# Approximate quantum error correction, covariance symmetry, and their
relation

Hao Dai<EMAIL_ADDRESS>Center for Quantum Information, Institute
for Interdisciplinary Information Sciences, Tsinghua University, Beijing
100084, P. R. China  Hefei National Laboratory, University of Science and
Technology of China, Hefei, Anhui 230088, P. R. China

###### Abstract

To perform reliable quantum computation, quantum error correction is
indispensable. In certain cases, continuous covariance symmetry of the
physical system can make exact error correction impossible. In this work we
study the approximate error correction and covariance symmetry from the
information-theoretic perspective. For general encoding and noise channels, we
define a quantity named infidelity to characterize the performance of the
approximate quantum error correction and quantify the noncovariance of an
encoding channel with respect to a general Lie group from the asymmetry
measure of the corresponding Choi state. In particular, when the encoding
channel is isometric, we derive a trade-off relation between infidelity and
noncovariance. Furthermore, we calculate the average infidelity and
noncovariance measure for a type of random code.

##  I introduction

Errors are inevitable in quantum computing and quantum error correction (QEC)
provides a method to realize fault-tolerant quantum computation [1, 2]. The
subject has been studied for decades and various correcting codes have been
developed [3, 4]. Beyond the quantum computation, QEC is closely connected
with a wide range of quantum topics, such as quantum metrology [5, 6, 7] and
quantum entanglement [8, 9, 10].

Symmetry is a ubiquitous property of the physical system and can put strong
constraints on the QEC. A no-go theorem, also known as the Eastin–Knill
theorem, claims that there does not exist a local error-detecting code in a
finite-dimensional system that allows for a set of universal logical gates to
act transversally on the physical system [11]. This theorem implies that the
continuous covariance symmetry and exact correction can be incompatible in
certain cases [12, 13], which has motivated the exploration of the relation
between covariance symmetry and approximate QEC. Several studies have focused
on the performance of quantum codes that are exactly covariant but correct
errors approximately [14, 15, 16, 17]. In particular, when the symmetry group
is the U(1) Lie group the corresponding generator in the physical system is a
Hamiltonian, covariant codes cannot correct errors perfectly if the physical
Hamiltonian satisfies the Hamiltonian-in-Kraus-span (HKS) condition[18, 19].
Under this special case, the relation between the covariance violation and the
inaccuracy of the approximate QEC has been investigated [21, 18].

In this work we study the approximate QEC and the covariance symmetry from an
information-theoretic perspective. For general encoding and noise channels in
the form of the Kraus representations, we evaluate the error-correcting
capability of the codes via a defined quantity called infidelity, which is
related to entanglement fidelity. When infidelity is equal to $0$, the errors
caused by the noise channel can be corrected exactly. We also quantify the
violation of covariance symmetry, which we term noncovariance, from the
asymmetry measures of the corresponding Choi state. We specifically explore
the infidelity and noncovariance measure for isometric encoding codes.
Moreover, we prove again that under the HKS condition, exact correctability
and covariance are incompatible. In addition, we investigate the generalized
Wigner-Yanase skew information and derive a sum uncertainty relation. By
virtue of the generalized skew information, we obtain a trade-off relation
between infidelity and noncovariance. Furthermore, we also calculate the
average infidelity and noncovariance measure for a type of random code.

The paper is organized as follows. In Sec. II we review the basic concepts
including QEC, Wigner-Yanase skew formation and asymmetry measures for states.
In Secs. III and IV we quantify the inaccuracy of the approximate QEC and the
noncovariance, respectively. In Sec. V we study the special case for the
isometric encoding channel. In Sec. VI we summarize and offer a suggestion for
future work.

##  II Preliminaries

In this section, to highlight the idea of our approach, we briefly review the
basic working knowledge and clarify some notation.

###  II.1 Quantum error correction

In a QEC procedure, the logical state is encoded into a higher-dimensional
physical system and redundancy is introduced to protect against errors. As a
starting point, we denote by $L$ the logical system and by $\mathcal{H}_{L}$
the relevant Hilbert space. The dimension of the Hilbert space is assumed to
be $d_{L}$ and the state space is denoted by $\mathcal{D}(\mathcal{H}_{L})$.
Similar definitions can be defined for other systems. The encoding is a
channel $\mathcal{E}$ from the logical system $L$ to the physical system $S$.
The subspace of system $S$,
$\mathcal{C}=\mathcal{E}(\mathcal{D}(\mathcal{H}_{L}))$, is known as code
space and the projector on the code space is denoted by $P$. After a noise
channel $\mathcal{N}(\rho)=\sum_{i=1}^{n}A_{i}\rho A_{i}^{\dagger}$ with
$\sum_{i=1}^{n}A_{i}^{\dagger}A_{i}=\mathbf{1}$, the encoding state is changed
and we can perform a corresponding decoding channel$\mathcal{R}$ to recover
the original state. An ideal QEC procedure can recover all states in the
logical system perfectly, that is,

$\mathcal{R}\circ\mathcal{N}\circ\mathcal{E}=\mathcal{I}$ (1)

with $\mathcal{I}$ being the identity map on logical system $L$.

The Knill-Laflamme condition is a necessary and sufficient condition for a
quantum code to achieve an exact correction [23]. For a given code
$\mathcal{E}$ with the projector on the code subspace $P$, the errors can be
corrected if and only if

$PA_{i}^{\dagger}A_{j}P=\alpha_{ij}P$ (2)

holds for a corresponding non-negative Hermitian matrix $(\alpha_{ij})$. Note
that when the Kraus operators of a noise channel can be described by a linear
span of $\\{A_{i}\\}$, the errors caused by this noise can also be corrected
exactly.

###  II.2 Wigner-Yanase skew information and its generalization

The conventional variance quantifies the total uncertainty of the observable
$H$ in the state $\rho$ and is defined as

$V(\rho,H)=\tr(\rho H^{2})-(\tr\rho H)^{2}.$ (3)

As a counterpart to it, the quantity

$I(\rho,H)=-\frac{1}{2}\tr[\sqrt{\rho},H]^{2}=\frac{1}{2}\norm{[\sqrt{\rho},H]}^{2}_{2}$
(4)

also known as the Wigner-Yanase skew information [24, 25, 26], can quantify
the quantum uncertainty of the observable $H$ in the state $\rho$. Here
$[X,Y]=XY-YX$ is the Lie product and
$\norm{X}_{p}=(\tr(XX^{\dagger})^{p/2})^{1/p}$ is the $p$-norm. For a pure
state, the skew information coincides with the variance.

The operator $H$ in Eq. (4) is required to be Hermitian and we can generalize
to non-Hermitian case [27]. For an arbitrary operator $K$ which can be non-
Hermitian, the generalized skew information is defined as

$I(\rho,K)=\frac{1}{2}\tr[\sqrt{\rho},K][\sqrt{\rho},K]^{\dagger}=\frac{1}{2}\norm{[\sqrt{\rho},K]}^{2}_{2}.$
(5)

In particular, for a pure state $\ket{\phi}$,

$I(\outerproduct{\phi}{\phi},K)=\frac{1}{2}\bra{\phi}KK^{\dagger}+K^{\dagger}K\ket{\phi}-\absolutevalue{\bra{\phi}K\ket{\phi}}^{2}.$
(6)

In addition, the generalized skew information can be expressed as a sum of the
original skew information

$I(\rho,K)=I(\rho,K^{\dagger})=I(\rho,\real(K))+I(\rho,\imaginary(K)),$ (7)

where $\real(K)=\frac{1}{2}(K+K^{\dagger})$ and
$\imaginary(K)=\frac{1}{2i}(K-K^{\dagger})$ represent the real and imaginary
components, respectively. Note that when $K$ is Hermitian, the generalized
skew information degenerates to the original ones.

The original skew information satisfies a series of uncertainty relations [28,
29, 30]. Here we give a sum uncertainty relation based on the generalized skew
information.

######  Lemma 1.

Let $K_{1},\cdots,K_{N}$ be a set of operators. For a state $\rho$, there is

$\sum_{j=1}^{N}I(\rho,K_{j})\geq\frac{1}{N}I(\rho,\sum_{j=1}^{N}K_{j}).$ (8)

###### Proof.

$\begin{split}I(\rho,\sum_{j=1}^{N}K_{j})&=\frac{1}{2}\norm{[\sqrt{\rho},\sum_{j=1}^{N}K_{j}]}_{2}^{2}\\\
&\leq\frac{1}{2}\Big{(}\sum_{j=1}^{N}\norm{[\sqrt{\rho},K_{j}]}_{2}\Big{)}^{2}\\\
&\leq\frac{N}{2}\sum_{j=1}^{N}\norm{[\sqrt{\rho},K_{j}]}_{2}^{2}\\\
&=N\sum_{j=1}^{N}I(\rho,K_{j}),\end{split}$ (9)

where the first inequality is from the triangle inequality of the norm and the
second inequality is from the Cauchy-Schwarz inequality. ∎

###  II.3 Asymmetry measures

Given a group $\mathbf{G}$, for any group element $g$, let $U(g)$ be the
unitary operator represented in the space $\mathcal{H}$. If a state $\rho$
remains unchanged under unitary transformations induced by $\mathbf{G}$,

$U(g)\rho U^{\dagger}(g)=\rho,\quad\forall g\in\mathbf{G},$ (10)

we say that the state is symmetric with respect to $\mathbf{G}$. In quantum
resource theory, the quantification of how much a state breaks this symmetry,
or the measure of asymmetry, is a significant problem. Different measures of
asymmetry have been proposed in the literature [31, 32, 33, 34]. For example,
some commonly used measures of asymmetry are based on skew information and von
Neumann entropy. Here we mainly focus on Lie groups and only review the
asymmetric measure given by skew information. Suppose the Lie algebra of the
Lie group $\mathbf{G}$ has an orthonormal base
$\\{H_{p}:p=1,\cdots,d_{\mathbf{G}}\\}$, where $d_{\mathbf{G}}$ is the
dimension of the Lie algebra. All generators can be written as linear
combinations of the elements in this base. The sum of the skew information,

$N_{\mathbf{G}}(\rho)=\sum_{p=1}^{d_{\mathbf{G}}}I(\rho,H_{p}),$ (11)

quantifies the asymmetry of the state $\rho$ with respect to the group
$\mathbf{G}$ [31, 34]. The asymmetry measure possesses the following desirable
properties.

(i) Here $N_{\mathbf{G}}(\rho)\geq 0$ and the equality holds if and only if
the state commutes with all generators, which indicates that the state is
symmetric with respect to $\mathbf{G}$.

(ii) For all $g$, there is

$N_{\mathbf{G}}(\rho)=N_{\mathbf{G}}(U(g)\rho U^{\dagger}(g)).$ (12)

(iii) Here $N_{\mathbf{G}}(\rho)$ is convex in the sense that

$N_{\mathbf{G}}(\sum_{i}\lambda_{i}\rho_{i})\leq\sum_{i}\lambda_{i}N_{\mathbf{G}}(\rho_{i}),$
(13)

where $\lambda_{i}\geq 0$ and $\sum_{i}\lambda_{i}=1$.

Items (i) and (iii) can be directly deduced from the properties of the skew
information [34]. We only need to prove item (ii). For all $g$, from the
unitary invariance of the skew information, we can obtain

$I(U(g)\rho U^{\dagger}(g),H_{p})=I(\rho,U^{\dagger}(g)H_{p}U(g)).$ (14)

For any unitary operator $U(g)$ and the generator $H_{p}$,
$U^{\dagger}(g)H_{p}U(g)$ is also a generator in Lie algebra [35].
Consequently, $\\{U^{\dagger}(g)H_{p}U(g):p=1,\cdots,d_{\mathbf{G}}\\}$ forms
an orthonormal base of the Lie algebra. Since the sum of the skew information
does not depend on the choice of the orthonormal base [36], item (ii) holds.

##  III approximate quantum error correction

The exact correctability is a strong restriction to practical codes. As a
result, we consider approximate QEC codes in some cases and many quantifiers
have been developed to evaluate the performance of the approximate error
correction[17, 15, 37, 38, 39, 40]. In an approximate QEC process, we need to
find a proper recovery channel such that the composite operation
$\mathcal{R}\circ\mathcal{N}\circ\mathcal{E}$ is close enough to the identity
map, which demonstrates that all states can be nearly recovered. To
characterize the performance of the approximate QEC codes, we first recall how
to quantify the “distance” between two channels.

For two states $\rho$ and $\sigma$ the distance is quantified by fidelity

$F(\rho,\sigma)=\norm{\sqrt{\rho},\sqrt{\sigma}}_{1}=\tr\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}.$
(15)

The fidelity and the trace distance are closely related. The two measures are
qualitatively equivalent since they satisfy the inequalities [41],

$1-\frac{1}{2}\norm{\rho-\sigma}_{1}\leq
F(\rho,\sigma)\leq\sqrt{1-\frac{1}{4}\norm{\rho-\sigma}^{2}_{1}}.$ (16)

For two channels $L$, $\Lambda$ and $\Lambda^{\prime}$ in the system, the
entanglement fidelity

$\begin{split}&F_{e}(\Lambda,\Lambda^{\prime})\\\
&=F\Big{(}(\Lambda_{L}\otimes\mathcal{I}_{R})(\outerproduct{\psi}{\psi}_{LR}),(\Lambda^{\prime}_{L}\otimes\mathcal{I}_{R})(\outerproduct{\psi}{\psi}_{LR})\Big{)}\end{split}$
(17)

measures the closeness between these two channels, where $R$ is the reference
system identical to system $L$ and
$\ket{\psi}_{LR}=1/\sqrt{d_{L}}\sum_{k=1}^{d_{L}}\ket{k}_{L}\ket{k}_{R}$ is
the maximally entangled state. As a special case, we take $\Lambda^{\prime}$
as the identity map and we obtain the entanglement fidelity of the channel
$\Lambda$,

$\begin{split}F_{e}(\Lambda)&=F_{e}(\Lambda,\mathcal{I})\\\
&=\sqrt{\bra{\psi}_{LR}(\Lambda_{L}\otimes\mathcal{I}_{R})(\outerproduct{\psi}{\psi}_{LR})\ket{\psi}_{LR}}.\end{split}$
(18)

With the above entanglement fidelity, now we can characterize the performance
of an encoding channel $\mathcal{E}$ under noise $\mathcal{N}$ by the quantity
defined as

$f_{e}(\mathcal{N}\circ\mathcal{E})=\underset{\mathcal{R}}{\max}F_{e}(\mathcal{R}\circ\mathcal{N}\circ\mathcal{E}).$
(19)

When $f_{e}(\mathcal{N}\circ\mathcal{E})=1$, we can find a channel
$\mathcal{R}$ such that all states are recovered perfectly.

The maximization problem in Eq. (19) is generally difficult since the
optimization is over all channels. Fortunately, we can study the problem from
the view of leaking information to the environment via the method of
complementary channels [42]. As shown in Fig. 1, the channel
$(\mathcal{N}\circ\mathcal{E})_{L\rightarrow S}$ has an isometry dilation
$V_{L\rightarrow SE}$ with environment system $E$ such that

$\mathcal{N}\circ\mathcal{E}(\rho_{L})=\tr_{E}\Big{(}V_{L\rightarrow
SE}\rho_{L}V_{L\rightarrow SE}^{\dagger}\Big{)}.$ (20)

Then the complementary channel is defined as

$\widehat{\mathcal{N}\circ\mathcal{E}}(\rho_{L})=\tr_{S}\Big{(}V_{L\rightarrow
SE}\rho_{L}V_{L\rightarrow SE}^{\dagger}\Big{)}.$ (21)

The optimization problem (19) has an equivalent form [13]

$f_{e}(\mathcal{N}\circ\mathcal{E})=\underset{\ket{\zeta}}{\max}F_{e}(\widehat{\mathcal{N}\circ\mathcal{E}},\mathcal{T}_{\zeta}),$
(22)

where $\mathcal{T}_{\zeta}(\cdot)=\tr(\cdot)\outerproduct{\zeta}{\zeta}$ is a
constant channel.

Figure 1: The $V_{L\rightarrow SE}$ is a Stinespring dilation of the channel
$(\mathcal{N}\circ\mathcal{E})_{L\rightarrow S}$ with an environment system
$E$. The input is a maximally entangled state of the logical system $L$ and
the reference system $R$. The output is denoted by $\ket{\Psi}_{RSE}$.

With the method of complementary channels, we give a lower bound of the
entanglement fidelity $f_{e}$ for generalized encoding and noise channels
which extends the results in Ref. [43].

######  Lemma 2.

Suppose the channel $(\mathcal{N}\circ\mathcal{E})_{L\rightarrow S}$ has a
Stinespring dilation $V_{L\rightarrow SE}$, as shown in Fig. 1. Let

$\ket{\Psi}_{SER}=(V_{L\rightarrow SE}\otimes\mathbf{1}_{R})\ket{\psi}_{LR},$
(23)

and denote by $\rho_{RE}$, $\rho_{R}$ and $\rho_{E}$ the reduced states of
$\ket{\Psi}_{RSE}$ on $RE$, $R$, and $E$, respectively. The quantity $f_{e}$
satisfies the inequality

$\begin{split}1-f_{e}(\mathcal{N}\circ\mathcal{E})&\leq\frac{1}{2}\norm{\rho_{RE}-\rho_{R}\otimes\rho_{E}}_{1}\\\
&\leq\frac{\sqrt{d_{L}d_{E}}}{2}\norm{\rho_{RE}-\rho_{R}\otimes\rho_{E}}_{2},\end{split}$
(24)

where $d_{E}$ is the dimension of the environment system $E$. The equality
holds if and only if $\rho_{RE}=\rho_{R}\otimes\rho_{E}$.

###### Proof.

From Eq. (22), we obtain

$\begin{split}&f_{e}(\mathcal{N}\circ\mathcal{E})\\\
&=\underset{\ket{\zeta}}{\max}F(\widehat{\mathcal{N}\circ\mathcal{E}}\otimes\mathcal{I}(\outerproduct{\psi}{\psi}_{LR}),\mathcal{T}_{\zeta}\otimes\mathcal{I}(\outerproduct{\psi}{\psi}_{LR}))\\\
&=\underset{\ket{\zeta}}{\max}F(\rho_{RE},\frac{\mathbf{1}_{R}}{d_{L}}\otimes\outerproduct{\zeta}{\zeta}_{E})\\\
&=\underset{\ket{\zeta}}{\max}F(\rho_{RE},\rho_{R}\otimes\outerproduct{\zeta}{\zeta}_{E})\\\
&\geq F(\rho_{RE},\rho_{R}\otimes\rho_{E})\\\ &\geq
1-\frac{1}{2}\norm{\rho_{RE}-\rho_{R}\otimes\rho_{E}}_{1},\end{split}$ (25)

where the last inequality is from Eq. (16).

Recall that for an operator $A$, the $1$-norm and the $2$-norm have the
relation

$\norm{A}_{1}\leq\sqrt{{\rm rank}(A)}\norm{A}_{2}.$ (26)

According to this relation and ${\rm
rank}(\rho_{RE}-\rho_{R}\otimes\rho_{E})\leq d_{L}d_{E}$, we obtain the
remaining inequality in Eq. (24). ∎

In general, the fidelity and the $1$-norm are difficult to calculate since we
need spectral decomposition. In comparison, the $2$-norm is easier to
calculate. After a tedious calculation of the $2$-norm in Eq. (24) presented
in Appendix A, we obtain a lower bound of $f_{e}$.

######  Observation 1.

Let $\mathcal{E}_{L\rightarrow S}$ and $\mathcal{N}_{S\rightarrow S}$ be the
encoding and noise channels, respectively. Suppose they have specific forms,

$\begin{split}\mathcal{E}(\rho)&=\sum_{s=1}^{m}E_{s}\rho E_{s}^{\dagger},\\\
\mathcal{N}(\sigma)&=\sum_{i=1}^{n}A_{i}\sigma A_{i}^{\dagger},\end{split}$
(27)

where $\sum_{s=1}^{m}E_{s}^{\dagger}E_{s}=\mathbf{1}_{L}$ and
$\sum_{i=1}^{n}A_{i}^{\dagger}A_{i}=\mathbf{1}_{S}$. Define
$O=\sum_{s=1}^{m}E_{s}E_{s}^{\dagger}$. The entanglement fidelity $f_{e}$ has
a lower bound,

$f_{e}(\mathcal{N}\circ\mathcal{E})\geq
1-\epsilon(\mathcal{N}\circ\mathcal{E}),$ (28)

where

$\begin{split}\epsilon(\mathcal{N}\circ\mathcal{E})&=\sqrt{\frac{mn}{4d_{L}}}\Big{(}\sum_{i,j=1}^{n}\tr(A_{i}^{\dagger}A_{j}OA_{j}^{\dagger}A_{i}O)\\\
&-\frac{1}{d_{L}}\sum_{i,j=1}^{n}\sum_{s,t=1}^{m}\absolutevalue{\tr(A_{i}^{\dagger}A_{j}E_{t}E_{s}^{\dagger})}^{2}\Big{)}^{1/2}\end{split}$
(29)

and we call $\epsilon$ the infidelity.

This observation gives a quantitative description of the performance of an
approximate QEC. When $\epsilon\ll 1$, the errors can be corrected
approximately. The defined infidelity $\epsilon$ also characterizes the
correlation between system $R$ and system $E$. As the environment becomes more
correlated with the reference system which contains the encoded quantum
information, more information leaks into the environment, which can result in
the degradation of the protected information.

##  IV Covariance symmetry

A channel $\mathcal{E}$ from system $L$ to system $S$ is called covariant with
group $\mathbf{G}$, if for all $g\in\mathbf{G}$ and all
$\rho\in\mathcal{D}(\mathcal{H}_{L})$ there is

$\mathcal{E}\Big{(}U_{L}(g)\rho
U_{L}(g)^{\dagger}\Big{)}=U_{S}(g)\mathcal{E}(\rho)U_{S}^{\dagger}(g),$ (30)

where $U_{L}(g)$ and $U_{S}(g)$ are unitary representations of group element
$g$ on space $\mathcal{H}_{L}$ and $\mathcal{H}_{S}$, respectively. We can
also say that the channel is symmetric with respect to $\mathbf{G}$. The
covariant channel is intimately connected with the symmetric state and the
Choi representation builds this bridge. More explicitly, the covariance
symmetry of a channel is equal to the group symmetry of the corresponding Choi
state [44]. Now we explain this equivalence relation in detail.

Recall that there exists a one-to-one correspondence between the channel and
the Choi state

$\begin{split}\Phi_{\mathcal{E}}&=(\mathcal{I}_{L}\otimes\mathcal{E}_{R\rightarrow
S})(\outerproduct{\psi}{\psi}_{LR}),\\\ \mathcal{E}_{L\rightarrow
S}(\rho)&=d_{L}\tr_{L}[(\rho_{L}^{T}\otimes\mathbf{1}_{S})\Phi_{\mathcal{E}}],\end{split}$
(31)

where $T$ represents the transposition. Suppose the channel $\mathcal{E}$ is
$\mathbf{G}$ covariant. Then for all $\rho$ and all $g$, we can obtain

$\begin{split}0=&\frac{1}{d_{L}}\mathcal{E}(\rho)-\frac{1}{d_{L}}U_{S}^{\dagger}(g)\mathcal{E}\Big{(}U_{L}(g)\rho
U_{L}^{\dagger}(g)\Big{)}U_{S}(g)\\\
=&\tr_{L}[(\rho_{L}^{T}\otimes\mathbf{1}_{S})\Phi_{\mathcal{E}}]\\\
-&U_{S}^{\dagger}(g)\tr_{L}\\{[U_{L}^{*}(g)\rho_{L}^{T}U_{L}^{T}(g)\otimes\mathbf{1}_{S}]\Phi_{\mathcal{E}}\\}U_{S}(g),\end{split}$
(32)

where $*$ is the conjugate operation. Therefore,

$\Phi_{\mathcal{E}}=[U_{L}^{T}(g)\otimes
U_{S}^{\dagger}(g)]\Phi_{\mathcal{E}}[U_{L}^{*}(g)\otimes U_{S}(g)]\forall g.$
(33)

This implies that the Choi state $\Phi_{\mathcal{E}}$ is symmetric with
respect to the unitary representation $\\{U_{L}^{*}(g)\otimes U_{S}(g)\\}$.
Hence, we can quantify the noncovariance of a channel from the asymmetry of
its Choi state. Concretely, noncovariance of the channel $\mathcal{E}$ is
defined as

$N_{\mathbf{G}}(\mathcal{E})=N_{\mathbf{G}}(\Phi_{\mathcal{E}})=\sum_{p=1}^{d_{\mathbf{G}}}I(\Phi_{\mathcal{E}},H_{p}),$
(34)

which has been thoroughly studied as discussed in Sec. II.

##  V Isometric encoding

In this section we investigate infidelity, noncovariance, and their trade-off
relation of a particular example.

###  V.1 Infidelity of the QEC

The isometric encoding channel is of the form $\mathcal{E}(\rho)=W\rho
W^{\dagger}$, with $W^{\dagger}W=\mathbf{1}_{L}$. The projector onto the
coding space is $P=WW^{\dagger}$ and the Choi state is

$\begin{split}\Phi_{\mathcal{E}}&=(\mathcal{I}_{L}\otimes\mathcal{E}_{R\rightarrow
S})(\outerproduct{\psi}{\psi}_{LR})\\\
&=\outerproduct{\tilde{\psi}}{\tilde{\psi}}_{LS},\end{split}$ (35)

where $\ket{\tilde{\psi}}_{LS}=(\mathbf{1}_{L}\otimes W_{R\rightarrow
S})\ket{\psi}_{LR}$. The noise channel is assumed to be of the general form
$\mathcal{N}_{S\rightarrow S}(\rho)=\sum_{i=1}^{n}A_{i}\rho A_{i}^{\dagger}$.

According to Observation 1, the square of infidelity is

$\begin{split}&\epsilon^{2}(\mathcal{N}\circ\mathcal{E})\\\
&=\frac{n}{4d_{L}}\sum_{i,j=1}^{n}\tr(PA_{j}^{\dagger}A_{i}PA_{i}^{\dagger}A_{j})-\frac{1}{d_{L}}\absolutevalue{\tr(PA_{i}^{\dagger}A_{j})}^{2}.\end{split}$

Notice that when the Knill-Laflamme condition is satisfied, namely,
$P(A_{i}^{\dagger}A_{j})P=\lambda_{ij}P$ holds for all $i$ and $j$ with some
constants $\lambda_{ij}$, then infidelity $\epsilon=0$ and perfect error
correction can be realized.

We define $K_{ij}=PA_{i}^{\dagger}A_{j}P$ and the infidelity can be written in
the form of the generalized skew information

$\begin{split}&\frac{4d_{L}\epsilon^{2}(\mathcal{N}\circ\mathcal{E})}{n}\\\
&=\sum_{i,j=1}^{n}\frac{1}{2}\tr(K_{ij}^{\dagger}K_{ij}+K_{ij}K_{ij}^{\dagger})-\sum_{i,j=1}^{n}\frac{1}{d_{L}}\absolutevalue{\tr
K_{ij}}^{2}\\\
&=\sum_{i,j=1}^{n}\frac{1}{2}\tr(W^{\dagger}(K_{ij}^{\dagger}K_{ij}+K_{ij}K_{ij}^{\dagger})W)\\\
&-\sum_{i,j=1}^{n}\frac{1}{d_{L}}\absolutevalue{\tr(W^{\dagger}K_{ij}W)}^{2}\\\
&=\sum_{i,j=1}^{n}\sum_{k,l=1}^{d_{L}}\frac{1}{2}\bra{k}W^{\dagger}(K_{ij}^{\dagger}K_{ij}+K_{ij}K_{ij}^{\dagger})W\ket{l}\innerproduct{k}{l}\\\
&-\sum_{i,j=1}^{n}\absolutevalue{\sum_{k,l=1}^{d_{L}}\bra{k}W^{\dagger}K_{ij}W\ket{l}\innerproduct{k}{l}}^{2}\\\
&=d_{L}\sum_{i,j=1}^{n}\Big{(}\frac{1}{2}\bra{\psi}\mathbf{1}_{L}\otimes
W^{\dagger}(K_{ij}^{\dagger}K_{ij}+K_{ij}K_{ij}^{\dagger})\mathbf{1}_{L}\otimes
W\ket{\psi}\\\ &-\absolutevalue{\bra{\psi}(\mathbf{1}_{L}\otimes
W^{\dagger})(\mathbf{1}_{L}\otimes K_{ij})(\mathbf{1}_{L}\otimes
W)\ket{\psi}}^{2}\Big{)}\\\
&=d_{L}\sum_{i,j=1}^{n}I\Big{(}\outerproduct{\tilde{\psi}}{\tilde{\psi}},\mathbf{1}_{L}\otimes
K_{ij}\Big{)}.\end{split}$

###  V.2 trade-off relation between infidelity and noncovariance

Consider a general Lie group $\mathbf{G}$ and denote the Lie algebra by
$\mathcal{L}_{\mathbf{G}}$. Here $\\{U_{L}^{*}(g)\\}$ is a unitary
representation of $\mathbf{G}$ in the logical space $\mathcal{H}_{L}$ and we
assume that the associated representation of the Lie algebra is $\pi_{L}$,
namely, $\\{\pi_{L}(X):X\in\mathcal{L}_{\mathbf{G}}\\}$ is the set of
generators for $\\{U_{L}^{*}(g)\\}$. Similarly, for the unitary representation
$\\{U_{S}(g)\\}$ in the physical system, suppose the associated representation
of the Lie algebra is $\pi_{S}$. Hence, in the Hilbert space
$\mathcal{H}_{L}\otimes\mathcal{H}_{S}$,
$\\{\pi_{L}(X)\otimes\mathbf{1}_{S}+\mathbf{1}_{L}\otimes\pi_{S}(X):X\in\mathcal{L}_{\mathbf{G}}\\}$
gives the representation of the Lie algebra with respect to the unitary
representation $\\{U_{L}^{*}(g)\otimes U_{S}(g)\\}$ [35], and we assume that
the set $\\{H_{L}^{p}\otimes\mathbf{1}_{S}+\mathbf{1}_{L}\otimes
H_{S}^{p}:p=1,\cdots,d_{\mathbf{G}}\\}$ constitutes an orthonormal base of the
Lie algebra $\mathcal{L}_{\mathbf{G}}$. The sum of skew information

$N_{\mathbf{G}}(\rho)=\sum_{p=1}^{d_{\mathbf{G}}}I(\rho,H_{L}^{p}\otimes\mathbf{1}_{S}+\mathbf{1}_{L}\otimes
H_{S}^{p}),$ (36)

quantifies the asymmetry of state $\rho$ with respect to $\mathbf{G}$ [34]. We
can obtain the noncovariance measure of a channel $\mathcal{E}$ from this
asymmetry measure, as defined in Eq. (34). Moreover, we find the following
relation by combining Lemma 1 with the expressions of infidelity and
noncovariance.

######  Observation 2.

For an isometric encoding channel $\mathcal{E}$ and noise channel
$\mathcal{N}$, noncovariance with respect to a Lie group $\mathbf{G}$ and
infidelity satisfy the trade-off relation

$\frac{4\epsilon^{2}(\mathcal{N}\circ\mathcal{E})}{n}+N_{\mathbf{G}}(\mathcal{E})\geq\frac{1}{n^{2}+d_{\mathbf{G}}}I\Big{(}\outerproduct{\tilde{\psi}}{\tilde{\psi}},K\Big{)},$
(37)

where $K=\sum_{i,j=1}^{n}\mathbf{1}_{L}\otimes
K_{ij}+\sum_{p=1}^{d_{\mathbf{G}}}(H_{L}^{p}\otimes\mathbf{1}_{S}+\mathbf{1}_{L}\otimes
H_{S}^{p})$.

Next we consider the special case of the U(1) group. In this case, we assume
that $U_{L}(g)=e^{-iH_{L}^{*}g}$ and $U_{S}(g)=e^{-iH_{S}g}$, where
$H_{L}^{*}$ and $H_{S}$ are Hamiltonians. Then

$\begin{split}U_{L}^{*}(g)\otimes U_{S}(g)&=e^{iH_{L}g}\otimes e^{-iH_{S}g}\\\
&=e^{-i(\mathbf{1}_{L}\otimes H_{S}-H_{L}\otimes\mathbf{1}_{S})g}.\end{split}$
(38)

Thus, the corresponding generated Hamiltonian of $U_{L}^{*}(g)\otimes
U_{S}(g)$ is $H=\mathbf{1}_{L}\otimes H_{S}-H_{L}\otimes\mathbf{1}_{S}$. The
noncovariance of the isometric encoding channel can be quantified by the skew
information

$N_{\mathbf{G}}(\mathcal{E})=I\Big{(}\outerproduct{\tilde{\psi}}{\tilde{\psi}},H\Big{)}.$
(39)

For the U(1) group, the HKS condition is sufficient for the nonexistence of
covariant and exact QEC codes [19]. Explicitly, if

$H_{S}\in{\rm span}\\{A_{i}^{\dagger}A_{j}:i,j=1,\cdots,n\\},$ (40)

all covariant codes cannot correct errors perfectly. Here we prove again this
no-go result.

Let $H_{S}=\sum_{i,j=1}^{n}\alpha_{ij}A_{i}^{\dagger}A_{j}$, with
$\alpha_{ij}\in\mathbb{C}$, and suppose the isometric encoding channel
$\mathcal{E}$ is covariant and corrects errors perfectly. Since
$N_{\mathbf{G}}(\mathcal{E})=0$ and $H$ is Hermitian, there exists a constant
$\lambda$ such that

$H(\mathbf{1}_{L}\otimes W)\ket{\psi}=\lambda(\mathbf{1}_{L}\otimes
W)\ket{\psi}.$ (41)

This implies that

$(\mathbf{1}_{L}\otimes P)H(\mathbf{1}_{L}\otimes P)(\mathbf{1}_{L}\otimes
W)\ket{\psi}=\lambda(\mathbf{1}_{L}\otimes W)\ket{\psi}.$ (42)

Consequently,

$\begin{split}0&=I\Big{(}\outerproduct{\tilde{\psi}}{\tilde{\psi}},(\mathbf{1}_{L}\otimes
P)H(\mathbf{1}_{L}\otimes P)\Big{)}\\\
&=I\Big{(}\outerproduct{\tilde{\psi}}{\tilde{\psi}},\sum_{i,j}\alpha_{ij}\mathbf{1}_{L}\otimes
K_{ij}-H_{L}\otimes P\Big{)}.\end{split}$ (43)

In addition, $\epsilon(\mathcal{N}\circ\mathcal{E})=0$ indicates that

$I(\outerproduct{\tilde{\psi}}{\tilde{\psi}},\alpha_{ij}\mathbf{1}_{L}\otimes
K_{ij})=0.$ (44)

Combining Eq. (43) with Eq. (44), we obtain

$\begin{split}0&\leq(n^{2}+1)I\Big{(}\outerproduct{\tilde{\psi}}{\tilde{\psi}},H_{L}\otimes
P\Big{)}\\\ &\leq
I\Big{(}\outerproduct{\tilde{\psi}}{\tilde{\psi}},-\sum_{i,j=1}^{n}\alpha_{ij}\mathbf{1}_{L}\otimes
K_{ij}+H_{L}\otimes P\Big{)}\\\
&+\sum_{i,j=1}^{n}I\Big{(}\outerproduct{\tilde{\psi}}{\tilde{\psi}},\alpha_{ij}\mathbf{1}_{L}\otimes
K_{ij}\Big{)}=0.\end{split}$ (45)

Therefore,

$(\mathbf{1}_{L}\otimes W^{\dagger})(H_{L}\otimes P)(\mathbf{1}_{L}\otimes
W)\ket{\psi}=\alpha\ket{\psi}$ (46)

holds for some constant $\alpha$. After a direct calculation, we have

$\bra{k}H_{L}\ket{l}=\alpha\delta_{kl},$ (47)

or equivalently, $H_{L}=H_{L}^{*}=\alpha\mathbf{1}_{L}$. This contradicts the
nontrivial assumption of a logical Hamiltonian.

###  V.3 Average infidelity and noncovariance for random codes

We consider a type of random code in which the encoding isometry has the
following expression

$W=U_{S}(\mathbf{1}_{L}\otimes\ket{0}_{A}),$ (48)

where $A$ is an ancillary system satisfying
$\mathcal{H}_{S}=\mathcal{H}_{L}\otimes\mathcal{H}_{A}$ and $U$ is a random
unitary under the Haar measure. Equivalently, the projector can be written as

$P=U_{S}(\mathbf{1}_{L}\otimes\outerproduct{0}{0}_{A})U_{S}^{\dagger}.$ (49)

For this type of random code, the average infidelity satisfies

$\begin{split}&\int_{\mathbf{U}(d_{S})}\epsilon^{2}(\mathcal{N}\circ\mathcal{E})d\mu(U)\\\
&=\frac{n(d_{L}^{2}-1)}{4d_{L}(d_{S}^{2}-1)}\\\
&\times\sum_{i,j=1}^{n}\Big{(}\tr(A_{j}^{\dagger}A_{i}A_{i}^{\dagger}A_{j})-\frac{1}{d_{S}}\absolutevalue{\tr(A_{i}^{\dagger}A_{j})}^{2}\Big{)},\end{split}$
(50)

where $\mathbf{U}(d_{S})$ represents the unitary group in system $S$ and $\mu$
is the Haar measure. When $\mathbf{G}$ is the U(1) group, the average
noncovariance is equal to

$\begin{split}&\int_{\mathbf{U}(d_{S})}N_{\mathbf{G}}(\mathcal{E})d\mu(U)\\\
&=\frac{d_{L}\tr(H_{L}^{2})-(\tr H_{L})^{2}}{d_{L}^{2}}\\\
&+\frac{(d_{L}d_{S}^{2}-d_{S})\tr(H_{S}^{2})-(d_{L}d_{S}-1)(\tr
H_{S})^{2}}{d_{L}d_{S}(d_{S}^{2}-1)}.\end{split}$ (51)

We leave the detailed calculation to Appendix B.

From Eqs. (50) and (51), we can see that if the dimension of the physical
system $d_{S}$ tends to infinity, the average infidelity tends $0$ while the
noncovariance tends to $[d_{L}\tr(H_{L}^{2})-(\tr H_{L})^{2}]/d_{L}^{2}$.

##  VI Conclusion and outlook

In this work we defined a quantity termed infidelity to characterize the
inaccuracy of an approximate QEC and also to quantify the noncovariance of an
encoding channel with respect to a general Lie group. With these two
quantities, we derived a trade-off relation between approximate QEC and
noncovariance in the special case that the encoding channel is isometric. For
a type of random code, we found that when the dimension of the physical system
is large enough, the errors can be corrected approximately while noncovariance
tends to a constant.

The information scrambling can protect encoding information against errors and
hence is closely connected with the capability of error correction [45, 46,
47]. For future work it would be interesting to explore the QEC ability and
the information scrambling quantitatively in systems with particular symmetry
via the infidelity we defined, which may help us design explicit covariant and
approximate QEC codes from scrambling circuits.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
Grant No. 12174216.

##  Appendix A Proof of Observation 1

The Stinespring isometry $V_{L\rightarrow SE}$ of the composite channel

$\mathcal{N}\circ\mathcal{E}(\rho)=\sum_{i,s}A_{i}E_{s}\rho
E_{s}^{\dagger}A_{i}^{\dagger},$ (52)

satisfies

$V_{L\rightarrow
SE}\ket{\varphi}_{L}=\sum_{i,s}A_{i}E_{s}\ket{\varphi}_{L}\otimes\ket{is}_{E}.$
(53)

Here $\\{\ket{is}_{E}\\}$ forms an orthonormal basis of the environment system
$E$ and the dimension $d_{E}=mn$, which is equal to the number of the Kraus
operators $\\{A_{i}E_{s}\\}$. Note that we omit the upper bound of the index
in the summation sign for convenience in this appendix.

The output state is

$\begin{split}\ket{\Psi}_{RSE}&=(\mathbf{1}_{R}\otimes V_{L\rightarrow
SE})\ket{\psi}_{RL}\\\ &=\frac{1}{\sqrt{d_{L}}}\sum_{k}V_{L\rightarrow
SE}\ket{k}_{L}\otimes\ket{k}_{R}\\\
&=\frac{1}{\sqrt{d_{L}}}\sum_{k,i,s}A_{i}E_{s}\ket{k}_{L}\otimes\ket{is}_{E}\otimes\ket{k}_{R}.\end{split}$
(54)

The reduced state in system $RE$ is

$\begin{split}\rho_{RE}=&\tr_{S}\outerproduct{\Psi}{\Psi}_{RSE}\\\
=&\frac{1}{d_{L}}\sum_{i,j,k,l,s,t}\tr_{S}[(A_{i}E_{s}\outerproduct{k}{l}E_{t}^{\dagger}A_{j}^{\dagger})_{S}\otimes\outerproduct{is}{jt}_{E}\otimes\outerproduct{k}{l}_{R}]\\\
=&\frac{1}{d_{L}}\sum_{i,j,k,l,s,t}\bra{l}E_{t}^{\dagger}A_{j}^{\dagger}A_{i}E_{s}\ket{k}\outerproduct{is}{jt}_{E}\otimes\outerproduct{k}{l}_{R}.\end{split}$
(55)

Then the reduced state in system $R$ is the maximally mixed state
$\mathbf{1}_{R}/d_{L}$ and

$\begin{split}\rho_{E}&=\tr_{R}\rho_{RE}\\\
&=\frac{1}{d_{L}}\sum_{k,i,j,s,t}\bra{k}E_{t}^{\dagger}A_{j}^{\dagger}A_{i}E_{s}\ket{k}\outerproduct{is}{jt}_{E}\\\
&=\frac{1}{d_{L}}\sum_{i,j,s,t}\tr(E_{t}^{\dagger}A_{j}^{\dagger}A_{i}E_{s})\outerproduct{is}{jt}_{E}.\end{split}$
(56)

To calculate the $2$-norm in Eq. (24), we map the states in system $RE$ to
states in system $LE$ through the isometric channel

$\begin{split}&\Lambda(\sum_{k,l,i,j,s,t}\alpha_{klijst}\outerproduct{is}{jt}_{E}\otimes\outerproduct{k}{l}_{R})\\\
&=\sum_{k,l,i,j,s,t}\alpha_{klijst}^{*}\outerproduct{is}{jt}_{E}\otimes\outerproduct{k}{l}_{L},\end{split}$
(57)

and then

$\norm{\Lambda(\rho_{RE})-\Lambda(\rho_{R}\otimes\rho_{E})}_{2}=\norm{\rho_{RE}-\rho_{R}\otimes\rho_{E}}_{2}.$
(58)

If we let

$\begin{split}D&=\Lambda(\rho_{RE})-\Lambda(\rho_{R}\otimes\rho_{E})\\\
&=\frac{1}{d_{L}}\sum_{i,j,s,t}\Big{(}E_{s}^{\dagger}A_{i}^{\dagger}A_{j}E_{t}-\tr(E_{s}^{\dagger}A_{i}^{\dagger}A_{j}E_{t})\frac{\mathbf{1}_{L}}{d_{L}}\Big{)}\otimes\outerproduct{is}{jt}_{E},\end{split}$
(59)

then we have

$\begin{split}\norm{\rho_{RE}-\rho_{R}\otimes\rho_{E}}_{2}^{2}&=\tr
DD^{\dagger}\\\
&=\frac{1}{d_{L}^{2}}\sum_{i,j,s,t}\Bigg{(}\tr(E_{s}^{\dagger}A_{i}^{\dagger}A_{j}E_{t}E_{t}^{\dagger}A_{j}^{\dagger}A_{i}E_{s})-\frac{1}{d_{L}}\tr(E_{s}^{\dagger}A_{i}^{\dagger}A_{j}E_{t})\tr(E_{t}^{\dagger}A_{j}^{\dagger}A_{i}E_{s})\Bigg{)}\\\
&=\frac{1}{d_{L}^{2}}\sum_{i,j}\tr(A_{i}^{\dagger}A_{j}OA_{j}^{\dagger}A_{i}O)-\frac{1}{d_{L}^{3}}\sum_{i,j,s,t}\absolutevalue{\tr(A_{i}^{\dagger}A_{j}E_{t}E_{s}^{\dagger})}^{2},\end{split}$
(60)

where $O=\sum_{t}E_{t}E_{t}^{\dagger}$.

##  Appendix B Calculation of average infidelity and noncovariance

In a Hilbert space $\mathcal{H}$ with dimension $d$, the uniform Haar measure
$\mu$ over unitary operator group $\mathbf{U}(d)$ remains invariant under both
left and right multiplication of any unitary operator $V\in\mathbf{U}(d)$ [48,
49, 50]. Mathematically,

$\mu(\mathcal{A})=\mu(\mathcal{A}V)=\mu(V\mathcal{A})$ (61)

holds for an arbitrary Borel subset $\mathcal{A}$ and arbitrary unitary $V$.
Here we recall some integral formulas over unitary groups, referring to Ref.
[48] for detailed proofs.

######  Lemma 3.

For Haar measure $\mu$, it holds that

  1. 1.

$\int_{\mathbf{U}(d)}UAU^{\dagger}d\mu(U)=\frac{\tr A}{d}\mathbf{1}_{d},$ (62)

  2. 2.

$\int_{\mathbf{U}(d_{A})}(U_{A}\otimes\mathbf{1}_{B})X_{AB}(U_{A}\otimes\mathbf{1}_{B})^{\dagger}d\mu(U_{A})=\frac{\mathbf{1}_{A}}{d_{A}}\otimes\tr_{A}X_{AB},$
(63)

  3. 3.

$\begin{split}&\int_{\mathbf{U}(d)}(U\otimes U)A(U\otimes
U)^{\dagger}d\mu(U)\\\ &=\Big{(}\frac{\tr
A}{d^{2}-1}-\frac{\tr(AF)}{d(d^{2}-1)}\Big{)}\mathbf{1}_{d^{2}}-\Big{(}\frac{\tr
A}{d(d^{2}-1)}-\frac{\tr(AF)}{d^{2}-1}\Big{)}F,\end{split}$ (64)

where $F$ is the swap operator.

  4. 4.

$\begin{split}&\int_{\mathbf{U}(d)}UAU^{\dagger}XUBU^{\dagger}d\mu(U)\\\
&=\frac{d\tr(AB)-\tr A\tr B}{d(d^{2}-1)}(\tr X)\mathbf{1}_{d}+\frac{d\tr A\tr
B-\tr(AB)}{d(d^{2}-1)}X.\end{split}$ (65)

We first calculate the average infidelity. According to the lemma 3, we have

$\begin{split}&\int_{\mathbf{U}(d_{S})}\tr(PA_{j}^{\dagger}A_{i}PA_{i}^{\dagger}A_{j})d\mu(U)\\\
&=\tr\int_{\mathbf{U}(d_{S})}U(\mathbf{1}_{L}\otimes\outerproduct{0}{0})U^{\dagger}A_{j}^{\dagger}A_{i}U(\mathbf{1}_{L}\otimes\outerproduct{0}{0})U^{\dagger}A_{i}^{\dagger}A_{j}d\mu(U)\\\
&=\frac{d_{S}d_{L}-d_{L}^{2}}{d_{S}(d_{S}^{2}-1)}\absolutevalue{\tr(A_{j}^{\dagger}A_{i})}^{2}+\frac{d_{S}d_{L}^{2}-d_{L}}{d_{S}(d_{S}^{2}-1)}\tr(A_{j}^{\dagger}A_{i}A_{i}^{\dagger}A_{j})\end{split}$
(66)

and

$\begin{split}&\int_{\mathbf{U}(d_{S})}\absolutevalue{\tr(PA_{i}^{\dagger}A_{j})}^{2}d\mu(U)\\\
&=\tr\int_{\mathbf{U}(d_{S})}U(\mathbf{1}_{L}\otimes\outerproduct{0}{0})U^{\dagger}A_{i}^{\dagger}A_{j}\otimes
U(\mathbf{1}_{L}\otimes\outerproduct{0}{0})U^{\dagger}A_{j}^{\dagger}A_{i}d\mu(U)\\\
&=\frac{d_{S}d_{L}^{2}-d_{L}}{d_{S}(d_{S}^{2}-1)}\absolutevalue{\tr(A_{i}^{\dagger}A_{j})}^{2}-\frac{d_{L}^{2}-d_{S}d_{L}}{d_{S}(d_{S}^{2}-1)}\tr(A_{j}^{\dagger}A_{i}A_{i}^{\dagger}A_{j}).\end{split}$
(67)

Thus, we can obtain

$\begin{split}&\int_{\mathbf{U}(d_{S})}\frac{4d_{L}\epsilon^{2}(\mathcal{N}\circ\mathcal{E})}{n}d\mu(U)\\\
&=\frac{d_{L}^{2}-1}{d_{S}^{2}-1}\sum_{i,j}\Big{(}\tr(A_{j}^{\dagger}A_{i}A_{i}^{\dagger}A_{j})-\frac{1}{d_{S}}\absolutevalue{\tr(A_{i}^{\dagger}A_{j})}^{2}\Big{)}.\end{split}$
(68)

Next, we calculate the average of noncovariance

$N_{\mathbf{G}}(\mathcal{E})=\bra{\tilde{\psi}}H^{2}\ket{\tilde{\psi}}-\bra{\tilde{\psi}}H\ket{\tilde{\psi}}^{2}.$
(69)

The first term is equal to

$\begin{split}&\int_{\mathbf{U}(d_{S})}\bra{\tilde{\psi}}H^{2}\ket{\tilde{\psi}}d\mu(U)\\\
&=\int_{\mathbf{U}_{d_{S}}}\bra{\psi^{\prime}}(\mathbf{1}_{L}\otimes
U)^{\dagger}H^{2}(\mathbf{1}_{L}\otimes U)\ket{\psi^{\prime}}d\mu(U)\\\
&=\frac{1}{d_{L}}\tr(H_{L}^{2})+\frac{1}{d_{S}}\tr(H_{S}^{2})-\frac{2\tr
H_{L}\tr H_{S}}{d_{L}d_{S}},\end{split}$ (70)

where $\ket{\psi^{\prime}}=1/\sqrt{d_{L}}\sum_{k}\ket{k}_{L}\ket{k0}_{S}$.

The second term is equal to

$\begin{split}&\int_{\mathbf{U}(d_{S})}\bra{\tilde{\psi}}H\ket{\tilde{\psi}}^{2}d\mu(U)\\\
&=\frac{1}{d_{L}^{2}}\int_{\mathbf{U}(d_{S})}\\{-\tr
H_{L}+\tr[U^{\dagger}H_{S}U(\mathbf{1}_{L}\otimes\outerproduct{0}{0})]\\}^{2}d\mu(U)\\\
&=\frac{(\tr H_{L})^{2}}{d_{L}^{2}}-\frac{2\tr
H_{L}}{d_{L}^{2}}\tr\int_{\mathbf{U}(d_{S})}(U^{\dagger}H_{S}U)(\mathbf{1}_{L}\otimes\outerproduct{0}{0})d\mu(U)\\\
&+\frac{1}{d_{L}^{2}}\tr\int_{\mathbf{U}(d_{S})}(U^{\dagger})^{\otimes
2}H_{S}^{\otimes 2}U^{\otimes
2}(\mathbf{1}_{L}\otimes\outerproduct{0}{0})^{\otimes 2}d\mu(U)\\\
&=\frac{(\tr H_{L})^{2}}{d_{L}^{2}}-\frac{2\tr H_{L}\tr
H_{S}}{d_{L}d_{S}}+\frac{d_{L}d_{S}-1}{d_{L}d_{S}(d_{S}^{2}-1)}(\tr
H_{S})^{2}+\frac{d_{S}-d_{L}}{d_{L}d_{S}(d_{S}^{2}-1)}\tr(H_{S}^{2})\end{split}$
(71)

Thus, the average noncovariance can be expressed as

$\begin{split}&\int_{\mathbf{U}(d_{S})}N_{\mathbf{G}}(\mathcal{E})d\mu(U)\\\
&=\frac{d_{L}\tr(H_{L}^{2})-(\tr
H_{L})^{2}}{d_{L}^{2}}+\frac{(d_{L}d_{S}^{2}-d_{S})\tr(H_{S}^{2})-(d_{L}d_{S}-1)(\tr
H_{S})^{2}}{d_{L}d_{S}(d_{S}^{2}-1)}.\end{split}$ (72)

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