# Bounding the quantum capacity with flagged extensions

Farzad Kianvash, Marco Fanizza, Vittorio Giovannetti
NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa,
Italy farzad.kianvash<EMAIL_ADDRESS>
###### Abstract

In this article we consider flagged extensions of channels that can be written
as convex combination of other channels, and find general sufficient
conditions for the degradability of the flagged extension. An immediate
application is a bound on the quantum and private capacities of any channel
being a mixture of a unitary operator and another channel, with the
probability associated to the unitary operator being larger than $1/2$. We
then specialize our sufficient conditions to flagged Pauli channels, obtaining
a family of upper bounds on quantum and private capacities of Pauli channels.
In particular, we establish new state-of-the-art upper bounds on the quantum
and private capacities of the depolarizing channel, BB84 channel and
generalized amplitude damping channel. Moreover, the flagged construction can
be naturally applied to tensor powers of channels with less restricting
degradability conditions, suggesting that better upper bounds could be found
by considering a larger number of channel uses.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Sufficient conditions for degradability of flagged extensions
  4. 4 General applications
    1. 4.1 Convex combination of a unitary operation with an arbitrary channel
  5. 5 Pauli channels
    1. 5.1 Qubit Pauli group
    2. 5.2 Qudit Pauli group
    3. 5.3 Flagged Pauli channels
    4. 5.4 Depolarizing channel
    5. 5.5 BB84 channel
  6. 6 Generalized amplitude damping
  7. 7 Discussion
  8. 8 Conclusions

##  1 Introduction

Protecting quantum states against noise is a fundamental requirement for
harnessing the power of quantum computers and technologies. In a transmission
line or in a memory, noise is modeled as a quantum channel, and several
accesses to the channel together with careful state preparation and decoding
can protect quantum information against noise. The quantum capacity of a
channel is the maximal amount of qubits which can be transmitted reliably, per
use of the channel. It can be expressed in terms of an entropic functional of
channels, the coherent information, which can be computed as a maximization
over quantum states sent through the channel. The quantum capacity [1, 2, 3]
of a channel can then be obtained as a limit for large $n$ of the coherent
information per use of the channel, for $n$ uses of the channel. A striking
feature of quantum Shannon theory [4, 5] is the potential super-additivity of
the coherent information [6, 7, 8, 9, 10, 11, 12, 13, 14], which hinders the
direct evaluation of the quantum capacity. In fact, this evaluation requires
an infinite number of optimizations on Hilbert spaces of dimension that grows
exponentially in the number of uses of the channel. The existence of an
algorithmically feasible evaluation of the quantum capacity remains as one of
the most important open problems in quantum Shannon theory, while finding
computable upper or lower bounds on the quantum capacity constitutes important
progress.

The phenomenon of superadditivity is not restricted to the quantum capacity,
as it shows up also for the classical capacity [15], the classical private
capacity [16] and the trade-off capacity region [17, 18]. In this paper we are
also interested in classical private capacity, which is the optimal rate for
classical communication protected by any eavesdropper [19, 20]. In general
this capacity is larger than the quantum capacity, but the upper bounds we
obtain in this paper hold for both capacities.

Extending previous results, in this work we formulate sufficient conditions to
obtain non-trivial upper bounds on the quantum capacity, using the so called
flagged extensions. A flagged extension of a channel that can be written as
convex combinations of other channels is such that the receiver gets, together
with the output of one of the channels in the convex combination, a flag
carrying the information about which of the channels acted. This technique is
particularly effective for a class of channels of physical significance, the
Pauli channels. A qubit Pauli channel describes random bit flip and phase flip
errors, which is a fundamental noise model; moreover, any qubit channel can be
mapped to a Pauli channel by a twirling map, which does not increase the
quantum capacity. With these new flagged extensions we improve the results of
[21, 22] for two important Pauli channels: the depolarizing channel and BB84
channel, which are both superadditive [6, 7], and their quantum capacities are
not known, despite a long history of efforts [7, 23, 24, 25, 26, 27, 8, 9, 28,
29, 30, 31, 32, 33, 14]. We also find new bounds for quantum capacity of the
generalized amplitude damping channel [34, 35, 36, 22], improving the results
of [22]. The bounds we obtain are not necessarily the best bounds available
with these techniques, being just good guesses among all the instances of
flagged channels that satisfy the sufficient conditions. In fact, we obtain an
infinite sequence of optimization problem depending on the number of uses of
the channel, each of which gives a bound on the capacity. It is not clear if a
phenomenon analogue to superadditivity appears in this scenario. Even with one
use of the channel, different choice of Kraus operators give different bounds.

These bounds are based on the most fruitful technique to obtain upper bounds
on the quantum capacity: finding a degradable extension of the channel (e.g.
[28, 30, 33].) In fact, degradable channels [37, 38] have the property that
the coherent information is additive, therefore the quantum capacity is
obtainable as the coherent information of the channel. Moreover, a fundamental
properties of capacities is that they are generically decreasing under
composition of channels, a fact that has a clear operational justification. It
is also known that the quantum capacity of a degradable channel is equal to
its private capacity, which is the optimal rate of classical communication
protected from any eavesdropper, explaining why the quantum capacity of a
flagged degradable extension of a channel is also an upper bound for the
private capacity of the original channel.

Moreover, when the channel is approximately degradable useful bounds can still
be obtained [31].

In a previous paper [21] we contributed to this line of work by considering a
flagged degradable extension of the depolarizing channel. While previous
constructions [28] used orthogonal flags, our contribution was to consider
non-orthogonal flags, showing that degradable extensions can be obtained even
in this less restricted setting, obtaining better bounds. In a subsequent work
[22] the author combined non-orthogonal flags with approximate degradability,
improving the bounds further by searching for the flagged extension with the
best bound from approximate degradability. However, from careful inspection it
seems that the advantage of the approximate degradability technique in this
context is that it finds exactly degradable extensions for a choice of flags
that our analysis did not cover. In fact in this work we extend the sufficient
conditions for degradability for flagged channels and we find even better
bounds by exploiting richer flag structures, while being to reproduce the
bounds already obtained with approximate degradability techniques. This bounds
could in principle be obtained by approximate degradability, by extending the
parameter space of the flags, but in practice the search in a larger space
would make them hard to find.

The outline of the paper is the following: after the preliminaries Section 2,
we show the derivation of the sufficient conditions for degradability of
flagged channels in Section 3. We then apply this result to obtain a general
bound on the quantum capacity of any channel which is the convex combination
of a unitary channel and any other channel in Section 4. In Section 5 we
rewrite our sufficient conditions for Pauli channels, where the bounds on the
quantum capacity appear to have a simpler form, and we show that two explicit
choices of degradable extensions give state-of-the-art bounds for the quantum
and private capacities of the depolarizing channel and the BB84 channel. In
Section 6 we apply the bound to the generalized amplitude damping channel. In
Section 7 we add some observations about the possibility of getting even
better bounds with this method.

##  2 Preliminaries

We consider a finite dimensional Hilbert space $\mathcal{H}$, and we denote
the space of linear operators on $\mathcal{H}$ as $\mathcal{L(H)}$. Then a
quantum channel
$\Lambda:\mathcal{L}(\mathcal{H}_{A})\rightarrow\mathcal{L}(\mathcal{H}_{B})$
is a Completely Positive Trace Preserving (CPTP) map with input system $A$ and
output system $B$. Any CPTP map can be written in Kraus representation

$\Lambda[\rho]=\sum_{i=1}^{r}K_{i}\rho K^{\dagger}_{i}\,,$ (1)

for some collection of Kraus operators $\\{K_{i}\\}_{i=1,...,r}$ satisfying
the normalization condition $\sum_{i=1}^{r}K^{\dagger}_{i}K_{i}=I$.
Equivalently, we can cast any quantum channel in the Stinespring
representation

$\Lambda[\rho]=\tr_{E}(V\rho V^{\dagger})\,,$ (2)

where $V$ is an isometry from $\mathcal{H}_{A}$ to
$\mathcal{H}_{B}\otimes\mathcal{H}_{E}$, the system $E$ formally acting as the
channel environment. For any Stinespring dilation one can define a
complementary channel of $\Lambda$, that is a channel
$\tilde{\Lambda}:\mathcal{L}(\mathcal{H}_{A})\rightarrow\mathcal{L}(\mathcal{H}_{E})$
defined as

$\tilde{\Lambda}[\rho]=\tr_{B}(V\rho V^{\dagger})\,.$ (3)

The possible Stinespring dilations and the complementary channels are related
by an isometry between their respective environments. If there exist a channel
$W$ such that $W\circ\Lambda=\tilde{\Lambda}$, we say that $\Lambda$ is
degradable [20].

The quantum capacity of a channel $Q(\Lambda)$ is the maximum asymptotic rate
at which quantum information can be transmitted reliably using quantum channel
$\Lambda$. In Refs. [1, 2, 3] it was showed that such quantity can be computed
as

$Q(\Lambda)=\lim_{n\rightarrow\infty}\frac{1}{n}Q_{n}(\Lambda)\;,\qquad
Q_{n}(\Lambda)=\max_{\rho}I_{c}(\Lambda^{\otimes n},\rho)\;,$ (4)

where $I_{c}(\Lambda,\rho):=S(\Lambda(\rho))-S(\tilde{\Lambda}(\rho))$ is the
coherent information functional. Not only the maximization over all input
states of $n$ uses of channel is hard, since the number of parameters grows
exponentially with $n$, one also needs to solve the optimization problem for
any $n$ in order to use the regularized formula to evaluate the capacity.
However, for degradable channels, the coherent information is additive [38],
and we get a single letter formula for the quantum capacity.

We also consider the private capacity of a channel $P(\Lambda)$ [19, 20],
which is the maximum rate of classical communication protected from any
eavesdropper. This capacity also has a regularized expression

$P(\Lambda)=\lim_{n\rightarrow\infty}\frac{1}{n}P_{n}(\Lambda)\,,\qquad
P_{n}(\Lambda)=\max_{{\cal E}_{n}}\\{\chi(\Lambda^{\otimes n}({\cal
E}_{n}))-\chi(\tilde{\Lambda}^{\otimes n}({\cal E}_{n}))\\}\;,$ (5)

where now the maximization has to be performed over all possible ensembles
${\cal E}_{n}$ of $n$ channel uses, and where $\chi(\Lambda^{\otimes n}({\cal
E}_{n}))$ is the Holevo information. In general $P_{n}(\Lambda)\geq
Q_{n}(\Lambda)$ and therefore $P(\Lambda)\geq Q(\Lambda)$. For degradable
channels $P(\Lambda)=Q(\Lambda)=Q_{1}(\Lambda)$ [40], therefore any upper
bound we find on the quantum capacity via a degradable extension is also an
upper bound for the private capacity.

##  3 Sufficient conditions for degradability of flagged extensions

We outline a systematic construction of degradable flagged extensions for any
convex combination of channels, i.e. channels of the form
$\Lambda=\sum_{i=1}^{l}p_{i}\Lambda_{i}$, with $\\{p_{i}\\}_{i=1,\cdots l}$ a
probability set and with $\Lambda_{i}$ channels themselves. We establish the
following:

######  Proposition 3.1 (Sufficient conditions for degradability of flagged
extensions).

Let $\Lambda$ be a channel acting on the quantum system $A$ and its flagged
extension

$\mathbb{\Lambda}=\sum_{i=1}^{l}p_{i}\Lambda_{i}\otimes\ket{\phi_{i}}\bra{\phi_{i}}\,,$
(6)

with $|\phi_{i}\rangle$ normalized states of an auxiliary system $F$. The map
$\mathbb{\Lambda}$ is degradable if there exists a choice of Kraus operators
$\\{K_{j}^{(i)}\\}_{j=1,...,r_{i}}$ for each channel $\Lambda_{i}$ such that,
given an arbitrary orthonormal basis $\\{|i\rangle\\}_{i}$ for the space of
$F$, we have

$\bra{i^{\prime}}\ket{\phi_{i}}\sqrt{p_{i}}K^{(i^{\prime})}_{j^{\prime}}K^{(i)}_{j}=\bra{i}\ket{\phi_{i^{\prime}}}\sqrt{p_{i^{\prime}}}K^{(i)}_{j}K^{(i^{\prime})}_{j^{\prime}}\qquad\forall
i,j,i^{\prime},j^{\prime}\,.$ (7)

###### Proof.

Observe that starting from a Kraus set $\\{K_{j}^{(i)}\\}_{j=1,...,r_{i}}$ of
the channel $\Lambda_{i}$, we can construct the following isometric
Stinespring dilation for such channel,

$V_{i}\ket{\psi}_{A}:=\sum_{j=1}^{r_{i}}K^{(i)}_{j}\ket{\psi}_{A}\ket{i}_{B}\ket{j}_{\bar{B}},$
(8)

for all $|\psi\rangle_{A}$ states of $A$, with the systems $B$ and $\bar{B}$
playing the role of the effective channel environment. A Stinespring
representation of the flagged channel (6) can then be obtained as

$V\ket{\psi}_{A}:=\sum_{i=1}^{l}\sqrt{p_{i}}V_{i}\ket{\psi}_{A}\ket{\phi_{i}}_{F}\,,$
(9)

which, via Eq. (3) allows to express the complementary counterpart of the
flagged channel as

$\tilde{\mathbb{\Lambda}}[\ket{\psi}_{A}\bra{\psi}]=\sum_{i,j}\sqrt{p_{i}p_{j}}{{}_{F}\langle}{\phi_{j}}|{\phi_{i}}\rangle_{F}\tr_{A}(V_{i}\ket{\psi}_{A}\bra{\psi}V^{\dagger}_{j})\;.$
(10)

Our goal is to find a channel such as $W$ that degrades the flagged channel to
its complementary channel i.e.
$W\circ\mathbb{\Lambda}=\tilde{\mathbb{\Lambda}}$. A natural candidate for the
Stinespring representation of the degrading channel is as following

$V^{\prime}\ket{\psi}_{A}\ket{i}_{F}:=V_{i}\ket{\psi}_{A}\,.$ (11)

Consider hence the following state

$\displaystyle V^{\prime}V\ket{\psi}_{A}$
$\displaystyle=\sum_{i=1}^{l}\sqrt{p_{i}}V^{\prime}V_{i}\ket{\psi}_{A}\ket{\phi_{i}}_{F}=\sum_{i=1}^{l}\sum_{i^{\prime}=1}^{l}\sqrt{p_{i}}\bra{i^{\prime}}\ket{\phi_{i}}V_{i^{\prime}}V_{i}\ket{\psi}_{A}$
$\displaystyle=\sum_{i=1}^{l}\sum_{i^{\prime}=1}^{l}\sum_{j=1}^{r_{i}}\sum_{j^{\prime}=1}^{r_{i^{\prime}}}\bra{i^{\prime}}\ket{\phi_{i}}\sqrt{p_{i}}K^{(i^{\prime})}_{j^{\prime}}K^{(i)}_{j}\ket{\psi}_{A}\ket{i}_{B}\ket{j}_{\bar{B}}\ket{i^{\prime}}_{B^{\prime}}\ket{j^{\prime}}_{\bar{B^{\prime}}}\,,$
(12)

where for ease of notation $\bra{i^{\prime}}\ket{\phi_{i}}$ stands for
${{}_{F}\langle}{i^{\prime}}|{\phi_{i}}\rangle_{F}$. By construction the
states of subsystem $B\bar{B}$ and $B^{\prime}\bar{B^{\prime}}$ are equal to
$\tilde{\mathbb{\Lambda}}[\ket{\psi}\bra{\psi}]$ and
$W\circ\mathbb{\Lambda}[\ket{\psi}\bra{\psi}]$ respectively. Therefore, a
sufficient condition for the degradability of $\mathbb{\Lambda}$ is that
$V^{\prime}V\ket{\psi}_{A}$ is invariant if we swap subsystem $B\bar{B}$ with
$B^{\prime}\bar{B^{\prime}}$. Writing this condition we get

$\displaystyle\sum_{i=1}^{l}\sum_{i^{\prime}=1}^{l}\sum_{j=1}^{r_{i}}\sum_{j^{\prime}=1}^{r_{i^{\prime}}}\bra{i^{\prime}}\ket{\phi_{i}}\sqrt{p_{i}}K^{(i^{\prime})}_{j^{\prime}}K^{(i)}_{j}\ket{\psi}_{A}\ket{i}_{B}\ket{j}_{\bar{B}}\ket{i^{\prime}}_{B^{\prime}}\ket{j^{\prime}}_{\bar{B^{\prime}}}$
$\displaystyle=\sum_{i=1}^{l}\sum_{i^{\prime}=1}^{l}\sum_{j=1}^{r_{i}}\sum_{j^{\prime}=1}^{r_{i^{\prime}}}\bra{i^{\prime}}\ket{\phi_{i}}\sqrt{p_{i}}K^{(i^{\prime})}_{j^{\prime}}K^{(i)}_{j}\ket{\psi}_{A}\ket{i^{\prime}}_{B}\ket{j^{\prime}}_{\bar{B}}\ket{i}_{B^{\prime}}\ket{j}_{\bar{B^{\prime}}}\,$
$\displaystyle=\sum_{i=1}^{l}\sum_{i^{\prime}=1}^{l}\sum_{j=1}^{r_{i}}\sum_{j^{\prime}=1}^{r_{i^{\prime}}}\bra{i}\ket{\phi_{i^{\prime}}}\sqrt{p_{i^{\prime}}}K^{(i)}_{j}K^{(i^{\prime})}_{j^{\prime}}\ket{\psi}_{A}\ket{i}_{B}\ket{j}_{\bar{B}}\ket{i^{\prime}}_{B^{\prime}}\ket{j^{\prime}}_{\bar{B^{\prime}}}\,,$
(13)

which implies (7) hence leading to the thesis. These are sufficient conditions
for the degradability of $\mathbb{\Lambda}$. In this way we reduced the
degradability condition to commutation conditions on the Kraus operators,
which generalize the result that channels with commuting Kraus operators are
degradable [37].

∎

##  4 General applications

###  4.1 Convex combination of a unitary operation with an arbitrary channel

Consider a channel with one unitary Kraus operator as following

$\Lambda[\rho]=(1-p)U\rho U^{\dagger}+p\sum_{j=1}^{r}K_{j}\rho
K_{j}^{\dagger}\,,$ (14)

where $K_{i}$ are the other Kraus operators. As the quantum capacity is
invariant under unitary transformations, we can just assume that the first
Kraus is identity and redefine the rest. With a slight abuse of notation we
can write

$\displaystyle\Lambda[\rho]=(1-p)\rho+p\sum_{j=1}^{r}K_{j}\rho
K_{j}^{\dagger}=(1-p)\rho+p\Lambda_{1}[\rho].$ (15)

It is a convex combination of two channels, and we define the flagged
extension as following

$\mathbb{\Lambda}[\rho]=(1-p)\rho\otimes\ket{\phi_{0}}\bra{\phi_{0}}+p\Lambda_{1}[\rho]\otimes\ket{\phi_{1}}\bra{\phi_{1}}\,.$
(16)

Writing the degradability conditions in Eq. (7) we get

$\displaystyle\bra{1}\ket{\phi_{0}}\sqrt{1-p}=\bra{0}\ket{\phi_{1}}\sqrt{p}\;,\qquad\bra{1}\ket{\phi_{1}}K_{j}K_{j^{\prime}}=\bra{1}\ket{\phi_{1}}K_{j^{\prime}}K_{j}\,.$
(17)

Since $K_{j}$ operators do not need to commute, we set
$\bra{1}\ket{\phi_{1}}=0$ and if $p\geq 1/2$ we get the following solution

$\ket{\phi_{1}}=\ket{0},\qquad\ket{\phi_{0}}=\sqrt{\frac{1-2p}{1-p}}\ket{0}+\sqrt{\frac{p}{1-p}}\ket{1}\,.$
(18)

Surprisingly, without any assumption on the form of $\Lambda_{1}$ we found a
regime for which the flagged channel in Eq. (16) is degradable. Therefore, we
get the following upper bound

$Q(\Lambda)\leq Q(\mathbb{\Lambda})=Q_{1}(\mathbb{\Lambda})\,.$ (19)

On the other hand, if $K_{j}=\sqrt{p_{j}}U_{j}$ for some unitaries $U_{j}$,
with $\sum_{i=1}^{r}p_{j}=1-p$, then we can consider the mixed unitary flagged
channel,

$\Lambda[\rho]=(1-p)\rho\otimes\outerproduct{\phi_{0}}{\phi_{0}}+\sum_{j=1}^{r}p_{j}U_{j}\rho
U_{j}^{\dagger}\otimes\outerproduct{\phi_{j}}{\phi_{j}}\;,$ (20)

and any flag choice such that

$\displaystyle\innerproduct{i}{\phi_{j}}=0\,\,\,\mathrm{if}\,i\neq
0\,\mathrm{and}\,j\neq 0\,\mathrm{and}\,i\neq j\;,$ (21)

gives a degradable extension. For the depolarizing channel and BB84 channel we
compute the upper bounds from degradable extensions with exactly this flag
structure. However, the degradability conditions are more general than this
and we give a more specialized treatment to these channels in the following.

##  5 Pauli channels

In this section we concentrate on an important subclass of mixed unitary
channels: Pauli channels, which describe random bit flip and phase flip errors
in qubits. We also consider the generalization of Pauli channels for qudits.
For this class of channels the structure of degradable flagged extensions is
quite rich and the upper bounds can be made more explicit. The following
treatment of generalized Pauli channels follows the phase-space description of
finite dimensional quantum mechanics [41, 42, 43, 44, 45, 46].

###  5.1 Qubit Pauli group

We start by recalling the Pauli group of one qubit:

$\mathcal{P}:=\\{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\\},$ (22)

where

$I=\left(\begin{matrix}1&0\\\ 0&1\end{matrix}\right),\qquad
X=\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right),\qquad
Y=\left(\begin{matrix}0&-i\\\ i&0\end{matrix}\right),\qquad
Z=\left(\begin{matrix}1&0\\\ 0&-1\end{matrix}\right).$

The Pauli group of $n$ qubits is obtained as the tensor product of $n$ copies
of the Pauli group of one qubit
$\mathcal{P}^{n}:=\\{\otimes_{j=1}^{n}\omega_{j}|\omega_{j}\in\mathcal{P}\\}$.
For our purposes it suffices to consider
$\mathsf{P}^{n}:=\mathcal{P}^{n}/C^{n}$, the quotient of the Pauli group with
its center $C^{n}:=\\{\pm I^{\otimes n},\pm iI^{\otimes n}\\}$. Each element
of $\mathsf{P}^{n}$ can be identified by a couple of $n$ bit-strings $x=(q,p)$
according to the definition

$P_{(q,p)}:=i^{-p\cdot q}\otimes_{j=1}^{n}Z^{p_{j}}X^{q_{j}}.$ (23)

It is then immediate to see that for any two
$x=(q,p),y=(q^{\prime},p^{\prime})$ we have $P_{x}P_{y}=(-1)^{\langle
x,y\rangle}P_{y}P_{x}$, where

$\langle x,y\rangle=p\cdot q^{\prime}-q\cdot p^{\prime}\mod 2\;,$ (24)

and that $\Tr[P_{x}]=\delta_{x,0}$.

###  5.2 Qudit Pauli group

The generalization to qudits is straightforward. The generalization of the
Pauli group for one qudit, is the group $\mathcal{W}_{d}$ generated by $\tau
I$ ($\tau:=e^{\frac{(d^{2}+1)\pi i}{d}}$), and the Weyl-Heisenberg operators
$X,Z$ acting as

$X\ket{j}=\ket{j+1}\,\,\,\mod d\qquad Z\ket{j}=e^{j\frac{2\pi
i}{d}}\ket{j}\qquad\,\,j={0,...,d-1}.$ (25)

For several qudits, likewise we set
$\mathcal{W}^{n}_{d}:=\\{\otimes_{j=1}^{n}\omega_{j}|\omega_{j}\in\mathcal{W}_{d}\\}.$
The center of this group is still a set of multiples of the identity
$C_{d}^{n}=\\{\tau^{j}I^{\otimes n}:j=0,...,D-1\\}$, where $D=d$ if $d$ is
even and $D=2d$ if $d$ is odd; we define
$\mathsf{W}^{n}_{d}:=\mathcal{W}_{d}^{n}/C_{d}^{n}$. Each element of
$\mathsf{W}^{n}_{d}$ can be identified by a couple of $n$ Dit-strings
$x=(q,p)\in\mathbb{Z}^{2n}_{D}$ according to the definition

$W_{(q,p)}:=e^{-\frac{(d^{2}+1)\pi i}{d}(p\cdot
q)}\otimes_{j=1}^{n}Z^{p_{j}}X^{q_{j}}.$ (26)

By close inspection it holds that

$W_{x}W_{y}=e^{\frac{2\pi i}{d}\langle x,y\rangle}W_{y}W_{x},$ (27)

where now

$\langle x,y\rangle=p\cdot q^{\prime}-q\cdot p^{\prime}\mod D.$ (28)

Moreover, for any $x,z\in\mathbb{Z}^{2n}_{D}$ we have

$W_{x+dz}=(-1)^{(d+1)\langle x,z\rangle}W_{x},$ (29)

and $\Tr[W_{x}]=\delta_{x,0}$.

###  5.3 Flagged Pauli channels

Each element of $\mathsf{W}^{n}_{d}$ is unitary matrix, therefore it describes
a reversible evolution of the system of $n$ qudits. Pauli channels are defined
as convex combination of Pauli unitaries:

$\Phi_{\bf w}[\rho]=\sum_{x\in\mathbb{Z}_{d}^{2n}}w_{x}W_{x}\rho W_{x},$ (30)

where now it suffices to sum over $\mathbb{Z}^{2n}_{d}$ instead of
$\mathbb{Z}^{2n}_{D}$ because of Eq. (29), and they are parametrically
described by the probability distribution ${\bf w}(x)=w_{x}$. For these
channels, our sufficient conditions for degradability are less stringent than
in general, because of the relations (27) occurring for any couple of Pauli
unitaries. To construct the flagged version of these channels, we note that
the flags live, without loss of generality, in a Hilbert space
$\mathcal{H}_{F}$ of dimension $d^{2n}$, with computational basis
$\\{\ket{x}\\}_{x\in\mathbb{Z}_{d}^{2n}}$. We also consider the space
$\mathcal{H}_{C}\otimes\mathcal{H}_{F}$, with
$\mathcal{H}_{C}\cong\mathcal{H}_{F}$, and denote the partial trace with
respect to $\mathcal{H}_{F}$ as $\Tr_{F}[\cdot]$.

Now consider a state on on $\mathcal{H}_{C}\otimes\mathcal{H}_{F}$ of the form

$\ket{\Psi}=\sum_{x\in\mathbb{Z}_{d}^{2n}}\sqrt{w_{x}}\ket{x}_{C}\otimes\ket{\phi_{x}}_{F}=\sum_{x\in\mathbb{Z}_{d}^{2n},y\in\mathbb{Z}_{d}^{2n}}\sqrt{w_{x}}\innerproduct{y}{\phi_{x}}\ket{x}_{C}\otimes\ket{y}_{F},$
(31)

This state defines the following flagged version of $\Phi_{\bf w}$:

$\Phi_{\Psi}[\rho]=\sum_{x\in\mathbb{Z}_{d}^{2n}}w_{x}W_{x}\rho
W_{x}\otimes\outerproduct{\phi_{x}}{\phi_{x}}.$ (32)

We define the projectors $\Pi_{j}$ on $\mathcal{H}_{C}\otimes\mathcal{H}_{F}$
projecting on $\mathrm{span}\\{\ket{x}\ket{y}-e^{j\frac{2\pi
i}{d}}\ket{y}\ket{x}:\langle x,y\rangle=j\,\mathrm{mod}\,d\\}$. With these
definitions, we are equipped to establish the following proposition:

######  Proposition 5.1 (Upper bound on the quantum capacity of Pauli
channels).

For any state $\ket{\Psi}\in\mathcal{H}_{C}\otimes\mathcal{H}_{F}$ satifying

$\displaystyle\tr[\Pi_{j}\outerproduct{\Psi}{\Psi}]=0\,\,\,\forall
j\in\\{0,...,d-1\\}\qquad\tr[\outerproduct{x}{x}\otimes
I\outerproduct{\Psi}{\Psi}]=w_{x}\qquad\forall x\in\mathbb{Z}_{d}^{2n}.$ (33)

we have that the quantum capacity of $\Phi_{\bf w}$ satisfies

$Q(\Phi_{\bf w})\leq n\log d-S({\bf
w})+S(\Tr_{C}[\outerproduct{\Psi}{\Psi}]).$ (34)

In particular, the optimal upper bound is obtained by minimizing
$S(\Tr_{C}[\outerproduct{\Psi}{\Psi}])$ with the constraints (33).

###### Proof.

Given a channel of the form (32), consider a state on on
$\mathcal{H}_{C}\otimes\mathcal{H}_{F}$ of the form

$\ket{\Psi}=\sum_{x\in\mathbb{Z}_{d}^{2n}}\sqrt{w_{x}}\ket{x}_{C}\otimes\ket{\phi_{x}}_{F}=\sum_{x\in\mathbb{Z}_{d}^{2n},y\in\mathbb{Z}_{d}^{2n}}\sqrt{w_{x}}\innerproduct{y}{\phi_{x}}\ket{x}_{C}\otimes\ket{y}_{F},$
(35)

Imposing this structure on a bipartite state on
$\mathcal{H}_{C}\otimes\mathcal{H}_{F}$ is equivalent to the condition

$\tr[\outerproduct{x}{x}\otimes I\outerproduct{\Psi}{\Psi}]=w_{x}\qquad\forall
x\in\mathbb{Z}_{d}^{2n}.$ (36)

Moreover, the degradability conditions can be rewritten as

$\displaystyle\tr[\Pi_{j}\outerproduct{\Psi}{\Psi}]=0\,\,\,\forall
j\in\\{0,...,d-1\\}.$ (37)

$Q_{1}=Q$ has a very simple form for flagged degradable Pauli channels. By
concavity of coherent information for degradable channels [39], and the
covariance property of (flagged) Pauli channels $\Phi_{\Psi}\circ
W=W\circ\Phi_{\Psi}$ for all $W\in\mathcal{W}_{d}^{n}$, the maximum of
coherent information corresponds to the maximally mixed state, which is
purified by the maximally entangled state
$\ket{\Xi}=\frac{1}{d^{n/2}}\sum_{j=0,...,d-1}\ket{j}\otimes\ket{j}$. It holds
that $\bra{\Xi}W_{x}W_{y}\otimes
I\ket{\Xi}=\frac{1}{d^{n}}\tr[W_{x}W_{y}]=\delta_{x,y}$, therefore

$\displaystyle Q_{1}$ $\displaystyle=S\left(\Phi_{\bf
w}\left[\frac{I}{d^{n}}\right)\right]-S(\Phi_{\bf w}\otimes
I[\outerproduct{\Xi}{\Xi}])=n\log
d+S(\sum_{x\in\mathbb{Z}_{d}^{2n}}w_{x}\outerproduct{\phi_{x}}{\phi_{x}})-S({\bf
w})$ $\displaystyle=n\log d-S({\bf w})+S(\Tr_{C}[\outerproduct{\Psi}{\Psi}]).$
(38)

∎

Note that the minimization problem suggested by Proposition (5.1) is non-
convex, therefore it’s hard to treat numerically. Its solution is also not
unique in general. However, some useful upper bounds on the quantum capacity
can be obtained, case by case, by simply minimizing over families of states
which satisfy the constraints, but can be expressed in terms of a few
parameters. In this way we obtain state-of-the art results for the
depolarizing channel and the BB84 channel. As a side comment, the treatment in
this section does not seem to cover the flag choice of [21, 22] for the
depolarizing channel. However, this is easily amended by splitting the Kraus
operator proportional to the identity in Eq. (30) into two Kraus operators
with suitable probabilities, and assigning a different flag to each of them,
respecting the sufficient conditions for degradability.

###  5.4 Depolarizing channel

The depolarizing channel on one qudit is

$\Lambda_{p}^{d}[\rho]:=(1-\frac{d^{2}-1}{d^{2}}p)I\rho
I+\frac{p}{d^{2}}\sum_{x\in\mathbb{Z}_{d}^{2}\setminus\\{0\\}}W_{x}\rho
W_{x}\;.$ (39)

The symmetries of this channel causes some potential redundancies on the
states that achieves the optimal upper bound according to Proposition (5.1).
Consider the unitary operation $U_{\sigma}$ indexed by permutations $\sigma\in
S_{d^{2}-1}$ which act permuting the orthogonal set
$\\{\ket{x}\\}_{x\in\mathbb{Z}_{d}^{2}\setminus\\{0\\}}$ while leaving
$\ket{0}$ invariant. Then, for any state $\ket{\Psi}$ satisfying the
constraints, $U_{\sigma}\otimes U_{\sigma}\ket{\Psi}$ also satisfies the
constraints, and it has the same entanglement entropy
$S(\Tr_{C}[\outerproduct{\Psi}{\Psi}])=S(\Tr_{C}[U_{\sigma}\otimes
U_{\sigma}\outerproduct{\Psi}{\Psi}(U_{\sigma}\otimes
U_{\sigma})^{\dagger}])$. We cannot establish if the minimization problem has
a unique solutions, but if this was the case, then we could restrict the
candidate states to those which are invariant under $U_{\sigma}\otimes
U_{\sigma}$ for every $\sigma\in S_{d^{2}-1}$. We just take this fact as a
suggestion for a guess, and we minimize
$S(\Tr_{C}[\outerproduct{\Psi}{\Psi}])$ on this restricted family of states.
This is convenient because $S(\Tr_{C}[\outerproduct{\Psi}{\Psi}])$ can be
determined analytically and we can reduce the problem to a one parameter
minimization.

######  Proposition 5.2.

For $\ket{\Psi}$ satisfying $\ket{\Psi}=U_{\sigma}\otimes
U_{\sigma}\ket{\Psi}$, we can parametrize $\ket{\Psi}$ with three complex
variables $\alpha=\innerproduct{0,0}{\Psi}$, $\beta=\innerproduct{0,x}{\Psi}$
for $x\neq 0$, $\gamma=\innerproduct{x,x}{\Psi}$ for $x\neq 0$, and

$S(\Tr_{C}[\outerproduct{\Psi}{\Psi}])=-(d^{2}-2)|\gamma|^{2}\log(|\gamma|^{2})-v_{+}\log
v_{+}-v_{-}\log v_{-}$ (40)

with

$v_{\pm}=\frac{1}{2}(|\alpha|^{2}+|\gamma|^{2}+2|\beta|^{2}(d^{2}-1)\pm\sqrt{(|\alpha|^{2}-|\gamma|^{2})^{2}+4(d^{2}-1)|\beta|^{2}|\alpha+\gamma^{*}|^{2}})$
(41)

###### Proof.

From the constraints we have that
$\beta=\innerproduct{0,x}{\Psi}=\innerproduct{x,0}{\Psi}$, and from the action
of a permutation $U_{xy}$ that exchanges $x,y\neq 0$ we have
$\bra{0,x}U_{xy}\otimes U_{xy}\ket{\Psi}=\innerproduct{0,y}{\Psi}$. From the
constraints we have that $\innerproduct{x,y}{\Psi}=e^{-\frac{2\pi i}{d}\langle
x,y\rangle}\innerproduct{y,x}{\Psi}$ for $x\neq y\,,\,\,x,y\neq 0$, then
$\bra{{x,y}}{U_{xy}\otimes
U_{xy}\ket{\Psi}}=\innerproduct{y,x}{\Psi}=e^{-\frac{2\pi i}{d}\langle
x,y\rangle}\innerproduct{y,x}{\Psi}=0$. Also,
$\bra{{x,x}}\ket{\Psi}=\bra{{x,x}}U_{xy}\ket{\Psi}=\bra{{y,y}}\ket{\Psi}=\gamma$
when $x,y\neq 0$. This completes the parametrization. The eigenvalues of
$\Tr_{C}[\outerproduct{\Psi}{\Psi}]$ can be determined from the singular
eigenvalues of the matrix $M_{xy}$ of coefficients of
$\ket{\Psi}=\sum_{x\in\mathbb{Z}_{d}^{2n},y\in\mathbb{Z}_{d}^{2n},}M_{xy}\ket{x}_{C}\otimes\ket{y}_{F}$.
We have that the coefficients of $M^{\dagger}M$ are

$\displaystyle M^{\dagger}M_{0,0}=|\alpha|^{2}+|\beta|^{2}(d^{2}-1)\qquad
M^{\dagger}M_{0,x}=\alpha\beta^{*}+\beta\gamma^{*},\,\,x\neq 0$ (42)
$\displaystyle\qquad M^{\dagger}M_{x,y}=|\beta|^{2},\,\,x\neq y,\,x,y\neq
0\qquad M^{\dagger}M_{x,x}=|\beta|^{2}+|\gamma|^{2},\,\,x\neq 0\;.$ (43)

Then $M^{\dagger}M-|\gamma|^{2}I$ has rank $2$ and the nonzero eigenvalues can
be determined by solving a quadratic equation.

∎

######  Proposition 5.3.

For $\ket{\Psi}$ satisfying $\ket{\Psi}=U_{\sigma}\otimes
U_{\sigma}\ket{\Psi}$, the minimization of
$S(\Tr_{C}[\outerproduct{\Psi}{\Psi}])$ is a one-parameter minimization
problem.

###### Proof.

From the expression of $S(\Tr_{C}[\outerproduct{\Psi}{\Psi}])$ in Eq. (40) and
from Eq. (41) it’s evident that the result does not depend on the phases of
$\alpha$, $\beta$ and $\gamma$ except for the term $|\alpha+\gamma^{*}|$,
which should be maximized. This happen without loss of generality if $\alpha$
and $\gamma^{*}$ are real and positive. Then the two constraints
$|\alpha|^{2}+(d^{2}-1)|\beta|^{2}=(1-\frac{d^{2}-1}{d^{2}}p)$ and
$|\beta|^{2}+|\gamma|^{2}=\frac{p}{d^{2}}$ eliminate the remaining two
parameters. ∎

The bound $Q_{\mathrm{fmin}}$ obtained from this one parameter minimization
can be combined with the no-cloning bound [24, 26, 28, 30]

$Q(\Lambda_{p}^{d})\leq\big{(}1-\tfrac{2p(d+1)}{d}\big{)}\log{d}\>.$ (44)

using the fact that the convex hull of upper bounds from degradable extensions
of the depolarizing channel is itself an upper bound [28, 30]. A comparison
between the most competitive upper bounds for $d=2$ is shown in Figure 1,
where we can see that the bound we obtained outperforms all previous bounds in
the whole parameter region. An improvement with respect to previous bounds can
be obtained also for generic $d$, and we show as an example the bound for
$d=4$ in Figure 2.

Figure 1: Bounds on the quantum capacity of the depolarizing channel for
$d=2$. Here $Q_{\mathrm{conv}}$ is the convex hull of the available upper
bounds from degradable extensions, $Q_{\mathrm{fmin}}$ is the new upper bound,
and $Q_{1}$ is the lower bound given by the coherent information of one use of
the channel. $Q_{LDS}$ is the bound from [33] and $Q_{W}$ is the bound from
[22]. Figure 2: Bounds on the quantum capacity of the depolarizing channel for
$d=4$. Here $Q_{\mathrm{conv}}$ is the convex hull of the available upper
bounds from degradable extensions, $Q_{\mathrm{fmin}}$ is the new upper bound,
and $Q_{1}$ is the lower bound given by the coherent information of one use of
the channel. $Q_{FKG}$ is the bound from [21] for pure flags evaluated at the
flag overlap allowed by the new degradability conditions.

###  5.5 BB84 channel

In this section we consider the channel that describes the famous quantum key
distribution protocol by Bennett and Brassard [47]. In its general form the
channel is

$\displaystyle
B_{p_{X},p_{Z}}[\rho]=(1-p_{X}-p_{Z}+p_{X}p_{Z})\rho+(p_{X}-p_{X}p_{Z})X\rho
X+(p_{Z}-p_{Z}p_{X})Z\rho Z+p_{X}p_{Z}Y\rho Y,$ (45)

As in [28] and [22] we restrict to the case $p_{X}=p_{Z}=p$. The flagged
extension we consider is

$\displaystyle B_{p,\Psi}[\rho]$
$\displaystyle=(1-p)^{2}\rho\otimes\outerproduct{\phi_{0}}{\phi_{0}}+p(1-p)X\rho
X\otimes\outerproduct{\phi_{1}}{\phi_{1}}+p(1-p)Z\rho
Z\outerproduct{\phi_{2}}{\phi_{2}}$ $\displaystyle+p^{2}Y\rho
Y\otimes\outerproduct{\phi_{3}}{\phi_{3}}\,.$ (46)

We choose the following parametrization for the flags

$\displaystyle\ket{\phi_{0}}$
$\displaystyle=\sqrt{1-2\alpha^{2}-\beta^{2}}\ket{0}+\alpha\ket{1}+\beta\ket{2}+\alpha\ket{3}$
$\displaystyle\ket{\phi_{1}}$
$\displaystyle=a\ket{0}+\sqrt{1-a^{2}-\gamma^{2}}\ket{1}-\gamma\ket{3}$
$\displaystyle\ket{\phi_{2}}$
$\displaystyle=a\ket{0}+\sqrt{1-a^{2}-\gamma^{2}}\ket{2}-\gamma\ket{3}$
$\displaystyle\ket{\phi_{3}}$
$\displaystyle=b\ket{0}+c\ket{1}+c\ket{2}+\sqrt{1-b^{2}-2c^{2}}\ket{3}\,,$
(47)

where the degradability conditions in Eq. (7) imply that
$\alpha=a\sqrt{\frac{p(1-p)}{(1-p)^{2}}}$, $\beta=\frac{bp}{1-p}$ and
$\gamma=c\sqrt{\frac{p}{1-p}}$. This is not the most general parametrization
for the flags, however because of the symmetry between the bit flip and phase
flip error in Eq. (45) we chose this parametrization. Any set of flags in the
form of Eq. (5.5) will result to a degradable extension of BB84 channel.
Therefore, to get the best upper bound for the quantum capacity or private
capacity of BB84 we should minimize the coherent information of its flagged
channel with respect to three free parameters $a,b,c$. We have compared the
result of the optimization problem with the previous bounds in FIgure 3. The
bound in [22] by Wang can be reproduced in our framework just by choosing
$a=b=1,\,c=0$.

Figure 3: Bounds on the quantum and private capacity of BB84 channel. $Q_{1}$
is the coherent information of BB84 channel. $Q_{fmin}$ is the new upper bound
obtained by the degradable extension. $Q_{W}$ is the upper bound obtained in
[22]. $Q_{SS}$ is the upper bound derived in [28].

##  6 Generalized amplitude damping

In this section we consider a bound on the quantum capacity of the generalized
amplitude damping (GAD) channel, which is a model of thermal loss on a qubit,
relevant for quantum superconducting processors [49]. The generalized
amplitude damping channel can be written as

$\mathcal{A}_{y,N}[\rho]=N\,\mathcal{A}_{y}[\rho]+(1-N)\,X\circ\mathcal{A}_{y}\circ
X[\rho],$ (48)

where $\mathcal{A}_{y,N}$ is the conventional amplitude damping channel, with
Kraus operators $K_{1}=(\outerproduct{0}{0}+\sqrt{1-y}\outerproduct{1}{1})$
and $K_{2}=\sqrt{y}\outerproduct{1}{0}$. While $\mathcal{A}_{y}$, and
$X\circ\mathcal{A}_{y}\circ X$ are degradable and their quantum capacity can
be computed [48], their convex combination is not, and its quantum capacity is
not determined. Previous upper bounds have been obtained by [35, 36, 34, 22].
In particular [22] used the following flagged extension together with
approximate degradability to get the tightest bound available:

$\mathcal{A}_{y,N}^{F}[\rho]=N\,\mathcal{A}_{y}[\rho]\otimes\outerproduct{0}{0}+(1-N)\,X\circ\mathcal{A}_{y}\circ
X[\rho]\otimes\outerproduct{1}{1},$ (49)

In fact this extension is exactly degradable: the output of a complementary
channel is

$\tilde{\mathcal{A}}_{y,N}^{F}[\rho]=N\,\tilde{\mathcal{A}}_{y}[\rho]\otimes\outerproduct{0}{0}+(1-N)\,\tilde{X\circ\mathcal{A}_{y}\circ
X}[\rho]\otimes\outerproduct{1}{1},$ (50)

and if the degrading map of $A_{y}$ is $W_{y}$, we have
$\tilde{\mathcal{A}}_{y,N}^{F}[\rho]=(W_{y}\otimes\outerproduct{0}{0}+X\circ
W_{y}\circ X\otimes\outerproduct{1}{1})\circ\mathcal{A}_{y,N}^{F}[\rho]$. The
quantum capacity of this extension can be evaluated to be
$Q({\mathcal{A}}_{y,N}^{F})\leq(1-N)I_{c}(A_{y},\rho)+NI_{c}(X\circ A_{y}\circ
X,\rho)=Q(A_{y})$. This simple bound seem to have not been pointed out
previously, and the actual quantum capacity $Q({\mathcal{A}}_{y,N}^{F})$ is
very close to it. Moreover, the structure of the GAD is such that one can get
better bounds from different degradable extensions, where we adapt the
argument by [28] on the depolarizing channel:

######  Proposition 6.1 (Combining bounds of degradable extensions of GAD).

For any collection of degradable extensions $\mathcal{A}_{y,N}^{ext,i}$,
$i=1,...,l$, for any $y_{0}$ the quantum capacity of $\mathcal{A}_{y_{0},N}$
is upper bounded by the convex hull of $Q(\mathcal{A}_{y_{0},N}^{ext,i})$,
$i=1,...,l$, as functions of the variable $N$.

###### Proof.

For any $N_{1},N_{2}$ such that $N=qN_{1}+(1-q)N_{2}$,
$1-N=1-qN_{1}+(1-q)N_{2}=q(1-N_{1})+(1-q)(1-N_{2})$, we have

$\mathcal{A}_{y,N}[\rho]=q(N_{1}\,\mathcal{A}_{y}[\rho]+(1-N_{1})\,X\circ\mathcal{A}_{y}\circ
X[\rho])+(1-q)(N_{2}\,\mathcal{A}_{y}[\rho]+(1-N_{2})\,X\circ\mathcal{A}_{y}\circ
X[\rho])$ (51)

If $\mathcal{A}_{y,N_{1}}^{ext,i}$ and $\mathcal{A}_{y,N_{2}}^{ext,j}$ are
degradable extensions of $\mathcal{A}_{y,N_{1}}$ and $\mathcal{A}_{y,N_{2}}$
respectively, then
$q\mathcal{A}_{y,N_{1}}^{ext,i}\otimes\outerproduct{0}{0}+(1-q)\mathcal{A}_{y,N_{2}}^{ext,j}\otimes\outerproduct{1}{1}$
is a degradable extension of $\mathcal{A}_{y,N}$ with quantum capacity less
than
$qQ(\mathcal{A}_{y,N_{1}}^{ext,i})+(1-q)Q(\mathcal{A}_{y,N_{2}}^{ext,j})$.

∎

In addition to the extension proposed by [22], we find two other degradable
extensions using the results of this paper. The first is obtained observing
that the following set is also a good choice of Kraus operators

$\displaystyle A_{1}$
$\displaystyle=\sqrt{N(1-N)}(\sqrt{1-y}+1)(\outerproduct{0}{0}+\outerproduct{1}{1})=\sqrt{N(1-N)}(\sqrt{1-y}+1)I$
(52) $\displaystyle A_{2}$ $\displaystyle=\sqrt{(1-N)y}\outerproduct{1}{0}$
(53) $\displaystyle A_{3}$
$\displaystyle=((1-N)-N\sqrt{1-y})\outerproduct{0}{0}+((1-N)\sqrt{1-y}-N)\outerproduct{1}{1}$
(54) $\displaystyle A_{4}$ $\displaystyle=\sqrt{Ny}\outerproduct{0}{1}$ (55)

We notice that $A_{1}$ is a rescaled unitary operator, therefore we can
directly apply the bound of Eq. (19) with $(1-p)=N(1-N)(\sqrt{1-y}+1)^{2}$.
This bound is applicable if $N(1-N)(\sqrt{1-y}+1)^{2}>1/2$. Moreover, at
$N=1/2$, the GAD becomes a Pauli channel:

$\displaystyle\mathcal{A}_{y,0.5}(\rho)$
$\displaystyle=\frac{1-y/2+\sqrt{1-y}}{2}\rho+\frac{1-y/2-\sqrt{1-y}}{2}Z\rho
Z+\frac{y}{4}(Y\rho Y+X\rho X)$ (56)

and we get a more refined bound $Q_{\mathrm{fmin}}(y)$, using the techniques
of the previous sections, in particular with the same flag structure of BB84
Eq. (5.5). Putting all together, we observe that the bound by [22] remains the
best one at high $y$, but at low $y$ it is beaten by the following bound
allowed by the convex hull argument:

$Q_{\mathrm{conv}}(y,N)=2NQ_{\mathrm{fmin}}(y)+(1-2N)Q(\mathcal{A}_{y})$ (57)

and using the full convex hull bound does not give substantial improvements.
We plot the results in Figure 4.

Figure 4: Bounds on the quantum capacity of the GAD, for three values of $N$.
$Q_{1}$ is the lower bound given by the coherent information of one use of the
GAD. $Q_{\mathrm{conv}}$ is the new upper bound, $Q_{W}$ is the upper bound
obtained by Wang [22]. Previous upper bounds [35, 36, 34, 22] are worse and
not plotted (see [22]).

##  7 Discussion

In the examples we provided we did not try to numerically optimize in the
whole parameter region allowed by the sufficient conditions for degradability.
Indeed, the minimization of the upper bound is not a convex optimization
problem and would require brute force search, but there are already many
parameters for Pauli channels and $d=2$, $n=1$. However, we stress the fact
that the family of upper bounds for the quantum and private capacity of a
channel $\Lambda$ is even larger in principle, as one can consider the flagged
extension of $\Lambda^{\otimes k}$: a degradable flagged extension of
$\Lambda$ gives also a degradable flagged extension of $\Lambda^{\otimes k}$
but the converse is not true. We tried to search for a better upper bound of
the quantum capacity of the depolarizing channel with two uses, by restricting
brute force search to certain parametrizations of the flags, but the attempts
we made did not show anything better than the $k=1$ bound. It is desirable to
further investigate if $k=1$ gives already the best upper bound or a
phenomenon similar to superadditivity shows up for flagged extensions.
Moreover, the extensions we obtained are explicitly dependent on the Kraus
representation chosen, which is not unique. We did not find a way to identify
an optimal choice of Kraus operators. Since for any channel the number of
Kraus operators can be increased arbitrarily by a suitable isometry, it is
conceivable that one needs to look at an unbounded space of flags to optimize
the upper bound of degradable extensions. As a side note, we point out that
the mixed flags extensions as considered in [21] can be treated in the
formalism of this paper, just by working with a proper choice of Kraus
operators, e.g. $K\rho
K^{\dagger}\otimes(q\outerproduct{0}{0}+(1-q)\outerproduct{1}{1})=\sqrt{q}K\rho\sqrt{q}K^{\dagger}\otimes\outerproduct{0}{0}+\sqrt{1-q}K\rho\sqrt{1-q}K^{\dagger}\otimes\outerproduct{1}{1}$
with the new Kraus operators $\sqrt{q}K$ and $\sqrt{1-q}K$.

##  8 Conclusions

We have introduced a method to construct degradable extension of quantum
channels which can be written as the convex sum of other channels. This method
is of general applicability, and we showed that it gives state-of-the-art
upper bounds on the quantum and private capacity of two important Pauli
channels, the depolarizing channel and the BB84 channel, and of the
generalized amplitude damping channel. By virtue of its simplicity, we believe
it can be used with success for many other channels too. The method could in
principle give better bounds by considering different Kraus representations
and flagged extensions of several uses of a channel, and it would be
interesting to study its limitations.

## Acknowledgments

We thank Felix Leditzky, Matteo Rosati and Xin Wang for valuable comments. We
acknowledge support from PRIN 2017 “Taming complexity with quantum
strategies”.

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$\displaystyle=\text{sigmoid}\left(\sum_{j=1}^{J}-\beta_{d}^{3}\langle\boldsymbol{w}_{j,\boldsymbol{\epsilon}}^{(t)},\boldsymbol{v}_{d}\rangle^{3}-\beta_{e}^{3}\langle\boldsymbol{w}_{j,\boldsymbol{\epsilon}}^{(t)},\boldsymbol{v}_{e}\rangle^{3}\right).$
(134)

The proof is the same as (Deng et al., 2023)[Lemma G.4].

##  Appendix C Additional Experimental Settings

###  C.1 Datasets and Training Details

Datasets. The CIFAR10 dataset (Krizhevsky et al., 2009) consists of 60,000 32
× 32 color images in 10 classes, with 6000 images per class. The CIFAR100
dataset (Krizhevsky et al., 2009) is just like the CIFAR10, except it has 100
classes containing 600 images each. For both of these datasets, the training
set has 50,000 images (5,000 per class for CIFAR10 and 500 per class for
CIFAR100) with the test set having 10,000 images. CINIC10 (Darlow et al.,
2018) represents an image classification dataset consisting of 270,000 images,
which is 4.5 times larger than CIFAR10. The dataset is created by merging
CIFAR10 with images extracted from the ImageNet database, specifically
selecting and downsampling images from the same 10 classes present in CIFAR10.
Tiny-ImageNet (Le & Yang, 2015) comprises 100,000 images distributed across
200 classes of ImageNet (Deng et al., 2009), with each class containing 500
images. These images have been resized to 64×64 dimensions and are in color.
The dataset consists of 500 training images, 50 validation images, and 50 test
images per class. The STL10 dataset (Coates et al., 2011) includes 5000 96x96
training labeled images, 500 per CIFAR10 class. The test set consists of 800
images per class, this counts up to 8,000 images in total.

Training on different datasets. Follow the setting from (Andriushchenko &
Flammarion, 2022; Andriushchenko et al., 2023), we trained Pre-Activation
ResNet18 on all datasets except for CIFAR100 which was trained with ResNet34.
We trained our models for 200 epochs with a batch size of 128 and used basic
data augmentations such as random mirroring and random crop. We used SGD with
the momentum parameter of 0.9 and set weight decay to 0.0005. We also fixed
$\rho=0.1$ for SAM unless further specified. For all datasets, we used a
learning rate schedule where we set the initial learning rate to 0.1. The
learning rate is decayed by a factor of 10 after 50% and 75% epochs, i.e., we
set the learning rate to 0.01 after 100 epochs and to 0.001 after 150 epochs.

Training with different architectures. We used the same training procedures
for Pre-Activation ResNet18, VGG19, and DenseNet121. We directly used the
official Pytorch (Paszke et al., 2019) implementation for VGG19 and
DenseNet121. For 3-layer MLPs, we used a hidden size of 512 with a dropout of
0.1 to avoid overfitting and set $\rho=0.01$. For ViT-S (Yuan et al., 2021),
we adopted a Pytorch implementation at https://github.com/lucidrains/vit-
pytorch. In particular, the hidden size, the depth, the number of attention
heads, and the MLP size are set to 768, 8, 8, and 2304, respectively. We
adjusted the patch size to 4 to fit the resolution of CIFAR10 and set both the
initial learning rate and $\rho$ to 0.01.

Computational resources. Each model is trained on 1 NVIDIA RTX A5000 GPU.

###  C.2 Other Implementation Details

When to separate the examples? We selected the best-separating epoch $t$ in
the set of $\\{4,5,6,7,8,10\\}$ for CIFAR10 and $\\{12,14,16,18,20,22\\}$ for
CIFAR100. Particularly, we separated examples of CIFAR10 at around epoch 8
while that of CIFAR100 is near epoch 20. Near this point, the gain in training
error diminishes significantly as shown in Figure 11(a), which shows a sign
that the model successfully learns easy features. In addition, we reported the
results for different separating epochs in Appendix D.6.

Forgetting score. To compute forgetting scores of training examples in each
dataset, we collected the same statistics as in (Toneva et al., 2018) but
computed at the end of each epoch. The reason is to make the statistics
consistent between two versions of the same difficult example which is
repeated in the upsampled dataset.

Hessian spectra. We approximated the density of the Hessian spectrum using the
Lanczos algorithm (Papyan, 2018; Ghorbani et al., 2019). The Hessian matrix is
approximated by 1000 examples (100 per class of CIFAR10). Then we extract the
top eigenvalues to calculate the maximum Hessian eigenvalue
$(\lambda_{\text{max}})$ and the bulk of spectra
$(\lambda_{\text{max}}/\lambda_{5})$ (Jastrzebski et al., 2020).

##  Appendix D Additional Results

Figure 6: The gap between contribution of easy and difficult features towards
the model output in SAM and GD. The toy datasets is generated from the
distribution in Definition 3.1 with $\beta_{d}=\beta_{e}=\alpha=1$.

###  D.1 An Extreme Case of Toy Datasets

We consider an extreme setting when two features have identical strength and
no missing easy features, i.e., $\beta_{e}=\beta_{d}=\alpha=1$, the gap
between LHS and RHS in Equation 11 is not small as shown in the figure 6. The
gap is consistently around 0.2-0.3 from epoch 250 onwards.

###  D.2 USEFUL is Useful for MLP

Figure 7: Test classification errors of 3-layer MLP on CIFAR10. The number
below each bar indicates the approximated cost to train the model and the tick
on top shows the standard deviation over three runs. USEFUL improves the
performance of SGD and SAM when training with 3-layer MLP. Figure 8:
Forgetting scores for training ResNet18 on CIFAR10. Forgetting scores measure
the learning difficulty of examples in training data. USEFUL approaches the
training dynamics of SAM, with more examples being forgotten infrequently and
fewer examples being forgotten frequently.

Figure 7 shows that for 3-layer MLP, USEFUL successfully reduces the test
error of both SGD and SAM by nearly 1%. Additionally, SGD+USEFUL yields better
performance than SAM alone.

###  D.3 SAM & USEFUL Reduce Forgetting Scores

We used forgetting scores (Toneva et al., 2018) to partition examples in
CIFAR10 into different groups. Forgetting scores count the number of times an
example is misclassified after being correctly classified during training and
is an indicator of the learning-difficulty of examples. Figure 8 illustrates
that SGD+USEFUL and SAM have fewer examples with high forgetting scores than
SGD does. This aligns with our theoretical analysis in Theorem 3.4 and results
on the toy datasets. By upsampling difficult examples in the dataset, they
contribute more to learning and hence SGD+USEFUL learns difficult features
faster than SGD.

Table 4: Test classification errors for training SAM and ASAM on the original
CIFAR10 and modified datasets by USEFUL. Results are averaged over 3 seeds.

| SAM | ASAM
---|---|---
+ |  4.23 $\pm$ 0.08 |  4.33 $\pm$ 0.19
\+ USEFUL | 4.04 $\pm$ 0.06 | 4.09 $\pm$ 0.10
\+ TA |  4.06 $\pm$ 0.08 |  3.93 $\pm$ 0.11
\+ TA + USEFUL | 3.49 $\pm$ 0.09 | 3.46 $\pm$ 0.01

Figure 9: Comparing test classification errors on Waterbirds. The number below each bar indicates the approximated cost to train the model and the tick on top shows the standard deviation over three runs. USEFUL boosts the performance of SGD and SAM on the balanced test set, showing its generalization to the OOD setting. In addition, the success of USEFUL in fine-tuning reveals its new application to the transfer learning setting.  Table 5: Comparison between USEFUL and upweighting loss regarding test classification errors. Upweighting loss doubled the loss for all examples in the difficult clusters found by USEFUL, which is different from dynamically upweighting examples during the training (Zhai et al., 2022). Results are averaged over 3 seeds. Method | SGD | SAM
---|---|---
| CIFAR10 | CIFAR100 | CIFAR10 | CIFAR100
Upweighting loss |  5.33 $\pm$ 0.09 |  26.70 $\pm$ 3.25 |  4.28 $\pm$ 0.02 |  21.75 $\pm$ 0.25
USEFUL | 4.79 $\pm$ 0.05 | 22.58 $\pm$ 0.08 | 4.04 $\pm$ 0.06 | 20.66 $\pm$ 0.44

###  D.4 USEFUL generalizes to other SAM variants.

In this experiment, we show that SAM can also generalize to other variants of
SAM. We chose ASAM, which is proposed by Kwon et al. to address the
sensitivity of parameter re-scaling (Dinh et al., 2017). Following the
recommended settings in ASAM, we trained it with a perturbation radius
$\rho=1.0$, which is 10 times that of SAM. Other settings are identical to the
standard settings in Appendix C. Table 4 demonstrates the results for training
ResNet18 on CIFAR10. Both SAM and ASAM can be combined with USEFULto improve
the test classification error. When using TA, ASAM shows a slightly better
performance than SAM.

###  D.5 USEFUL generalizes to OOD

While our main contribution is providing a novel and effective method to
improve the in-distribution generalization performance, we conducted new
experiments confirming the benefits of our method to improving out-of-
distribution (OOD) generalization performance. As a few very recent works (Cha
et al., 2021; Wang et al., 2023) showed the benefit of SAM in improving OOD
performance, it is expected that USEFUL also extend to this setting. As can be
seen in Figure 9, both SAM and USEFUL effectively improve the performance on
the balanced test set though the model is trained on the spurious training
set. When training ResNet18 with SGD from scratch, USEFUL decreases the
classification errors by 5.8%. In addition, it can be successfully applied to
the pre-trained ResNet50 (on ImageNet), opening up a promising application to
the transfer learning setting.

Table 6: Test classification errors of SGD for different partition methods. Results are averaged over 3 seeds. Partition method | CIFAR10 | CIFAR100
---|---|---
Quantile |  5.27 $\pm$ 0.10 |  23.49 $\pm$ 0.82
Misclassification |  4.98 $\pm$ 0.17 |  23.86 $\pm$ 0.70
USEFUL | 4.79 $\pm$ 0.05 | 22.58 $\pm$ 0.08
Figure 10: Alation studies of training ResNet18 on CIFAR10. In each
experiment, we used the standard training settings while (left) varying
training batch size or (middle) varying learning rate, or (right) varying
upsampling factor.

(a) Training error trajectories

(b) Test classification error

Figure 11: Separating epoch analysis. (left) Red lines indicate our optimal
choice of $t$ to separate examples at and restart training. The early epoch
that can best separate the examples is when the change in training error
starts to shrink. (right) Red points indicate our optimal choice of $t$.

###  D.6 Ablation studies

USEFUL vs Upweighting loss. We compare USEFUL with upweighting loss of
examples in the difficult clusters. As can be seen in Table 5, when coupling
with either SGD or SAM, USEFUL clearly outperforms upweighting loss on both
CIFAR datasets. It is worth mentioning that upweighting loss is different from
iteratively importance sampling methods such as GRW (Zhai et al., 2022), which
dynamically upweights examples during the training by factors that depend on
the loss value. In addition, GRW is dedicated to the distribution shift
setting while our paper considers the in-distribution setting.

Data selection method. In this experiment, we compare clustering with other
methods for partitioning data. The first baseline is to upsample misclassified
examples (Misclassification) while the second baseline is to upsample all
examples whose training errors are larger than the median value (Quantile).
All the three methods are performed at the same epoch $t$. Table 6 shows that
USEFUL selects a better set of upsampling examples, leading to the best model
performance.

Training batch size. Figure 10 left shows the gap between USEFULand SGD when
changing the batch size. Our method consistently improves the performance,
proving its effectiveness is not simply captured by the gradient variance
caused by the training batch size.

Learning rate. The small learning rate is a nuance in our theoretical results
to guarantee that easy and difficult features are separable in early training.
In general, a small learning rate is required for most theoretical results on
gradient descent and its convergence and is a standard theoretical assumption
(Roux et al., 2012; Schmidt et al., 2017; Wen et al., 2022; Chen et al.,
2023). In practice, both for separating easy vs difficult examples and for
training on the upsampled data, we used the standard learning rate that
results in the best generalization performance for both SGD and SAM following
prior work (Kwon et al., 2021; Zheng et al., 2021; Andriushchenko &
Flammarion, 2022). While the theoretical requirement for the learning rate is
always smaller than the one that is used in practice, empirically a larger
learning rate does not yield better generalization for the problems we
considered in contrast to other settings (Li et al., 2019; Puli et al., 2024).
As shown in Figure 10 middle, increasing the learning rate has an adverse
effect on the model performance. Indeed, for a fair comparison, the algorithms
should be trained with hyperparameters that empirically yield the best
performance; otherwise, the conclusions are not valid. Nevertheless, USEFUL
always improves the test error across different learning rates.

Upsampling factor. We empirically found the upsampling factor of 2 to
consistently work well across different datasets and architectures. Using
larger upsampling factors results in a too-large discrepancy between the
training and test distribution and does not work better, as is expected and
discussed in Section 1. As illustrated in Figure 10 right, while all factors
from 2 to 5 bring performance improvement, the upsampling factor of 2 yields
the best performance, as it reduces the simplicity bias with minimum change to
the rest of the data distribution.

Separating epoch. Fig 11(a) shows that the early epoch that can best separate
the examples is when the change in training error starts to shrink. At this
time, examples that are learned can be separated from the rest by clustering
model outputs as analyzed in our theoretical results in Section 3.3. Figure
11(b) demonstrates the performance of USEFUL when separating examples at
different epochs in early training. Too early or too late epochs do not
cluster examples well, i.e., some examples with easy features fall into the
difficult clusters and vice versa. This ablation study shows that upsampling
correct examples and with enough amount is important for our method to achieve
its best.


# Improved constraints for axion-like particles from 3-photon events at
$e^{+}e^{-}$ colliders

Aleksandr Pustyntsev  Institut für Kernphysik and $\text{PRISMA}^{+}$ Cluster
of Excellence, Johannes Gutenberg Universität, D-55099 Mainz, Germany  Marc
Vanderhaeghen  Institut für Kernphysik and $\text{PRISMA}^{+}$ Cluster of
Excellence, Johannes Gutenberg Universität, D-55099 Mainz, Germany

(August 28, 2024)

###### Abstract

Axions and axion-like particles (ALPs) are one of the most widely discussed
extensions of the Standard Model when it comes to the strong CP problem and
dark matter candidates. Current experiments are focused on the indirect
searches of invisible pseudoscalars in a wide parameter range. In this paper
we investigate limits on ALP mass, and its couplings to photons and leptons
from 3-photon annihilation at $e^{+}e^{-}$ colliders. We provide detailed
calculations and apply them to the particular kinematics of the Belle II
experiment, covering the ALP mass range from few hundred MeV to around 10 GeV.
Our results, which improve upon previous analyses by also including the ALP
coupling to electrons, show that such future analyses will allow to
significantly extend the ALP search range and impose much more stringent
restrictions on their couplings.

Axions and ALPs and $e^{+}e^{-}$ experiments

##  I Introduction

Initially proposed in 1977, the Peccei-Quinn theory so far is considered to be
the most compelling strong CP problem resolution Peccei:1977hh ; Peccei:1977ur
. In this model a CP-violating phase is dynamically driven to zero, giving
rise to a new pseudoscalar particle called axion Weinberg:1977ma ;
Wilczek:1977pj . During the last four decades numerous attempts have been made
to find a signal of this particle, including both lab searches and
astronomical observations Graham:2015ouw ; Irastorza:2018dyq . Current
constraints show that the QCD axion (in case it exists) must be very weakly
interacting and thus is called "invisible", which forces one to concentrate on
the possible indirect detection of this particle Rosenberg:2000wb ;
Sikivie:2020zpn .

A key property of the QCD axion is the linear proportionality between its
couplings to the Standard Model particles and the axion mass DiLuzio:2020wdo .
The exact relation being model-dependent and usually refers to KSVZ
PhysRevLett.43.103 ; Shifman:1979if  or DFSZ mechanisms Dine:1981rt ;
Zhitnitsky:1980tq . With the current limits taken into account, both scenarios
result in a very small axion mass, $m_{a}\lesssim 10^{-3}$ eV Peccei:2006as .
However, recent studies reopen the possibility of MeV mass range for the QCD
axion not yet excluded by experiments Alves:2017avw . Given the significance
of the problem it is important to assure that there are no loopholes left at
this scale and to reinvestigate the parameter space in the MeV to GeV range.

In addition to the QCD axion mechanism, various Standard Model extensions with
axion-like particles were proposed Bagger:1994hh ; Arvanitaki:2009fg ;
Hook:2014cda . The main difference to the original model is that ALPs are not
restricted to a linear mass-coupling relation. During the past few years there
was an increasing interest in the MeV to GeV range Dolan:2017osp ;
Millea:2015qra ; Jaeckel:2015jla ; Beacham:2019nyx . Furthermore, ALPs are
considered to be promising dark matter candidates as being both very long-
living and weakly-interacting with the mass unconstrained by their
interactions with other particles Arias:2012az ; Giannotti:2015kwo .

In this work we investigate the mass and coupling constraints of ALPs in the
MeV to GeV range from 3-photon events in $e^{+}e^{-}$ annihilation. We focus
on the kinematical setting of the Belle II experiment. Section 2 provides a
general overview of the given formalism with the discussion of couplings,
matrix elements and cross sections. Section 3 illustrates the main results and
provides predictions for Belle II kinematics and constraints which follow from
the calculated processes. Section 4 summarizes our work.

##  II ALP formalism

In this work we assume that ALPs in the MeV to GeV mass range couple
predominantly to photons and electrons, i.e. decay only to visible states.

The following section provides a short review of the relevant ALP
interactions. The parameter space includes three variables - the ALP mass
$m_{a}$ and its couplings to photons and electrons, which are denoted by
$g_{a\gamma\gamma}$ and $g_{aee}$, respectively. We detail the calculations of
ALP contributions to 2- and 3-photon annihilation of $e^{+}e^{-}$ pairs.

###  II.1 Interaction with photons

In this section we analyze the interaction of ALPs with the electromagnetic
field. The corresponding effective Lagrangian has the form

$\mathcal{L}_{a\gamma\gamma}=-\frac{g_{a\gamma\gamma}}{4}aF^{\mu\nu}\tilde{F}_{\mu\nu},$
(1)

$a$ stands for the pseudoscalar ALP field, $F^{\mu\nu}$ is the electromagnetic
field tensor with the corresponding dual pseudotensor
$\tilde{F}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\lambda\sigma}{F}^{\lambda\sigma}$,
$g_{a\gamma\gamma}$ is the coupling constant of the dimension
$\mbox{GeV}^{-1}$.

The matrix element for the $a\to 2\gamma$ decay shown on Fig. 1 is given by

$\begin{split}M_{a\to\gamma\gamma}\left(k_{1},k_{2}\right)&=-ig_{a\gamma\gamma}k_{1,\kappa}k_{2,\beta}\varepsilon^{\kappa\beta\mu\nu}\\\
&\times\epsilon^{*}_{\mu}\left(k_{1},\lambda_{1}\right)\epsilon^{*}_{\nu}\left(k_{2},\lambda_{2}\right),\end{split}$
(2)

where $\epsilon_{\mu}\left(k_{1},\lambda_{1}\right)$ and
$\epsilon_{\mu}\left(k_{2},\lambda_{2}\right)$ are the polarization vectors of
the photons with 4-momenta $k_{1}$, $k_{2}$ and helicities $\lambda_{1}$,
$\lambda_{2}$, respectively. Summing over the final helicities, we obtain

$\sum_{f}\left|M_{a\to\gamma\gamma}\left(k_{1},k_{2}\right)\right|^{2}=2g^{2}_{a\gamma\gamma}\left(k_{1}k_{2}\right)^{2}.$
(3)

Figure 1: ALP decay to two photons in the lowest order.

The corresponding decay width is then obtained as

$\Gamma_{a\gamma\gamma}=\frac{g^{2}_{a\gamma\gamma}m_{a}^{3}}{64\pi}.$ (4)

###  II.2 Interaction with leptons

We next discuss the ALP-fermion coupling and the corresponding decay rate. The
generic interaction of ALPs with fermions is of the form

$\mathcal{L}_{aff}=-\frac{g_{aff}}{2m_{f}}\partial_{\mu}a\bar{f}\gamma^{5}\gamma^{\mu}f,$
(5)

where $f$ stands for the fermion field, $m_{f}$ denotes its mass and $g_{aff}$
is the dimensionless coupling constant.

From $\mathcal{L}_{aff}$ it is clear that lepton universality requires the
large enhancement of ALP coupling to muon, namely

$g_{a\mu\mu}\approx\frac{m_{\mu}}{m_{e}}g_{aee}.$ (6)

In this paper we follow Alves and Wiener work Alves:2017avw  and consider ALPs
coupled only to electrons in order to avoid effects induced by this enhanced
coupling, such as $\left(g-2\right)_{\mu}$ corrections on the muon anomalous
magnetic moment.

At tree level $\mathcal{L}_{aff}$ can be equivalently reduced to a
pseudoscalar coupling

$\mathcal{L}_{aff}=-ig_{aff}\bar{f}\gamma^{5}f.$ (7)

We are interested in the $a\to e^{+}e^{-}$ decay shown on Fig. 2, which has
the amplitude

$M_{a\to
e^{+}e^{-}}=g_{aee}\bar{u}\left(p_{-},s_{-}\right)\gamma^{5}v\left(p_{+},s_{+}\right),\\\
$ (8)

where $u\left(p_{-},s_{-}\right)$ and $v\left(p_{+},s_{+}\right)$ are the
bispinors describing electron and positron with momenta $p_{\pm}$ and
helicities $s_{\pm}$, respectively.

Figure 2: ALP decay to the lepton-antilepton pair in the lowest order.

In the domain of interest we can assume $m_{e}\ll m_{a}$. After summing over
the final helicities, the squared amplitude for this process is given by

$\sum_{f}\left|M_{a\to e^{+}e^{-}}\right|^{2}=2g_{aee}^{2}m_{a}^{2}.$ (9)

The corresponding decay width then has the form

$\Gamma_{aee}=\frac{g_{aee}^{2}m_{a}}{8\pi}.$ (10)

In the absence of interaction with other fields, the total ALP decay width is
assumed to consist of two contributions

$\Gamma_{a}=\Gamma_{aee}+\Gamma_{a\gamma\gamma}.$ (11)

###  II.3 ALP production at $e^{+}e^{-}$ colliders

An ALP contributes to the 2-photon annihilation of $e^{+}e^{-}$ through the
diagram shown on Fig. 3. The matrix element is

$\begin{split}M_{e^{+}e^{-}\to\gamma\gamma}&=ig_{aee}\frac{\bar{v}\left(p_{+},s_{+}\right)\gamma^{5}u\left(p_{-},s_{-}\right)}{s-m_{a}^{2}+im_{a}\Gamma_{a}}\\\
&\times M_{a\to\gamma\gamma}\left(k_{1},k_{2}\right),\end{split}$ (12)

with $M_{a\to\gamma\gamma}\left(k_{1},k_{2}\right)$ given in Eq. (2).

As a function of $m_{a}$, this cross section is significantly different from
zero only in a small region around $m_{a}^{2}=s=4E^{2}$, where $E$ denotes the
initial electron (positron) energy in the center-of-momentum frame. Thus, for
a fixed collider energy, the $e^{+}e^{-}\to\gamma\gamma$ process is not
providing constraints on ALP parameters in a broad $m_{a}$,
$g_{a\gamma\gamma}$, $g_{aee}$ parameter space in $e^{+}e^{-}$ annihilation.
Therefore, in the following we investigate 3-photon final states.

Figure 3: $e^{+}e^{-}$ annihilation into two photons through an intermediate
ALP.

Fig. 4 shows the contribution of ALP-photon coupling resulting in 3-photon
events. The corresponding amplitudes are given by

$\begin{split}&M_{e^{+}e^{-}\to\gamma\gamma\gamma\,\left(ALP_{1}\right)}=i\frac{H_{e^{+}e^{-}\to\gamma^{*}\to
a\gamma}\left(k_{1}\right)}{K_{23}^{2}-m_{a}^{2}+im_{a}\Gamma_{a}}\\\ &\times
M_{a\to\gamma\gamma}\left(k_{2},k_{3}\right)+\mbox{crossed terms},\end{split}$
(13)

where $H_{e^{+}e^{-}\to\gamma^{*}\to a\gamma}$ stands for $e^{+}e^{-}\to
a\gamma_{i}$ amplitude

$\begin{split}H_{e^{+}e^{-}\to\gamma^{*}\to
a\gamma}\left(k_{i}\right)&=-ieg_{a\gamma\gamma}\,\varepsilon_{\alpha\beta\mu\gamma}q^{\alpha}k_{i}^{\beta}\epsilon^{\gamma}\left(k_{i},\lambda_{i}\right)\\\
&\times\frac{\bar{v}\left(p_{+},s_{+}\right)\gamma^{\mu}u\left(p_{-},s_{-}\right)}{s},\end{split}$
(14)

with $e$ being the positron charge and the internal photon 4-momentum
$q=p_{+}+p_{-}$. We denote the ALP 4-momenta as $K_{23}=k_{2}+k_{3}$.

Figure 4: $e^{+}e^{-}$ annihilation into three photons involving the
$g_{a\gamma\gamma}$ coupling. Graphs obtained from these by crossing are not
shown, but are evaluated too.

It is generally assumed that ALPs are long-lived particles, i.e. their decay
width $\Gamma_{a}$ is a small quantity, typically much smaller than the
experimental resolution of the invariant mass of the two-photon system in
which the ALP decays. Thus the integration over the phase space gives the main
contribution only in a very small range of variables where the invariant mass
of the photon pair produced by the ALP is close to $m_{a}^{2}$.

In such kinematics the interference terms become unobservable and can be
omitted. After the integration over the phase space the total cross section
can be represented as a cross section obtained from only Feynman diagrams
shown in Fig. 4 multiplied by a factor of three to account for the 3 channels.
Thus we obtain

$\begin{split}&\bar{\sum_{i}}\sum_{f}\left|H_{e^{+}e^{-}\to\gamma^{*}\to
a\gamma}\left(k_{1}\right)\right|^{2}\\\
&=\frac{2e^{2}g^{2}_{a\gamma\gamma}}{s^{2}}\left[\left(k_{1}p_{+}\right)^{2}+\left(k_{1}p_{-}\right)^{2}\right]\left(p_{-}p_{+}\right),\end{split}$
(15)

where $\bar{\sum_{i}}\sum_{f}$ denotes the average over initial helicities
states and the sum over final.

We next discuss the 3-photon production in $e^{+}e^{-}$ annihilation which
results from the ALP-electron coupling contribution, shown in Fig. 5. The
corresponding amplitudes can be expressed as

Figure 5: $e^{-}e^{+}$ annihilation into three photons involving the $g_{aee}$
coupling. Graphs which are obtained by crossing are not shown, but are
evaluated too.

$\begin{split}&M_{e^{+}e^{-}\to\gamma\gamma\gamma\,\left(ALP_{2}\right)}\\\
&=i\frac{H_{e^{+}e^{-}\to a\gamma,1}\left(k_{1}\right)+H_{e^{+}e^{-}\to
a\gamma,2}\left(k_{1}\right)}{K_{23}^{2}-m_{a}^{2}+im_{a}\Gamma_{a}}\\\
&\times M_{a\to\gamma\gamma}\left(k_{2},k_{3}\right)+\mbox{crossed
terms},\end{split}$ (16)

where $H_{e^{+}e^{-}\to a\gamma,i}$ denote amplitudes for the corresponding
$2\to 2$ process

$\displaystyle\begin{split}H_{e^{+}e^{-}\to
a\gamma,1}\left(k_{i}\right)&=eg_{aee}\,\epsilon_{\eta}^{*}\left(k_{i},\lambda_{i}\right)\\\
&\times\bar{v}\left(p_{+},s_{+}\right)\gamma^{\eta}\frac{\hat{l}_{i}}{l_{i}^{2}}\gamma^{5}u\left(p_{-},s_{-}\right),\end{split}$
(17) $\displaystyle\begin{split}H_{e^{+}e^{-}\to
a\gamma,2}\left(k_{i}\right)&=eg_{aee}\,\epsilon_{\lambda}^{*}\left(k_{i},\lambda_{i}\right)\\\
&\times\bar{v}\left(p_{+},s_{+}\right)\gamma^{5}\frac{\hat{f}_{i}}{f_{i}^{2}}\gamma^{\lambda}u\left(p_{-},s_{-}\right),\end{split}$
(18)

with the internal electron momenta

$l_{i}=k_{i}-p_{+},\quad f_{i}=p_{-}-k_{i}.$ (19)

It is worth noticing that there is no interference between this set of
diagrams and the diagrams shown in Fig. 4. Using the same arguments as before,
we conclude that for the cross section calculation, we only need to evaluate
the two topologies shown in Fig. 5, as

$\begin{split}&\bar{\sum_{i}}\sum_{f}\left|H_{e^{+}e^{-}\to
a\gamma,1}\left(k_{1}\right)+H_{e^{+}e^{-}\to
a\gamma,2}\left(k_{1}\right)\right|^{2}\\\
&=e^{2}g_{aee}^{2}\left(\frac{p_{-}k_{1}}{p_{+}k_{1}}+\frac{p_{+}k_{1}}{p_{-}k_{1}}+2\frac{\left(p_{+}K_{23}\right)\left(p_{-}K_{23}\right)}{\left(p_{-}k_{1}\right)\left(p_{+}k_{1}\right)}\right).\end{split}$
(20)

###  II.4 Cross section and observables

The cross section of $e^{+}e^{-}\to\gamma\gamma\gamma$ process is given by the
expression

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\frac{1}{3!}\int
d_{LIPS}\left(2\pi\right)^{4}\delta^{\left[4\right]}\left(p_{-}+p_{+}-k_{1}-k_{2}-k_{3}\right)$
(21) $\displaystyle\times$
$\displaystyle\frac{1}{2s}\bar{\sum_{i}}\sum_{f}\left|M_{e^{+}e^{-}\to\gamma\gamma\gamma}\right|^{2},$

where $d_{LIPS}$ stands for the Lorentz-invariant phase space of the three
final photons

$d_{LIPS}=\frac{d^{3}k_{1}}{2\omega_{1}\left(2\pi\right)^{3}}\frac{d^{3}k_{2}}{2\omega_{2}\left(2\pi\right)^{3}}\frac{d^{3}k_{3}}{2\omega_{3}\left(2\pi\right)^{3}}.$
(22)

After the integration with the delta function, the phase space can be
expressed as

$\begin{split}&\,\,d_{LIPS}\left(2\pi\right)^{4}\delta^{\left[4\right]}\left(p_{-}+p_{+}-k_{1}-k_{2}-k_{3}\right)\\\
&=\frac{1}{2^{8}\pi^{5}}\frac{\omega_{1}\omega_{2}}{2E+\omega_{1}\left(\cos{\theta_{12}}-1\right)}d\omega_{1}d\Omega_{1}d\Omega_{2},\end{split}$
(23)

with $\theta_{12}$ denoting the angle between $\textbf{k}_{1}$ and
$\textbf{k}_{2}$ momenta. The remaining phase space is parameterized as

$d\Omega_{1}d\Omega_{2}=2\pi d\phi\,d\cos{\theta_{1-}}d\cos{\theta_{2-}},$
(24)

where $\theta_{i-}$ is the angle between $\textbf{p}_{-}$ and
$\textbf{k}_{i}$, which leads to

$\cos{\theta_{12}}=\sin{\theta_{1-}}\sin{\theta_{2-}}\cos{\phi}+\cos{\theta_{1-}}\cos{\theta_{2-}}.$
(25)

Furthermore, in the center-of-momentum frame it holds

$\begin{cases}\omega_{1}+\omega_{2}+\omega_{3}=2E,\\\
\textbf{k}_{1}+\textbf{k}_{2}+\textbf{k}_{3}=0,\end{cases}$ (26)

allowing to express $\omega_{2}$ as

$\omega_{2}=\frac{2E\left(E-\omega_{1}\right)}{2E+\omega_{1}\left(\cos{\theta_{12}}-1\right)}.$
(27)

For the ALP-associated process, the photon which is opposite to the ALP in
center-of-momentum frame is denoted by $k_{1}$. In this case, we can remove
the integration over $d\omega_{1}$ using the definition of the delta function

$\begin{split}&\frac{1}{\left(K_{23}^{2}-m_{a}^{2}\right)^{2}+\left(m_{a}\Gamma_{a}\right)^{2}}\to\frac{\pi}{m_{a}\Gamma_{a}}\delta\left(K_{23}^{2}-m_{a}^{2}\right)\\\
&=\frac{\pi}{m_{a}\Gamma_{a}}\frac{1}{4E}\,\delta\left(\omega_{1}-\frac{4E^{2}-m^{2}_{a}}{4E}\right).\end{split}$
(28)

Due to the resonant behavior of the amplitude, one photon is always emitted
with a fixed energy

$\omega=\frac{4E^{2}-m^{2}_{a}}{4E}.$ (29)

We note that, as the branching fraction, represented by the ratio
$\Gamma_{a\gamma\gamma}/\Gamma_{a}$, is always less than $1$, the total cross-
section of the process under investigation with an intermediate ALP may
actually become smaller with a non-zero value of $g_{aee}$, compared to when
this quantity is equal to zero.

Figure 6: Angular distributions of $e^{+}e^{-}\to\gamma\gamma\gamma$
annihilation for a process with an intermediate ALP in Belle II kinematics for
two values of $m_{a}$ and two values of $g_{aee}$. The value
$g_{a\gamma\gamma}=10^{-4}\,\mbox{GeV}^{-1}$ is used for all curves.

After the integration over the full phase space, the cross section of the
$2\to 3$ process with the intermediate ALP can be written in the compact form

$\sigma_{e^{+}e^{-}\to\gamma\gamma\gamma}=\sigma_{e^{+}e^{-}\to
a\gamma}\times\frac{\Gamma_{a\gamma\gamma}}{\Gamma_{a}}.$ (30)

If $g_{aee}=0$, the ALP decays directly to photons and this formula can be
simplified further (notably, it is independent of $s$ if $m_{a}^{2}\ll s$) as

$\sigma_{e^{+}e^{-}\to a\gamma}=\frac{\alpha
g^{2}_{a\gamma\gamma}}{24}\left(1-\frac{m_{a}^{2}}{s}\right)^{3},$ (31)

where $\alpha\equiv e^{2}/4\pi$.

For a realistic detector, one has to integrate the cross section formula over
the phase space, restricted by the experimental setup, as discussed below.

##  III Results and discussion

The ALP signal detection strategy can be based on searches for a narrow peak
in the squared mass distribution $m_{\gamma\gamma}$ of photon pairs, or a
narrow peak in the photon energies distributions, due to the fact that the
photon which accompanies the ALP is always monoenergetic in this process.

If no significant ALP signal is observed, it is possible to constrain ALP
parameters in the corresponding mass range.

In this section we illustrate our results with exclusion plots for the
kinematics of an $e^{+}e^{-}$ collider. For this purpose, we first split the
total $e^{+}e^{-}\to\gamma\gamma\gamma$ cross section into three terms

$\sigma_{ALP+B}=\sigma_{ALP}+\sigma_{B}+\sigma_{int},$ (32)

with $\sigma_{ALP}$ referring to the ALP-associated process (shown in Fig. 4
and 5), while $\sigma_{B}$ is the background. The interference term
$\sigma_{int}$ does not contribute since the ALP decay width $\Gamma_{a}$ is
assumed to be much smaller than the experimental resolution of the invariant
mass of the final photon pair.

The dominant part of the background originates from QED 3-photon annihilation
Dolan:2017osp , i.e. $\sigma_{B}=\sigma_{QED}$. The aimed sensitivity is then
expressed by the formula

$\frac{\sigma_{ALP}}{\sigma_{QED}}=\frac{N}{\sqrt{L\cdot\sigma_{QED}}},$ (33)

where $L$ denotes the integrated luminosity and $N$ is the number of standard
deviations that determines whether or not a fluctuation is considered as a
signal. We conventionally set $N=2$, which refers to 95% confidence level.

In our study we neglect the potential hadronic background from $\pi^{0}$,
$\eta$ and $\eta^{\prime}$ mesons. In a complete analysis, however, those
contributions must be included. Therefore, the parameter space for $m_{a}$ in
the vicinity of the $\pi^{0}$, $\eta$ and $\eta^{\prime}$ masses can be
expected to be modified.

The $e^{+}e^{-}\to a\gamma\to\gamma\gamma\gamma$ cross section is a function
of three variables. For purposes of illustration, we use the available
independent constraints for $g_{aee}$ to show two-dimensional projections of
$g_{a\gamma\gamma}$ as a function of $m_{a}$.

Experimental searches in the MeV to GeV region are mostly focused on ALP-muon
interaction Beacham:2019nyx  and therefore not able to constrain $g_{aee}$.
However, it is possible to convert constraints on visibly decaying dark
photons to limits on the ALP-electron mixing Alves:2017avw . Indeed, the
processes of $X\to e^{+}e^{-}$ and $a\to e^{+}e^{-}$ achieve comparable signal
strengths in case of $g_{Xee}\sim g_{aee}$. This relation, of course, is only
approximate, since the two processes have different angular distributions, but
using it one can estimate $g_{aee}\lesssim 10^{-4}$ Beacham:2019nyx .

In the following, we discuss the reach on $m_{a}$, $g_{aee}$ and
$g_{a\gamma\gamma}$ which can be obtained from
$e^{+}e^{-}\to\gamma\gamma\gamma$ data that are already available from the
Belle II experiment or are expected from future running.

###  III.1 Belle II kinematics

Figure 7: QED background distributions for the softest, middle and hardest
photons in the $e^{+}e^{-}\to\gamma\gamma\gamma$ process in Belle II
kinematics.

To obtain the exclusion plots for Belle II kinematics, we start by discussing
the detector acceptance.

Belle II is an asymmetric collider, for which electron and positron have
energies of $7\,\mbox{GeV}$ and $4\,\mbox{GeV}$, respectively. This requires a
boost with a relative velocity $\beta\approx 0.27$ to the center-of-momentum
frame, where particles have energies of $E=5.29\,\mbox{GeV}$. The angular
coverage of the electromagnetic calorimeter in the lab frame is
$12.4^{\circ}<\theta<155.1^{\circ}$.

The angular region $37.3^{\circ}<\theta<123.7^{\circ}$ provides the best
energy resolution, avoiding regions close to detector gaps, and offers the
lowest beam background levels Belle-II:2020jti . Following the work of
Dolan:2017osp , we set the photon energy selection threshold of
$0.25\,\mbox{GeV}$ in the center-of-momentum frame. Our analysis requires all
three photons to be in this acceptance range and, unless otherwise specified,
these experimental cuts are used for all the plots shown below.

The angular distributions for the ALP process are presented in Fig. 6 for two
different values of $m_{a}$ and two different values of $g_{aee}$. For a given
$g_{aee}$, there is more than an order of magnitude difference between
$m_{a}=0.3\,\mbox{GeV}$ and $m_{a}=3\,\mbox{GeV}$ curves due to the fact that
for a relatively light ALP the decay width is dominated by the $a\to
e^{+}e^{-}$ channel, see Eqs. (4) and (10). For the particular case of
$g_{a\gamma\gamma}=10^{-4}\,\mbox{GeV}^{-1}$ and $g_{aee}=10^{-4}$, one
obtains

$\displaystyle\frac{\Gamma_{a\gamma\gamma}}{\Gamma_{a}}\approx
0.01,\quad\mbox{for}\quad m_{a}=0.3\,\mbox{GeV},$
$\displaystyle\frac{\Gamma_{a\gamma\gamma}}{\Gamma_{a}}\approx
0.53,\quad\mbox{for}\quad m_{a}=3\,\mbox{GeV}.$

###  III.2 QED background

We next discuss the QED background process. The cross section of leading order
QED $e^{+}e^{-}$ annihilation in 3 photons in the massless electron limit is
given by Eidelman:1978rw

$\begin{split}&\bar{\sum_{i}}\sum_{f}\left|M_{e^{+}e^{-}\to\gamma\gamma\gamma\,\left(QED\right)}\right|^{2}=s\left(4\pi\alpha\right)^{3}\\\
&\times\frac{\sum_{i=1}^{3}\left(p_{+}k_{i}\right)\left(p_{-}k_{i}\right)\left[\left(p_{+}k_{i}\right)^{2}+\left(p_{-}k_{i}\right)^{2}\right]}{\prod_{i=1}^{3}\left(p_{+}k_{i}\right)\left(p_{-}k_{i}\right)}.\end{split}$
(34)

For the total cross section an additional factor $1/3!$ must be added due to
the 3 identical bosons in the final state.

Fig. 7 shows the corresponding QED background angular and energy
distributions. In contrast to the ALP-related process (see Fig. 6), which
exhibits a rather uniform angular distribution, the QED three-photon
annihilation is characterized by an enhanced angular distribution in both the
forward and backward directions. The presence of a distinct peak in the photon
energy distribution would serve as an indication of ALP creation.

###  III.3 Belle II results from 2018 data set

In the 2018 data run Belle II achieved an integrated luminosity of
$445\,\mbox{pb}^{-1}$ Belle-II:2020jti , which was used for the ALP searches
in a simplified way by converting the cross section limit to the coupling
using Eq. (31). The latter formula does not take into consideration the fact
that all three photons in the ALP-associated process must be detected in the
acceptance range of the electromagnetic calorimeter.

Figure 8: Left panel: Belle II constraints for $g_{a\gamma\gamma}$ based on
the 2018 data set, with the analytical result shown by the black dashed curve.
Right panel: projected results on the $\left(m_{a},g_{a\gamma\gamma}\right)$
reach for the future data collection at Belle II corresponding to
$50\,\mbox{ab}^{-1}$ of integrated luminosity.

We require three resolved photons with energies higher than $0.65\,\mbox{GeV}$
in the center-of-momentum frame as a criteria for this event selection. These
requirements are slightly different from those used in the Belle II report
Belle-II:2020jti , where the selection of photons with energies above
$0.65\,\mbox{GeV}$ (for $m_{a}>4\,\mbox{GeV}$) and $1\,\mbox{GeV}$ (for
$m_{a}\leq 4\,\mbox{GeV}$) in the lab frame was performed. The difference is
negligible since $g_{a\gamma\gamma}$ is sensitive to $\sigma_{QED}^{-1/4}$.

Our result based on Eq. (31) is shown on Fig. 8 (left panel) by the black
curve. It shows a good agreement with the analysis of Belle-II:2020jti  in the
higher ALP mass region. In the lower mass region some deviations are seen.
This is expected because in the case of a light ALP the invariant mass of a
photon pair also becomes low, i.e. two photons travel in a very narrow cone
with each other, oppositely to the third photon. This produces very asymmetric
kinematics and the QED background becomes suppressed. In our analysis we do
not take this into consideration, but more detailed investigation can be
performed in future work.

###  III.4 Belle II projection from upcoming data collection

Belle II is expected to reach an integrated luminosity of $50\,\mbox{ab}^{-1}$
after around 10 years of running. The resulting constraints which such future
data would yield were investigated in Dolan:2017osp  for the case where ALPs
are only coupled to photons (i.e. for $g_{aee}=0$). Fig. 8 right panel shows
the projected sensitivity in two scenarios: ALPs coupled only to photons and
ALPs coupled to photons and electrons with different $g_{aee}$ coupling
strength.

Our results for the scenario $g_{aee}=0$ are in reasonably good agreement with
the exclusion limits deduced in Dolan:2017osp  in the high $m_{a}$ region. For
lower values of $m_{a}$ one expects a similar deviation as for the 2018 Belle
II data discussed above. Fig. 8 also shows that the inclusion of a non-zero
interaction of ALPs with electrons significantly affects the final result,
especially in ALP mass range $m_{a}\lesssim 2\,\mbox{GeV}$. The assumption
$g_{aee}=0$ generally leads to an overestimated $g_{a\gamma\gamma}$ limit,
which may be incorrect if the ALP has other decay channels besides the
2-photon mode. In such case, more detailed models with additional parameters
are required to constrain invisible particles in a more rigorous way.

##  IV Conclusion

In this paper we discussed ALPs coupled to electrons and photons in a minimal
way. The contributions of ALP states to 2- and 3-photon $e^{+}e^{-}$
annihilation events were calculated. In this way, we obtained new constraints
for possible ALPs in the MeV to GeV mass range, which can be tested at
$e^{+}e^{-}$ colliders. Results were shown for Belle II kinematics both from
existing data and from forthcoming data with projected integrated luminosity
of $50\,\text{ab}^{-1}$.

Our results indicate that the $g_{a\gamma\gamma}$ limits can be vastly
affected in the presence of an additional decay mode, especially in the lower
$m_{a}$ region. Using current best limits for $g_{aee}$, it is possible to
improve the $g_{a\gamma\gamma}$ limits by at least an order of magnitude,
which allows to significantly narrow down the search area for potential ALPs
and to test the possible solution of the strong CP problem in the MeV to GeV
mass range. This result can be improved further if a better way to constrain
$g_{aee}$ independently is available.

There are many possible ways to further extend this work. First of all, a more
precise background modeling and experimental analysis must be performed in
order to refine the exclusion plots, especially around the $\pi^{0}$, $\eta$
and $\eta^{\prime}$ masses. The comparison with the Belle II 2018 data
analysis shows that with more detailed background analysis it is possible to
even further improve $g_{a\gamma\gamma}$ constraints in the $m_{a}^{2}\ll s$
case.

Furthermore, the ALP coupling to photons can be replaced with a more general
model of ALPs interacting with the electroweak sector of the Standard Model as
discussed e.g. in Dolan:2017osp . This implies the interaction of ALPs with
Z-bosons, which was not taken into consideration here. In the presence of such
coupling, the exclusion plots will be modified accordingly.

Additionally, in this paper we assumed that ALPs interact only with the
electrons and photons. Hidden decay channels were not considered. However, the
contribution of light dark matter particles of sub-GeV masses may change the
obtained constraints if the pair production threshold is surpassed. At the
same time, the inclusion of such particles makes it more complicated to set
any constraints, as new free parameters appear.

Finally, we restricted ourselves with ALPs which are not coupled to muons (and
taus). However, lepton universality leads to an increase in the coupling
constant $g_{a\mu\mu}$ by around two orders of magnitude compared to
$g_{aee}$. It may notably affect the results if $m_{a}$ is larger than
$2m_{\mu}$. The parameter space in such case will get additional restrictions
from the requirement of compatibility with current $\left(g-2\right)_{\mu}$
data Buen-Abad:2021fwq ; Muong-2:2023cdq .

## Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), in part through the Research Unit [Photon-photon
interactions in the Standard Model and beyond, Projektnummer 458854507 - FOR
5327], and in part through the Cluster of Excellence [Precision Physics,
Fundamental Interactions, and Structure of Matter] (PRISMA+ EXC 2118/1) within
the German Excellence Strategy (Project ID 39083149).

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