where $f(n)$ is a upper bound for each matrix element $g_{\sigma\nu}^{ij}$ in
the block $\mathscr{B}_{n}$, and $r(n)$ is the rank of the block.

At higher order, we can continue to expand the second class of Lindbladian
operator and one can repeat these calculations following the same procedure.
Errors of the form $E_{\sigma}^{m}$ then induce information decay at
$O(\frac{1}{N^{2m}})$, with magnitude proportional to $(\lambda t)^{m}$.

In summary, we find that both the first class of errors and the second class
of errors contribute to the information decay with some rate function
$K_{m}(t)\propto\frac{t^{m}}{N^{2m}}$ as long as the assumptions of Theorem 1
are satisfied. For $N^{2}\gg\lambda t\gg 1$,

$I(S:R)=2\ln{d}-\sum_{m=1}^{\infty}K_{m}(t).$ (128)

##  Appendix B aKL condition for single trace errors

###  B.1 non-local model

In this section, we present the aKL equations for the most general single-
trace error operators. Since the logical states (Eq.35) are symmetric in
$a_{1}^{\dagger}$ and $a_{2}^{\dagger}$, we only need to consider the single-
trace operators involving $a_{1}$ and $a_{1}^{\dagger}$. In the following, we
will only keep track of the terms up to $O(\frac{1}{N^{2}})$.

Let us define a class of error operators

$E_{n,m}^{P}=\frac{:\operatorname{Tr}(P\\{a_{1}^{\dagger
n}a_{1}^{m}\\}):}{N^{\frac{n+m}{2}}},$ (129)

where $P$ denotes a particular permutation of the operators inside the curly
brackets. Acting it on the logical states, one obtains,

$E_{n,m}^{P}|\tilde{i}\rangle=f|\mu^{i}\rangle+\frac{g}{N^{2}}|\mu\rangle+O(\frac{1}{N^{4}}).$
(130)

To extract these coefficients, we calculate the overlap

$\langle\tilde{i}|E_{n^{\prime},m^{\prime}}^{P^{\prime}\dagger}E_{n,m}^{P}|\tilde{j}\rangle,$
(131)

which is essentially a four-point function that factorizes into,

$\langle\tilde{O}_{i}^{\dagger}E^{P^{\prime}\dagger}_{n^{\prime},m^{\prime}}E_{n,m}^{P}\tilde{O}_{j}\rangle=\langle\tilde{O_{i}}^{\dagger}\tilde{O_{j}}\rangle\langle
E^{P^{\prime}\dagger}_{n^{\prime},m^{\prime}}E_{n,m}^{P}\rangle+O(\frac{1}{N}).$
(132)

This is the only way to factorize into disconnected correlators, because
$O_{i}$ and $O_{j}$ contain $a_{2}^{\dagger}$, while $E_{n,m}^{P}$’s do not.
The two point function $\langle
E^{P^{\prime}\dagger}_{n^{\prime}m^{\prime}}E_{n,m}^{P}\rangle$ vanishes,
unless $m,m^{\prime}=0$, which is the case that we discussed in the main text.
Moreover, the sub-leading term can be $O(\frac{1}{N})$ only if
$m^{\prime}=n^{\prime}=0$, hence the only non-trivial four-point function
contribution is when $m=n$. In this case, we define
$Z_{n}^{P}=\frac{:\operatorname{Tr}(P\\{a^{\dagger n}a^{n}\\}):}{N^{n}}$ and
find that,

$\langle\tilde{j}|Z_{n}^{P}|\tilde{i}\rangle=\frac{c_{n}^{P}}{N}+O(\frac{1}{N^{3}}),$
(133)

for some constant $c_{n}^{P}$, which is non-zero only when $n=2$. This is
consistent with the aKL condition (Eq. (18)).

So far, we have discussed the order one and $\frac{1}{N}$ term in Eq. (133).
All the other error operators can only produce order $\frac{1}{N^{2}}$ terms
when acting on the logical states. This completes the proof of aKL condition.

As a special example, consider the decay mode, $E_{0,2}$, which lower the
energy when acting on the logical states. It also satisfies the aKL condition,

$\begin{split}\langle\tilde{1}|E_{0,2}^{\dagger}E_{0,2}|\tilde{1}\rangle-\langle\tilde{2}|E_{0,2}^{\dagger}E_{0,2}|\tilde{2}\rangle=\frac{8}{N^{2}}+O(\frac{1}{N^{4}})\\\
\langle\tilde{2}|E_{0,2}^{\dagger}E_{0,2}|\tilde{1}\rangle=\frac{4\sqrt{2}}{N^{3}}+O(\frac{1}{N^{5}}).\end{split}$
(134)

So the decay mode belongs to the second class of errors classified in Appendix
A, and lead to a spontaneous decay of quantum information at zero temperature
with a rate proportional to $\frac{1}{N^{2}}$.

###  B.2 Local model

In this section, we prove the aKL condition for the single-trace errors of the
form

$E_{\\{n_{i}\\}}=\frac{:\operatorname{Tr}(P\\{a_{1}^{\dagger
n_{1}}a_{2}^{\dagger
n_{2}}a_{1}^{n_{3}}a_{2}^{n_{4}}\\}):}{N^{\frac{\sum_{k=1}^{4}n_{k}}{2}}}$
(135)

with logical states defined as

$\begin{split}|\tilde{\uparrow}\rangle&=\frac{\operatorname{Tr}(a_{1}^{\dagger})}{\sqrt{N}}|0\rangle_{12}\\\
|\tilde{\downarrow}\rangle&=\frac{\operatorname{Tr}(a_{2}^{\dagger})}{\sqrt{N}}|0\rangle_{12}.\end{split}$
(136)

In the main text, we derived a general formula for the case when
$n_{3}=n_{4}=0$. Moreover, since the logical states are annihilated by more
than two $a_{1}$ or $a_{2}$, and the error operators are normal ordered, we
are left to discuss the case when $n_{3}+n_{4}=1$. Without loss of generality,
we will only consider the case with $n_{3}=1,\ n_{4}=0$. Then the relevant
overlaps are,

$\begin{split}&\langle\tilde{\uparrow}|E^{\dagger}_{(n-1,m,0,0)}E_{(n,m,1,0)}|\tilde{\uparrow}\rangle=\frac{n}{N^{2}}+O(\frac{1}{N^{4}})\\\
&\langle\tilde{\uparrow}|E^{\dagger}_{(n,0,1,0)}E_{(n,0,1,0)}|\tilde{\uparrow}\rangle=\frac{n}{N^{2}}+O(\frac{1}{N^{4}})\\\
&\langle\tilde{\downarrow}|E^{\dagger}_{(n,m,0,0)}E_{(n,m,1,0)}|\tilde{\downarrow}\rangle=\langle\tilde{\downarrow}|E^{\dagger}_{(n,0,1,0)}E_{(n,0,1,0)}|\tilde{\downarrow}\rangle=0\\\
&\langle\tilde{\downarrow}|E^{\dagger}_{(n,m-1,0,0)}E_{(n,m,1,0)}|\tilde{\uparrow}\rangle=\frac{m}{N^{2}}+O(\frac{1}{N^{4}}).\end{split}$
(137)

Exceptions of these formulae consist of special cases when $n=1,\ m=0$ and
$n=0,m=1$, where one can derive that,

$\begin{split}&\langle\tilde{\uparrow}|\frac{\operatorname{Tr}(a_{1}^{\dagger}a_{1})}{N}|\tilde{\uparrow}\rangle=\frac{1}{N},\
\
\langle\tilde{\downarrow}|\frac{\operatorname{Tr}(a_{1}^{\dagger}a_{1})}{N}|\tilde{\downarrow}\rangle=0\\\
&\langle\tilde{\downarrow}|\frac{\operatorname{Tr}(a_{2}^{\dagger}a_{1})}{N}|\tilde{\uparrow}\rangle=\frac{1}{N}.\end{split}$
(138)

These are $O(\frac{1}{N})$ overlap, which are consistnet with Eq. (18). So we
complete the proof of aKL (18) for general single-trace errors in this local
model.

###  B.3 Global symmetry

In this section, we prove the aKL condition for the model in Section 4. The
logical space is spanned by the two states of unit norm

$\begin{split}&|1\rangle=\frac{\operatorname{Tr}(a^{\dagger
L})^{2}}{N^{L}\xi(L,L)}\\\ &|2\rangle=\frac{\operatorname{Tr}(a^{\dagger
L+1})\operatorname{Tr}(a^{\dagger L-1})}{N^{L}\xi(L+1,L-1)},\end{split}$ (139)

where the normalization constant are
$\xi(L,L)=\sqrt{2}L(1+O(\frac{1}{N^{2}}))$ and
$\xi(L+1,L-1)=\sqrt{L^{2}-1}(1+O(\frac{1}{N^{2}}))$. Note that these states
are not orthogonal but have overlap

$\langle 1|2\rangle=\frac{L^{2}\sqrt{L^{2}-1}}{\sqrt{2}N^{2}}.$ (140)

To check the aKL condition, we insert the error operators
$E_{n}=\frac{\operatorname{Tr}(a^{\dagger n})}{N^{\frac{n}{2}}\xi(n)}$ with
$\xi(n)$ being the normalization constant, and derive the following equations

$\begin{split}&\langle 1|E_{m}^{\dagger}E_{n}|1\rangle-\langle
2|E_{m}^{\dagger}E_{n}|2\rangle=\delta_{mn}[\frac{2n}{N^{2}}+O(\frac{1}{N^{4}})]\\\
&\langle
1|E_{m}^{\dagger}E_{n}|2\rangle=\delta_{mn}[\frac{L^{2}\sqrt{L^{2}-1}}{\sqrt{2}N^{2}}+O(\frac{1}{N^{4}})].\end{split}$
(141)

Note that the leading term in the second equation is a consequence of the non-
orthogonality between our definition of $|1\rangle$ and $|2\rangle$; it can be
easily removed via a change of basis (c.f. equation 65). The above equation
also holds for $E_{n}=\frac{\operatorname{Tr}(a^{n})}{N^{\frac{n}{2}}\xi(n)}$.

For more general operators $E_{(n,m)}:=\frac{\operatorname{Tr}(a^{\dagger
n}a^{m})}{N^{\frac{m+n}{2}}}$, with $n\neq m$, one can show that

$\begin{split}&\langle
1|E_{n^{\prime},m^{\prime}}^{\dagger}E_{n,m}|1\rangle-\langle
2|E_{n^{\prime},m^{\prime}}^{\dagger}E_{n,m}|2\rangle=\delta_{n^{\prime}-m^{\prime},n-m}[\frac{2}{N^{2}}+O(\frac{1}{N^{4}})]\\\
&\langle
1|E_{n^{\prime},m^{\prime}}^{\dagger}E_{n,m}|2\rangle=\delta_{n^{\prime}-m^{\prime},n-m}[\frac{L^{2}\sqrt{L^{2}-1}}{\sqrt{2}N^{2}}+O(\frac{1}{N^{4}})],\end{split}$
(142)

which indicates the aKL condition holds.

For the special case of $n=m$, the operator also incurs no phase error,
because

$\langle 1|\frac{\operatorname{Tr}(a^{\dagger
n}a^{n})}{N^{n}}|1\rangle=\langle 2|\frac{\operatorname{Tr}(a^{\dagger
n}a^{n})}{N^{n}}|2\rangle=\frac{2L}{N}.$ (143)

We thereby conclude that the aKL conditions hold for all the single trace
operators.

###  B.4 Spin model

The approximate Knill-Laflamme condition for the spin model can be calculated
using the spin algebra. Recall that we can construct two states that span the
code subspace in a way analogous to our definition in the oscillator model,

$\begin{split}&|1\rangle=\frac{\operatorname{Tr}(S^{+L})^{2}}{N^{L}\eta(L,L)}\\\
&|2\rangle=\frac{\operatorname{Tr}(S^{+L-1})\operatorname{Tr}(S^{+L+1})}{N^{L}\eta(L+1,L-1)},\end{split}$
(144)

where $\eta(L,L^{\prime})$ is the normalization constant. They have the same
leading order behavior as $\xi(L,L^{\prime})$ in the harmonic oscillator
model, but have different forms in sub-leading terms. The overlap of these two
states is

$\langle 1|2\rangle=\frac{f(L)L\sqrt{L^{2}-1}}{\sqrt{2}N^{2}},$ (145)

where $f(L)$ is some complicated function of $L$. Again, we insert error
operators of the form $E_{n}=\frac{\operatorname{Tr}(S^{\pm
n})}{N^{\frac{n}{2}}\eta(n)}$ and
$E_{n,m}=\frac{\operatorname{Tr}(S^{+n}S^{-m})}{N^{\frac{m+n}{2}}}$ into the
correlator, and derive the aKL equations for the spin model,

$\begin{split}&\langle 1|E_{m}^{\dagger}E_{n}|1\rangle-\langle
2|E_{m}^{\dagger}E_{n}|2\rangle=\delta_{mn}[\frac{2n}{N^{2}}+O(\frac{1}{N^{4}})]\\\
&\langle 1|E_{n^{\prime},m^{\prime}}^{\dagger}E_{n,m}|1\rangle-\langle
2|E_{n^{\prime},m^{\prime}}^{\dagger}E_{n,m}|2\rangle=\delta_{n^{\prime}-m^{\prime},n-m}[\frac{2}{N^{2}}+O(\frac{1}{N^{4}})].\end{split}$
(146)

Again, the bit flip error only appears at $O(\frac{1}{N^{4}})$

$\langle
1|E_{m}^{\dagger}E_{n}|2\rangle=\delta_{mn}[\frac{f(L)L\sqrt{L^{2}-1}}{\sqrt{2}N^{2}}+O(\frac{1}{N^{4}})],$
(147)

where we recall the first term can be removed by choosing an orthonormal
basis. This completes our proof for the aKL condition.

##  Appendix C  Master equation in Lindblad form

Suppose the coupling between system and thermal reservoir is

$\begin{split}H=&H_{S}+H_{B}+V_{I}\\\ V_{I}=&\sum_{a}E_{a}\otimes
b_{a}+h.c.,\end{split}$ (148)

where $H_{S}$ and $H_{B}$ are the system and bath Hamiltonian respectively,
and $b_{a}$’s are the bath modes.

In the interaction picture, time evolution is described by

$\begin{split}\dot{\rho}(t)=&-i[V_{I}(t),\rho(t)]\\\
=&-i[V_{I}(t),\rho_{0}]-[V_{I}(t),\int_{0}^{t}dt^{\prime}[V_{I}(t^{\prime}),\rho(t^{\prime})]].\end{split}$
(149)

The essential simplification is enabled by the Born-Markov approximation,
which allows one to approximate the full density matrix as
$\rho(t)\sim\rho_{S}(t)\otimes\rho_{B}$ at all times. After tracing out the
thermal state, we arrive at,

$\dot{\rho}_{IS}(t)=\int_{0}^{t}dt^{\prime}\sum_{ab}[E_{b}(t^{\prime})\rho_{IS}(t)E_{a}^{\dagger}(t)-E_{a}^{\dagger}(t)E_{b}(t^{\prime})\rho_{IS}(t)]\operatorname{Tr}(\rho_{B}b_{a}^{\dagger}(t)b_{b}(t^{\prime}))+h.c..$
(150)

For sufficiently long time $t$, we can treat the initial time as $-\infty$. We
also assume that the bath modes are uncorrelated. Then we define

$\int_{-\infty}^{t}dt^{\prime}\operatorname{Tr}(\rho_{B}b_{a}^{\dagger}(t)b_{b}(t^{\prime}))e^{i\omega(t-t^{\prime})}:=\frac{\gamma(\omega)}{2}\delta_{ab}.$
(151)

For operators $E_{a}(t)$ with the decomposition,

$E_{a}(t)=\sum_{k}E_{a}^{(k)}e^{-i\epsilon_{a}^{(k)}t},$ (152)

Eq (150) simplifies as,

$\dot{\rho}_{IS}(t)=\sum_{a,k}\frac{\gamma(\epsilon_{a}^{(k)})}{2}[E_{a}^{(k)}\rho_{IS}(t)E_{a}^{\dagger}(t)-E_{a}^{\dagger}(t)E_{a}^{(k)}\rho_{IS}(t)]e^{-i\epsilon_{a}^{(k)}t}+h.c..$
(153)

For example, consider the error operator $E_{ijkl}=a^{\dagger i}_{j}a^{\dagger
k}_{l}$ in Section 4. It can be decomposed as,

$\begin{split}E_{ijkl}(t)=&\widetilde{a^{\dagger i}_{j}a^{\dagger
k}_{l}}e^{-2i\mathcal{J}t}+[\frac{\widetilde{(a^{\dagger
2})^{i}_{l}}}{N}\delta^{k}_{j}+\frac{\widetilde{(a^{\dagger
2})^{k}_{j}}}{N}\delta^{i}_{l}+\frac{\operatorname{Tr}(a^{\dagger})}{N}\tilde{a}^{\dagger
i}_{j}\delta^{k}_{l}+\frac{\operatorname{Tr}(a^{\dagger})}{N}\tilde{a}^{\dagger
k}_{l}\delta^{i}_{j}]e^{-i\mathcal{J}t}\\\
&+\frac{\operatorname{Tr}(a^{\dagger
2})}{N^{2}}\delta^{i}_{l}\delta^{k}_{j}+\frac{\operatorname{Tr}(a^{\dagger})^{2}}{N^{2}}\delta^{i}_{j}\delta^{k}_{l},\end{split}$
(154)

where the operator under the tilde vanishes when contracting with the
kronecker delta, e.g. $a^{\dagger i}_{j}\delta_{i}^{j}=0$. Consequently, the
tensor coefficients are almost orthogonal to each other, i.e.,

$\sum_{ijkl}E_{ijkl}^{\dagger(m)}E_{ijkl}^{(n)}\propto\delta^{mn}+O(\frac{1}{N}),$
(155)

where the $O(\frac{1}{N})$ term denotes operators that has $O(\frac{1}{N})$
expectation value with respect to any low energy state.

More generally, a tensor product of individual creation and annihilation
operators can always be decomposed into different components obtained by index
contractions, which are orthogonal to each other up to $O(\frac{1}{N})$, in
the sense of Eq. (155). As a result, the Lindbladian operator can always be
written as a diagonal sum of components that correspond to different
contractions,

$\sum_{m}\gamma(\epsilon^{(m)})\sum_{a}E_{a}^{(m)}\rho
E^{\dagger(m)}_{a}=\sum_{ma}\gamma(\epsilon^{(m)})E_{a}^{(m)}\rho
E_{a}^{\dagger(m)}.$ (156)

##  Appendix D Expectation value of the Casimir

In this section, we evaluate the expectation value of the penalty Hamiltonian
$H_{G}$ (in Eq. (59)) for a thermal state $\rho_{T}$ at temperature
$T<\frac{\mathcal{J}}{2\log{N}}$,

$\operatorname{Tr}(\rho_{T}H_{G})=\sum_{m=0}^{\infty}d_{m}m\mathcal{J}e^{-\frac{m(\mathcal{J}+\omega)}{T}},$
(157)

where $m$ is the highest weight of a particular representation, with
degeneracy $d_{m}$ and energy $m\mathcal{J}$. Examples of non-singlet states
are the $N^{2}$ number of $a^{\dagger i}_{j}|0\rangle$’s in the adjoint
representation with energy $\mathcal{J}$, and the $N^{4}$ number of
$a^{\dagger i}_{j}a^{\dagger k}_{l}|0\rangle$’s in the tensor product
representation with energy $2\mathcal{J}$.

In the oscillator model, we expect $d_{m}$ to scale as $d_{m}\sim N^{2m}$.
This factor cancels with the factor $e^{\frac{-m\mathcal{J}}{T}}$, since
$\mathcal{J}>2T\log{N}$. Therefore, we estimate the energy expectation value
to be

$\begin{split}\operatorname{Tr}(\rho_{T}H_{G})<&\sum_{m}m\mathcal{J}e^{\frac{-m\omega}{T}}.\\\
\end{split}$ (158)

This is an $O(1)$ number times the energy scale $\mathcal{J}$. So we conclude
that, for a thermal state with temperature $T<\frac{\mathcal{J}}{2\log{N}}$,
the energy expectation value of $H_{G}$ per oscillator is $O(1)$ times
$\frac{\mathcal{J}}{N^{2}}$.