take $r=3$ as an example, the part
of Knill Laflamme conditions are

$PL_{i}L_{j}L_{k}P\propto P,~{}PL^{\dagger}_{i}L_{j}L_{k}P\propto
P,~{}PL^{\dagger}_{i}L^{\dagger}_{j}L_{k}P\propto
P,~{}PL^{\dagger}_{i}L^{\dagger}_{j}L^{\dagger}_{k}P\propto P$ (53)

One can then write down the following terms which could contribute to
$b^{(3)}$,

$\displaystyle b^{(3)}_{1}\propto$ $\displaystyle\sum_{i,j,k}$
$\displaystyle[D\textrm{Tr}(PL_{i}L_{j}L_{k}PL^{\dagger}_{i}L^{\dagger}_{j}L^{\dagger}_{k})-\textrm{Tr}(PL_{i}L_{j}L_{k})\textrm{Tr}(PL^{\dagger}_{i}L^{\dagger}_{j}L^{\dagger}_{k})]$
$\displaystyle b^{(3)}_{2}\propto$ $\displaystyle\sum_{i,j,k}$
$\displaystyle[D^{2}\textrm{Tr}(PL_{i}PL^{\dagger}_{i}L_{j}L_{k}PL^{\dagger}_{j}L^{\dagger}_{k})-\textrm{Tr}(PL_{i})\textrm{Tr}(PL^{\dagger}_{i}L_{j}L_{k})\textrm{Tr}(PL^{\dagger}_{j}L^{\dagger}_{k})]+h.c.$
$\displaystyle b^{(3)}_{3}\propto$ $\displaystyle\sum_{i,j,k}$
$\displaystyle[D^{2}\textrm{Tr}(PL^{\dagger}_{i}L_{j}PL^{\dagger}_{j}L_{k}PL^{\dagger}_{k}L_{i})-\textrm{Tr}(PL^{\dagger}_{i}L_{j})\textrm{Tr}(PL^{\dagger}_{j}L_{k})\textrm{Tr}(PL^{\dagger}_{k}L_{i})]$
$\displaystyle b^{(3)}_{4}\propto$ $\displaystyle\sum_{i,j,k}$
$\displaystyle[D^{3}\textrm{Tr}(PL^{\dagger}_{i}PL_{j}PL^{\dagger}_{j}L_{k}PL^{\dagger}_{k}L_{i})-\textrm{Tr}(PL^{\dagger}_{i})\textrm{Tr}(PL_{j})\textrm{Tr}(PL^{\dagger}_{j}L_{k})\textrm{Tr}(PL^{\dagger}_{k}L_{i})]$
$\displaystyle b^{(3)}_{5}\propto$ $\displaystyle\sum_{i,j,k}$
$\displaystyle[D^{4}\textrm{Tr}(PL^{\dagger}_{i}PL_{j}PL^{\dagger}_{j}PL_{k}PL^{\dagger}_{k}L_{i})-\textrm{Tr}(PL^{\dagger}_{i})\textrm{Tr}(PL_{j})\textrm{Tr}(PL^{\dagger}_{j})\textrm{Tr}(PL_{k})\textrm{Tr}(PL^{\dagger}_{k}L_{i})]$
$\displaystyle b^{(3)}_{6}\propto$ $\displaystyle\sum_{i,j,k}$
$\displaystyle[D^{5}\textrm{Tr}(PL^{\dagger}_{i}PL_{j}PL^{\dagger}_{j}PL_{k}PL^{\dagger}_{k}PL_{i})-\textrm{Tr}(PL^{\dagger}_{i})\textrm{Tr}(PL_{j})\textrm{Tr}(PL^{\dagger}_{j})\textrm{Tr}(PL_{k})\textrm{Tr}(PL^{\dagger}_{k})\textrm{Tr}(PL_{i})]$

Denoting $\Delta^{(3)}_{\min}=\min\\{\Delta,\Delta^{(2)},\Delta^{(3)}\\}$ and
assuming $\Delta^{(3)}_{\min}>1/2$, we can argue that each term above is
bounded above by $b^{(3)}_{i}\leq O(n^{1-2\Delta^{(3)}_{\min}})$. In fact, if
$\Delta^{(3)}$ is the smallest, only the first term $b^{(3)}_{1}$ saturates
the bound. The rest of terms are controlled by $\Delta$ and $\Delta^{(2)}$ as
in previous orders of perturbation theory. For example one can show that
$b^{(3)}_{6}\propto n^{3-6\Delta}\leq O(n^{1-2\Delta^{(3)}_{\min}})$ and
$b^{(3)}_{4}\leq O(n^{1-2\Delta})\leq O(n^{1-2\Delta^{(3)}_{\min}})$. Now we
turn to $b^{(3)}_{1}$. We will argue that the largest contribution comes from
when $i,j,k$ are all close to each other, where we can fuse
$L_{i},L_{j},L_{k}$ to one error with scaling dimension $\Delta^{(3)}$ and
thus gives $O(n^{1-2\Delta^{(3)}})=O(n^{1-2\Delta^{(3)}_{\min}})$. If the
three points are $O(n)$ far away, then their fusion gives rise to a factor of
$n^{-3\Delta}$. Each points can take $O(n)$ locations, so we have in total
$n^{3}\times(n^{-3\Delta})^{2}=n^{3-6\Delta}$ contributions to $b^{(3)}_{1}$.
Likewise, if two points are close and the other point is $O(n)$ far away, the
contribution to $b^{(3)}_{1}$ would be
$n^{2}\times(n^{-\Delta^{(2)}-\Delta})^{2}=n^{2-2\Delta^{(2)}-2\Delta}<n^{2-4\Delta^{(3)}_{\min}}<n^{1-2\Delta^{(3)}_{\min}}$.

Now we can generalize the above argument to arbitrary orders. Let
$\Delta_{\min}:=\min_{r}\\{\Delta^{(r)}\\}$ be the minimal scaling dimension
and the minimum is achieved in the $r$-th order (if there are multiple orders
which achieve the minimum, then we choose $r$ to be the lowest order). The
leading contribution to $b^{(r)}$ is given by

$b^{(r)}\propto\sum_{i_{1},i_{2},\cdots
i_{r}}D\textrm{Tr}(PL_{i_{1}}L_{i_{2}}\cdots
L_{i_{r}}PL^{\dagger}_{i_{1}}L^{\dagger}_{i_{2}}\cdots
L^{\dagger}_{i_{r}})-\textrm{Tr}(PL_{i_{1}}L_{i_{2}}\cdots
L_{i_{r}})\textrm{Tr}(PL^{\dagger}_{i_{1}}L^{\dagger}_{i_{2}}\cdots
L^{\dagger}_{i_{r}}).$ (54)

Following a similar argument as $b^{(3)}_{1}$ above, if $\Delta^{(r)}$
achieves the minimum, then dominant contribution comes from when all $i$’s are
close to each other. This the gives $b^{(r)}\propto n^{1-2\Delta^{(r)}}$ and
consequently $\nu=(1-2\Delta^{(r)})/r$.

## Proving at least $O(\log\log n)$ logical qubits

The main result that we will prove is that

$b(n)=\frac{1}{D^{2}}\sum_{i=1}^{n}[D\textrm{Tr}(L_{i}PL^{\dagger}_{i}P)-\textrm{Tr}(L_{i}P)\textrm{Tr}(L^{\dagger}_{i}P)]\rightarrow
0$ (55)

if the jump operator $L_{i}$ corresponds to a CFT primary operator $O$ with
scaling dimension $\Delta_{0}>1/2$ and

$P=\sum_{m,\bar{m}=0}^{M}|\partial^{m}\bar{\partial}^{\bar{m}}\phi\rangle\langle\partial^{m}\bar{\partial}^{\bar{m}}\phi|$
(56)

for $M=\mathrm{polylog}(n)$. Recall that the condition $b(n)\rightarrow 0$
implies that the code can correct the dephasing noise at a finite threshold
(assuming the scaling hypothesis). With this choice of code subspace we have
$D=M^{2}$. Thus, this ensures at least $k=O(\log\log n)$ logical qubits. In
order for proof, one first observes that

$b(n)\leq\frac{1}{D}\textrm{Tr}(L_{i}PL^{\dagger}_{i}P)$ (57)

so we only need to prove

$\sum_{i=1}^{n}\textrm{Tr}(L_{i}PL^{\dagger}_{i}P)<O(M^{2})$ (58)

The matrix element of $L_{i}$ in the code subspace must take the following
form

$|\langle\partial^{p}\bar{\partial}^{\bar{p}}\phi|L_{i}|\partial^{q}\bar{\partial}^{\bar{q}}\phi\rangle|=\left(\frac{2\pi}{n}\right)^{\Delta_{0}}C_{O\phi\phi}f(p,q)f(\bar{p},\bar{q})$
(59)

due to conformal symmetry. The dependence on $i$ is eliminated because of
translation symmetry. Thus, we only need to prove

$\sum_{p,q=0}^{M}f(p,q)^{2}=o(n^{2\Delta_{0}-1}).$ (60)

Below we compute $f(p,q)$ using conformal field theory techniques. First, let
us write the overlap in terms of correlation functions by the Weyl
transformation,

$\langle\partial^{p}\bar{\partial}^{\bar{p}}\phi|L_{i}|\partial^{q}\bar{\partial}^{\bar{q}}\phi\rangle=\left(\frac{2\pi}{n}\right)^{\Delta_{0}}\frac{\langle(\partial^{p}\bar{\partial}^{\bar{p}}\phi(0))^{\dagger}O(z_{1}=1)\partial^{q}\bar{\partial}^{\bar{q}}\phi(0)\rangle}{\sqrt{N_{p,\bar{p}}N_{q,\bar{q}}}},$
(61)

where

$N_{p,\bar{p}}=\langle(\partial^{p}\bar{\partial}^{\bar{p}}\phi(0))^{\dagger}\partial^{p}\bar{\partial}^{\bar{p}}\phi(0)\rangle$
(62)

is a normalization constant. We will compute the denominator first, which goes
as below

$\displaystyle N_{p,\bar{p}}$ $\displaystyle=$
$\displaystyle\lim_{z\rightarrow\infty,z_{0}\rightarrow
0}\langle(-z^{2}\partial)^{p}(-\bar{z}^{2}\bar{\partial})^{\bar{p}}(z^{2h}\bar{z}^{2\bar{h}}\phi(z,\bar{z}))\partial^{p}_{0}\bar{\partial}^{\bar{p}}_{0}\phi(z_{0},\bar{z}_{0})\rangle$
(63) $\displaystyle=$ $\displaystyle\lim_{z\rightarrow\infty,z_{0}\rightarrow
0}(-z^{2}\partial)^{p}(-\bar{z}^{2}\bar{\partial})^{\bar{p}}\partial^{p}_{0}\bar{\partial}^{\bar{p}}_{0}\langle
z^{2h}\bar{z}^{2\bar{h}}\phi(z,\bar{z})\phi(z_{0},\bar{z}_{0})\rangle$ (64)
$\displaystyle=$ $\displaystyle\lim_{z\rightarrow\infty,z_{0}\rightarrow
0}[(-z^{2}\partial)^{p}(z^{2h}\partial^{p}_{0}(z-z_{0})^{-2h})]\times c.c.$
(65) $\displaystyle=$
$\displaystyle\lim_{z\rightarrow\infty}\frac{\Gamma(p+2h)}{\Gamma(2h)}[(-z^{2}\partial)^{p}(z^{2h}z^{-2h-p})]\times
c.c.$ (66) $\displaystyle=$
$\displaystyle\lim_{z\rightarrow\infty}\frac{\Gamma(p+2h)}{\Gamma(2h)}[(-z^{2}\partial)^{p}z^{-p}]\times
c.c.$ (67) $\displaystyle=$ $\displaystyle\lim_{w\rightarrow
0}\frac{\Gamma(p+2h)}{\Gamma(2h)}[\partial^{p}_{w}w^{p}]\times c.c.$ (68)
$\displaystyle=$ $\displaystyle p!\frac{\Gamma(p+2h)}{\Gamma(2h)}\times c.c.$
(69)

In this calculation we have used the fact that
$\phi(0)^{\dagger}=\lim_{z\rightarrow\infty}z^{2h}\bar{z}^{2\bar{h}}\phi(z,\bar{z})$
and in the last step we have substitute $w=1/z$ in the limit. Note that here
$c.c.$ means substitute $p$ with $\bar{p}$ and $h$ with $\bar{h}$.

In a similar fashion, we can compute the numerator as

$\displaystyle\langle(\partial^{p}\bar{\partial}^{\bar{p}}\phi(0))^{\dagger}O(z_{1}=1)\partial^{q}\bar{\partial}^{\bar{q}}\phi(0)\rangle$
(70) $\displaystyle=$ $\displaystyle\lim_{z\rightarrow\infty,z_{0}\rightarrow
0}(-z^{2}\partial)^{p}(-\bar{z}^{2}\bar{\partial})^{\bar{p}}\partial^{q}_{0}\bar{\partial}^{\bar{q}}_{0}\langle
z^{2h}\bar{z}^{2\bar{h}}\phi(z,\bar{z})O(z_{1}=1)\phi(z_{0},\bar{z}_{0})\rangle$
(71) $\displaystyle=$ $\displaystyle
C_{O\phi\phi}\lim_{z\rightarrow\infty,z_{0}\rightarrow
0}(-z^{2}\partial)^{p}[z^{2h}\partial^{q}_{0}((z-z_{0})^{h_{0}-2h}(z-1)^{-h_{0}}(1-z_{0})^{-h_{0}})]\times
c.c.$ (72) $\displaystyle=$ $\displaystyle
C_{O\phi\phi}\left[\sum_{k=0}^{\min(p,q)}\frac{q!}{(q-k)!}\frac{\Gamma(h_{0}+q-k)\Gamma(2h-h_{0}+k)\Gamma(h_{0}+p-k)}{\Gamma(h_{0})^{2}\Gamma(2h-h_{0})}\right]\times
c.c.$ (73)

We have omitted the details of evaluating the limit and derivatives. Now we
can identify the expression of $f(p,q)$,

$f(p,q)=\frac{\Gamma(2h)}{\sqrt{p!q!\Gamma(p+2h)\Gamma(q+2h)}}\sum_{k=0}^{\min(p,q)}\frac{q!}{(q-k)!}\frac{\Gamma(h_{0}+q-k)\Gamma(2h-h_{0}+k)\Gamma(h_{0}+p-k)}{\Gamma(h_{0})^{2}\Gamma(2h-h_{0})}.$
(74)

We now show that, if $h,h_{0}$ are $O(1)$ numbers, then

$f(p,q)\leq\mathrm{poly}(p,q)$ (75)

We use the fact that $(n+\lfloor x\rfloor-1)!\leq\Gamma(n+x)\leq(n+\lfloor
x\rfloor)!$ for integer $n\geq 2$ and positive real number $x$, we can
estimate that $\Gamma(h_{0}+q-k)=\mathrm{poly}(q)\times(q-k)!$. In a similar
way, we make the substitution $\Gamma(2h-h_{0}+k)=\mathrm{poly}(k)\times k!$.
Thus we estimate the term in the sum as

$\frac{q!}{(q-k)!}\frac{\Gamma(h_{0}+q-k)\Gamma(2h-h_{0}+k)\Gamma(h_{0}+p-k)}{\Gamma(h_{0})^{2}\Gamma(2h-h_{0})}=\mathrm{poly}(p,q)\times
q!k!(p-k)!$ (76)

The sum in $k$ can be upper and lower bounded because
$\sum_{k=0}^{p}k!(p-k)!=O(1)\times p!$, thus we obtain
$f(p,q)\leq\mathrm{poly}(p,q)$. As long as $M=\mathrm{polylog}(n)$, the LHS
Eq. (60) scales smaller than any power law. This completes the proof.

## Free fermion CFT code

We further justify our $\nu=1-2\Delta$ prediction with an example where the
coherent information can be computed exactly. We consider the free Majorana
fermion CFT realized by

$H=i\sum_{i=1}^{2n}c_{i}c_{i+1},$ (77)

where $\\{c_{i},c_{j}\\}=2\delta_{ij}$ represents the Majorana fermions. This
model is dual to the TFIM under the Jordan-Wigner transformation. We take the
Neveu-Schwarz boundary condition $c_{2i+1}=-c_{1}$. The code subspace is taken
to be spanned by the ground state (corresponding to the identity operator
$\mathbb{I}$) and the first excited state (corresponding to the chiral fermion
operator $\Psi$). Note that the two codewords have different fermionic parity.

We consider the amplitude damping channel as in Ref. [22]. The Lindbladian

$\mathcal{L}(\rho)=\sum_{i=1}^{n}\left[L_{i}\rho
L^{\dagger}_{i}-\frac{1}{2}\\{L^{\dagger}_{i}L_{i},\rho\\}\right]$ (78)

has local jump operators

$L_{j}=a_{j},$ (79)

where $a_{j}=(c_{2j-1}+ic_{2j})/2$ is the complex fermion annihilation
operator. Now we can integrate the Lindbladian to give a channel
$\mathcal{N}_{p}=e^{t\mathcal{L}}$, where $p=1-e^{-t}$ is the damping
probability. In terms of Kraus representation, the single-site channel can be
written as

$\mathcal{N}^{[j]}_{p}(\rho)=K_{0}\rho K^{\dagger}_{0}+K_{1}\rho
K^{\dagger}_{1}$ (80)

where $K_{1}=\sqrt{p}a$ and $K_{0}=I+(\sqrt{1-p}-1)a^{\dagger}a$. The jump
operator has scaling dimension $\Delta_{a}=1/2$ and thus one expect that
$\nu=1-2\Delta_{a}=0$. Thus, the coherent information only depends on $p$ and
does not depend on $n$. It turns out that here the coherent information can be
computed using correlation matrix techniques, since $|\psi_{RQ}\rangle$ is a
Gaussian state and $\mathcal{N}^{[j]}_{p}$ is a Gaussian channel. We indeed
observe that the $I_{c}$ only depends on $p$, see Fig. 6.

Figure 6: Coherent information for the free fermion CFT code under amplitude
damping noise

This means that the fermionic amplitude damping noise is not correctable for
$p\neq 0$ in the thermodynamic limit, as $I_{c}<\log D$. However, as opposed
to the case where $\Delta<1/2$, the coherent information is a continuous
function of $p$ and does not jump immediately at $p=0$. Thus, for small $p$
the noise is still approximately correctable for any size $n$.

## Renyi coherent information for CFT code

In this section we consider the Renyi coherent information, defined by

$I^{(n)}_{c}=S^{(n)}_{Q}-S^{(n)}_{QR},$ (81)

where

$S^{(n)}(\rho):=\frac{1}{1-n}\log\textrm{Tr}(\rho^{n}),$ (82)

We show that the Renyi coherent information still has a scaling collapse
$I^{(n)}_{c}=f(pN^{\nu^{(n)}})$, where $\nu^{(n)}$ could in general be
different across different $n$. Note that in the section we will use $N$ to
denote the total number of physical qubits and $n$ to denote the Renyi index.
We show that for each Renyi index $n$ there is a notion of relevant,
irrelevant and marginal dephasing based on defect RG flow in the CFT. This
defect RG flow is identical to what we encounter in Ref. [22]. We will first
review the result in Ref. [22] and show that the RG flow of coherent
information is identical to the defect RG flow. We will present the numerical
result on the Ising model and show that the $Z$ dephasing is marginal at
$n=2$, which means $\nu^{(2)}=0$ and thus $I^{(2)}_{c}$ is independent of $n$.
Recall that we have previously shown that $Z$ dephasing is correctable with
$\nu=-1$ in the replica limit $n\rightarrow 1$. Thus, this section also serves
as an example where the Renyi index $n=2$ gives a qualitatively different
prediction from the replica limit.

###  .7 Review of channeling quantum criticality

We consider the entropy of a subsystem $A$ of a CFT eigenstate
$|\phi_{\alpha}\rangle$ under decoherence channel $\mathcal{N}$. Here we have
generalized the approach in Ref. [22] which only considers the CFT ground
state. The mixed state of interest is
$\rho_{\alpha}=\mathcal{N}(|\phi_{\alpha}\rangle\langle\phi_{\alpha}|)$. The
$n-$th momentum of the reduced density matrix is

$\displaystyle\textrm{Tr}(\rho^{n}_{\alpha,A})$ $\displaystyle=$
$\displaystyle\textrm{Tr}(\rho_{\alpha}^{\otimes
n}\tau^{(n)}_{A}\mathbb{I}_{B})$ (83) $\displaystyle=$
$\displaystyle\textrm{Tr}((|\phi_{\alpha}\rangle\langle\phi_{\alpha}|)^{\otimes
n}\mathcal{N}^{*\otimes n}(\tau^{(n)}_{A})\mathbb{I}_{B})$ (84)
$\displaystyle=$ $\displaystyle\langle\phi^{\otimes
n}_{\alpha}|\mathcal{B}^{(n)}_{\mathcal{N},A}\mathbb{I}_{B}|\phi^{\otimes
n}_{\alpha}\rangle$ (85) $\displaystyle=$ $\displaystyle\langle\phi^{\otimes
2n}_{\alpha}|\mathcal{B}^{(n)}_{\mathcal{N},A}\mathbb{I}_{B}\rangle$ (86)

where $B$ is the complement of $A$, $\tau^{(n)}$ is the forward permutation on
replicas and $\mathcal{B}^{(n)}_{\mathcal{N}}=\mathcal{N}^{*\otimes
n}(\tau^{(n)})$. In the last step we have made use of the duality between
operators and states in doubled Hilbert space and the overlap is taken within
$2n$ copies of CFT. In terms of the path integral, this can be represented by
a three-point correlation function of a bulk operator and two boundary
condition changing operators. Under RG flow, the boundary state
$|\mathcal{B}^{(n)}_{\mathcal{N}}\rangle$ flows to the same conformal boundary
condition regardless of the bulk insertion. This means that the RG flow of the
channel defined in Ref. [22] can be applied to all low-energy eigenstates. In
particular, if the channel $\mathcal{N}$ flows to complete dephasing, then the
Renyi negativity of all low-energy states becomes area law under the channel.
If the channel is irrelevant (i.e., flows to the identity channel), then the
entanglement scaling of all low-energy eigenstates at long-distances remains
unchanged (except non-universal constants).

###  .8 RG flow of the Renyi coherent information

So far we have shown that the RG flow of quantum channels are identical for
all low-energy states. One may naturally expect that the code property of the
low-energy subspace subject to uniform decoherence follows the same RG flow.
If the channel flows to complete dephasing, then the error cannot be
corrected. If the channel flows to the identity channel, then the error can be
corrected. The two cases correspond to relevant and irrelevant dephasing,
respectively. The correctability of the error is characterized by the loss of
coherent information. The subtlety is that Ref. [22] works at integer Renyi
index $n\geq 2$ but the code property is only related to the replica limit
$n\rightarrow 1$.

We will fix a Renyi index $n\geq 2$ below. We compute the Renyi coherent
information of the CFT code and show that it decreases to zero for relevant
dephasing and remains invariant for irrelevant dephasing. For marginal case,
the Renyi coherent information does not depend on system size $N$ and is a
continuous function of $p$.

we first couple the CFT to the reference qudit,

$|\psi\rangle_{QR}=\frac{1}{\sqrt{D}}\sum_{\alpha=1}^{D}|\phi_{\alpha}\rangle_{Q}\otimes|\alpha\rangle_{R}$
(87)

The state subjected to decoherence is
$\rho_{QR}=\mathcal{N}_{Q}(|\psi\rangle\langle\psi|)$. The Renyi coherent
information is

$\displaystyle
I^{(n)}_{c}=\frac{1}{n-1}\log\frac{\textrm{Tr}\rho^{n}_{QR}}{\textrm{Tr}\rho^{n}_{Q}}$
(88)

In terms of the channeled twist operator $\mathcal{B}^{(n)}_{\mathcal{N}}$,
the numerator and demominator can be expressed as

$\displaystyle\textrm{Tr}\rho^{n}_{QR}$ $\displaystyle=$
$\displaystyle\frac{1}{D^{n}}\sum_{\alpha_{1},\cdots\alpha_{n}=1}^{D}\left\langle\bigotimes_{i}\phi_{\alpha_{\tau(i)}}\right|\mathcal{B}^{(n)}_{\mathcal{N}}\left|\bigotimes_{i}\phi_{\alpha_{i}}\right\rangle.$
(89) $\displaystyle\textrm{Tr}\rho^{n}_{Q}$ $\displaystyle=$
$\displaystyle\frac{1}{D^{n}}\sum_{\alpha_{1},\cdots\alpha_{n}=1}^{D}\left\langle\bigotimes_{i}\phi_{\alpha_{i}}\right|\mathcal{B}^{(n)}_{\mathcal{N}}\left|\bigotimes_{i}\phi_{\alpha_{i}}\right\rangle,$
(90)

where $\tau(i)=i+1~{}\mathrm{mod}~{}n$ forward permutes the replica index. If
the channel flows to identity channel, then $\mathcal{B}^{(n)}_{\mathcal{N}}$
flows to the twist operator $\tau^{(n)}$. In this case the Eq. (89) gets
contributions from all terms and Eq. (90) only gets contributions from the
diagonal terms (where all $\alpha_{i}$’s are equal). Thus,
$\textrm{Tr}\rho^{n}_{QR}=1$ and $\textrm{Tr}\rho^{n}_{Q}=D^{1-n}$, and

$I^{(n)}_{c}=\log D,~{}~{}~{}\mathrm{irrelevant~{}dephasing.}$ (91)

If instead the channel flows to complete dephasing, say, in the direction of
$\sigma_{\vec{n}}\equiv\vec{n}\cdot\vec{\sigma}$, then the on each site

$\mathcal{B}^{(n)}_{\mathcal{N}}=\sum_{i=+,-}|i^{\otimes n}\rangle\langle
i^{\otimes n}|,~{}~{}~{}\mathrm{complete~{}dephasing.}$ (92)

where $|\pm\rangle$ is the eigenvector of $\sigma_{\vec{n}}$ with eigenvalue
$\pm 1$. Since $\mathcal{B}^{(n)}_{\mathcal{N}}$ is a completely symmetric
tensor with respect to permutation of the $2n$ replicas, each term in Eq. (89)
and Eq. (90) is equal. Thus, the Renyi coherent information Eq. (88) vanishes
for all $n\geq 2$, i.e.,

$I^{(n)}_{c}=0,~{}~{}~{}\mathrm{complete~{}dephasing.}$ (93)

Finally, the extreme case where we have complete depolarization (erasure)
gives $\mathcal{B}_{\mathcal{N}}=\mathbb{I}$. Thus, only diagonal terms
contribute to Eq. (89) and

$I^{(n)}_{c}=-\log D,~{}~{}~{}\mathrm{complete~{}erasure.}$ (94)

Thus we have established that for a given Renyi index $n$, if the dephasing
channel is relevant, then $I^{(n)}_{c}=0$ in the thermodynamic limit; if the
dephasing channel is irrelevant, then $I^{(n)}_{c}=\log D$ in the
thermodynamic limit.

###  .9 Numerical result for the Ising model

Taking the $|I\rangle$ and $|\sigma\rangle$ as the codeword states, we
consider the Renyi-2 coherent information under $X$,$Y$ or $Z$ dephasing for
the Ising CFT code. We obtain the following results (see Fig. 7).

Figure 7: Renyi coherent information of the Ising CFT code under $X$ (left),
$Y$ (center) and $Z$ (right) dephasing

The result indicates that the $X,Y$ and $Z$ dephasings are relevant,
irrelevant and marginal respectively, which can be seen in the following way.
For $X$ dephasing, increasing the system size decreases the coherent
information. For $Y$ dephasing, increasing the system size increases
$I^{(2)}_{c}$ to $\log 2$ at small $p$. The $Z$ dephasing is the marginal case
where $I^{(2)}_{c}$ does not depend on $N$. This is in accordance with the
result of Ref. [22] which considers the Renyi negativity and the $g$ function.
However, it is important to note that only Von-Neumann coherent information
indicates the code properties. As we already stress, the $Z$ dephasing here
offers an example which is marginal for $n\geq 2$ but irrelevant for
$n\rightarrow 1$.