# Enhanced Optimal Quantum Communication
by Generalized Phase Shift Keying Coherent Signal

Min Namkung  Jeong San Kim<EMAIL_ADDRESS>Department of Applied
Mathematics and Institute of Natural Sciences, Kyung Hee University, Yongin
17104, Republic of Korea

###### Abstract

It is well known that the maximal success probability of the binary quantum
communication can be improved by using a sub-Poissonian non-standard coherent
state as an information carrier. In the present article, we consider the
quantum communication with $N$-ary phase shift keying ($N$-PSK) signal for an
arbitrary positive integer $N>1$. By using non-standard coherent state, we
analytically provide the maximal success probability of the quantum
communication with $N$-PSK. Unlike the binary case, we show that even super-
Poissonianity of non-standard coherent state can improve the maximal success
probability of $N$-PSK quantum communication.

††preprint: APS/123-QED

##  I Introduction

In optical communication, a sender encodes a message in an optical signal and
sends it to a receiver who detectes the signal to decode the message
g.cariolaro . Thus, the success probability of the optical communication is
determined by the physical and statistical properties of the optical signal
together with the structure of the receiver’s measurement device. In classical
optical communication, the receiver can use an on-off detector to decode a
sender’s message encoded in on-off keying signal c.w.helstrom ; k.tsujino ,
and a homodyne detector for binary phase shift keying signal j.g.proakis .
However, the maximal success probability for decoding encoded messages by
using conventional measurements such as the on-off and the homodyne detectors
cannot exceed the standard quantum limit.

One of the goals in quantum communication is to design a novel measurement so
that the maximal success probability to decode messages can surpass standard
quantum limit i.a.burenkov . According to the quantum theory, optical signal
is described as a density operator on a Hilbert space and a measurement is
descrived as a positive-operator-valued measure (POVM), therefore the quantum
communication is described as a quantum state discrimination protocol
s.m.barnett ; j.a.bergou .

Minimum error discrimination j.bae ; d.ha  is one representative state
discrimination strategy used in various quantum communication protocols. When
one bit message is encoded by binary coherent states, minimum error
discrimination between the binary coherent states can be implemented via the
Dolinar receiver s.j.dolinar . However, when several bits are encoded and
sequentially sent, the photon number detector used for the discrimination may
not efficiently react along the received states i.a.burenkov . For this
reason, $N$-ary coherent states such as $N$-amplitude shift keying ($N$-ASK)
signal c.w.helstrom  and $N$-phase shift keying ($N$-PSK) signal j.g.proakis
have been considered to send $\log_{2}N$ bit messages.

According to a recent work e.m.f.curado , the maximal success probability (or
Helstrom bound) of discriminating a message encoded in 2-PSK signal composed
of non-standard coherent states (NS-CS) with a novel quantum measurement can
be improved by the sub-Poissonianity of the NS-CS. Moreover, the experimental
method for implementing the quantum measurement reaching for the Helstrom
bound has recently been proposed m.namkung . Since the negative Mandel
parameter to quantify the sub-Poissonianity is considered as a resource in a
non-classical light s.dey , this result implies that the sub-Poissonianity can
be a resource for improving the performance of the quantum communication.

In the present article, we consider the quantum communication with $N$-PSK
signal for arbitrary an arbitrary positive integer $N>1$. By using non-
standard coherent state, we analytically provide the maximal success
probability of the quantum communication with $N$-PSK. Unlike the binary case,
we show that even super-Poissonianity of non-standard coherent state can
improve the maximal success probability of $N$-PSK quantum communication: The
Helstrom bound can be improved by considering the sub-Poissonian NS-CS for
$N=3$, meanwhile the super-Poissonian NS-CS can improve the Helstrom bound for
$N=4$ and $N=8$.

For $N>2$, $N$-PSK signal enables us to transmit a $\log_{2}N$-bit message per
a signal pulse, which is a better information exchange rate than binary-PSK.
Moreover it is also known that $N$-PSK signal can provide an improved
information exchange rate between the sender and receiver even though the
receiver’s measurement is slow i.a.burenkov . However, the maximal success
probability of discriminating a message encoded in $N$-PSK signal generally
decreases as $N$ is getting large. Thus our results about the possible
enhancement of the maximal success probability in $N$-PSK quantum
communication by NS-CS is important and even necessary to design efficient
quantum communication schemes.

The present article is organized as follows. In Section 2, we briefly review
the problem of minimum error discrimination among $N$ symmetric pure states.
In Section 3, we provide the analytical Helstrom bound of $N$-PSK signal
composed of NS-CS. In Section 4, we investigate the Helstrom bound of $N$-PSK
signal composed of optical spin coherent states (OS-CS), Perelomov coherent
states (P-CS), Barut-Girardello coherent states (BG-CS) and modified Susskind-
Glogower coherent states (mSG-CS), and discuss the relation between the sub-
Poissonianity of the non-classical light and the performance of the $N$-PSK
quantum communication. Finally, in Section 5, we propose the conclusion of the
present article.

##  II Preliminaries: Minimum Error Discrimination among Symmetric Pure States

In quantum communication, Alice (sender) prepares her message
$x\in\\{1,\cdots,N\\}$ with a prior probability
$q_{x}\in\\{q_{1},\cdots,q_{N}\\}$, encodes the message in a quantum state
$\rho_{x}\in\\{\rho_{1},\cdots,\rho_{N}\\}$, and sends the quantum state to
Bob (receiver). Bob performs a quantum measurement described as a POVM
$\\{M_{1},\cdots,M_{N}\\}$ to discriminate the encoded message. In the POVM,
$M_{x}$ is a POVM element with respect to a result $x$.

For a given ensemble $\mathcal{E}=\\{q_{x},\rho_{x}\\}_{x=1}^{N}$ of Alice and
a POVM $\mathcal{M}=\\{M_{x}\\}_{x=1}^{N}$ of Bob, the success probability of
the quantum communication between Alice and Bob is described by the success
probability of the state discrimination,

$P_{s}(\mathcal{E},\mathcal{M})=\sum_{x=1}^{N}q_{x}\mathrm{tr}\left\\{\rho_{x}M_{x}\right\\},$
(1)

One way to optimize the efficiency of quantum communication is to consider a
POVM that maximizes the success probability in Eq. (1). In this case, the
maximization of the success probability in Eq. (1) is equivalent to the
minimization of the error probability

$P_{e}(\mathcal{E},\mathcal{M})=1-P_{s}(\mathcal{E},\mathcal{M})=\sum_{x=1}^{N}\sum_{y\not=x}q_{x}\mathrm{tr}\left\\{\rho_{x}M_{y}\right\\}.$
(2)

Minimum error discrimination is to minimize the error probability in Eq. (2)
over all possible POVMs $\mathcal{M}$ of Bob.

For a given ensemble $\mathcal{E}$, it is known that the following inequality
is a necessary and sufficient condition for POVM $\mathcal{M}$ minimizing the
error probability c.w.helstrom ; s.m.barnett2 ,

$\displaystyle\sum_{z=1}^{N}q_{z}\rho_{z}M_{z}-q_{x}\rho_{x}\geq 0,\ \ \forall
x\in\\{1,\cdots,N\\}.$ (3)

Moreover, it is known that the following equality is a useful necessary
condition to characterize the structure of the POVM,

$\displaystyle M_{x}(q_{x}\rho_{x}-q_{y}\rho_{y})M_{y}=0,\ \ \forall
x,y\in\\{1,\cdots,N\\}.$ (4)

If every quantum state $\rho_{x}$ is pure, that is,
$\rho_{x}=|\psi_{x}\rangle\langle\psi_{x}|$, the optimal POVM is given by a
rank-1 projective measurement c.w.helstrom . In other word,
$M_{x}=|\pi_{x}\rangle\langle\pi_{x}|$ for every $x\in\\{1,\cdots,N\\}$.

Now, we focus on the minimum error discrimination among a specific class of
pure states, called symmetric pure states.

###### Definition 1

a.chefles  For a positive integer $N$, the distinct pure states
$|\psi_{1}\rangle,\cdots,|\psi_{N}\rangle$ are called symmetric, if there
exists a unitary operator $V$ such that

$|\psi_{x}\rangle=V^{x-1}|\psi_{1}\rangle$ (5)

for $x=1,2,\cdots,N$ and

$V^{N}=\mathbb{I},$ (6)

where $\mathbb{I}$ is an identity operator on a subspace spanned by
$\\{|\psi_{1}\rangle,\cdots,|\psi_{N}\rangle\\}$.

The Gram matrix composed of the symmetric pure states in Definition 1 is

$G=\left(\langle\psi_{x}|\psi_{y}\rangle\right)_{x,y=1}^{N}.$ (7)

From a straightforward calculation, the eigenvalues of the Gram matrix in Eq.
(7) are in forms of

$\lambda_{p}=\sum_{k=1}^{N}\langle\psi_{j}|\psi_{k}\rangle e^{-\frac{2\pi
i(p-1)(j-k)}{N}},\ \ p=1,2,\cdots,N,$ (8)

for any choice of $j\in\\{1,2,\cdots,N\\}$. We note that the set
$\\{\lambda_{p}\\}_{p=1}^{N}$ is invariant under the choice of $j$ due to the
symmetry of the pure state $\\{|\psi_{1}\rangle,\cdots,|\psi_{N}\rangle\\}$.
The following proposition provides the maximal success probability of the
minimum error discrimination among the symmetric pure states in Definition 1.

###### Proposition 1

c.w.helstrom  Let $\mathcal{E}_{sym}$ be an equiprobable ensemble of symmetric
pure states $|\psi_{1}\rangle,\cdots,|\psi_{N}\rangle$. Then, the maximal
success probability is given as

$P_{hel}(\mathcal{E}_{sym})=\frac{1}{N^{2}}\left(\sum_{p=1}^{N}\sqrt{\lambda_{p}}\right)^{2},$
(9)

where $\lambda_{p}$ are the eigenvalues of the Gram matrix composed of
$\\{|\psi_{1}\rangle,\cdots,|\psi_{N}\rangle\\}$ in Eq. (7).

Eq. (9) is also called Helstrom bound, and $1-P_{hel}(\mathcal{E}_{sym})$ is
called minimum error probability.

##  III Optimal Communication with
Phase Shift Keying (PSK) Signal

In quantum optical communication, phase shift keying (PSK) signal is expressed
as equiprobable symmetric pure states g.cariolaro . In this section, we derive
the maximal success probability of the quantum communication with PSK signal
composed of generalized coherent states. First, we provide definition of the
generalized coherent state which encapsulates standard coherent state (S-CS)
and non-standard coherent state (NS-CS) as special cases.

###### Definition 2

e.m.f.curado  If a pure state takes the form

$|\alpha,\vec{h}\rangle=\sum_{n=0}^{\infty}\alpha^{n}h_{n}(|\alpha|^{2})|n\rangle,\
\ \alpha\in\mathbb{C},$ (10)

where $\\{|n\rangle|n\in\mathbb{Z}^{+}\cup\\{0\\}\\}$ is Fock basis and
$\vec{h}$ is a tuple of real-valued functions
$h_{n}:[0,R^{2}]\rightarrow\mathbb{R}$ satisfying

$\displaystyle\sum_{n=0}^{\infty}u^{n}\left\\{h_{n}(u)\right\\}^{2}=1,$ (11)
$\displaystyle\sum_{n=0}^{\infty}nu^{n}\left\\{h_{n}(u)\right\\}^{2}$ (12)
$\displaystyle\ \ \ \ \ \ \ \ \textrm{ is a strictly increasing function of
}u,$ $\displaystyle\int_{0}^{R^{2}}duw(u)u^{n}\left\\{h_{n}(u)\right\\}^{2}=1$
(13) $\displaystyle\ \ \ \ \ \ \ \ \textrm{ for a real-valued function
}w:[0,R^{2}]\rightarrow\mathbb{R}^{+},$

then the pure state is called generalized coherent state. If every real-valued
function $h_{n}$ in Eq. (10) takes the form

$h_{n}(u)=\frac{1}{\sqrt{n!}}e^{-\frac{1}{2}u},\ \ \forall
n\in\mathbb{Z}^{+}\cup\\{0\\},$ (14)

then Eq. (10) is called standard coherent state (S-CS) r.j.glauber .
Otherwise, Eq. (10) is called non-standard coherent state (NS-CS).

Several examples of NS-CS have been introduced such as optical spin coherent
state (OS-CS) a.m.perelomov , Perelomov coherent state (P-CS) a.m.perelomov ,
Barut-Girardello coherent state (BG-CS) a.o.barut  and modified Susskind-
Glogower coherent state (mSG-CS) j.-p.gazeau .

###### Example 1

For a given non-negative integer $\widetilde{n}$, if $h_{n}$ takes the form

$\displaystyle
h_{n}(u)=\sqrt{\frac{\widetilde{n}!}{n!(\widetilde{n}-n)!}}(1+u)^{-\frac{\widetilde{n}}{2}},$
(15)

for $0\leq n\leq\widetilde{n}$ and $h_{n}(u)=0$ for $n>\widetilde{n}$, then
the generalized coherent state in Eq. (10) is called optical spin coherent
state (OS-CS).

###### Example 2

For all non-negative integer $n$ and a real number $\varsigma$ with
$\varsigma\geq 1/2$, if $h_{n}$ takes the form

$\displaystyle
h_{n}(u)=\frac{1}{\mathcal{N}(u)}\sqrt{\frac{\Gamma(2\varsigma)}{n!\Gamma(2\varsigma+n)}},$
(16)

then the generalized coherent state in Eq.(10) is called Barut-Girardello
coherent state (BG-CS). Here, $\Gamma$ is the Gamma function of the first kind
and $\mathcal{N}(u)$ is a normalization factor

$\mathcal{N}(u)=\Gamma(2\varsigma)u^{1/2-u}I_{2\varsigma-1}(2\sqrt{u}),$ (17)

where $I_{\nu}$ is the modified Bessel function of the first kind.

###### Example 3

For all non-negative integer $n$, if $h_{n}$ takes the form

$\displaystyle
h_{n}(u)=\sqrt{\frac{n+1}{\mathcal{\bar{N}}(u)}}\frac{1}{u^{\frac{n+1}{2}}}J_{n+1}(2\sqrt{u}),$
(18)

then the generalized coherent state in Eq. (10) is called modified Susskind-
Glogower coherent state (mSG-CS). Here, $J_{n}$ is the Bessel function of the
first kind and $\mathcal{\bar{N}}(u)$ is a normalization factor

$\displaystyle\mathcal{\bar{N}}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{u}\Big{[}2u\left\\{J_{0}(2\sqrt{u})\right\\}^{2}$
$\displaystyle-$
$\displaystyle\sqrt{u}J_{0}(2\sqrt{u})J_{1}(2\sqrt{u})+2u\left\\{J_{1}(2\sqrt{u})\right\\}^{2}\Big{]}.$

###### Example 4

For all non-negative integer $n$, $\varsigma$ and an integer or half-integer
with $\varsigma\geq 1/2$, if $h_{n}$ takes the form

$\displaystyle
h_{n}(u)=\sqrt{\frac{(2\varsigma-1+n)!}{n!(2\varsigma-1)!}}(1-u)^{\varsigma},$
(20)

then the generalized coherent state in Eq. (10) is called Perelomov coherent
state (P-CS).

We mainly focus on which NS-CS provided in the examples can give the advantage
to the $N$-ary PSK quantum communication. For this reason, we define the
$N$-ary generalized PSK ($N$-GPSK) signal as follows.

###### Definition 3

If an equiprobable ensemble $\mathcal{E}_{gcs}$ consists of generalized
coherent states,

$\left\\{|\alpha_{x},\vec{h}\rangle|x\in\\{1,2,\cdots,N\\}\right\\},$ (21)

where $N\in\mathbb{Z}^{+}$ and $\alpha_{x}\in\mathbb{C}$ such that

$\alpha_{x}=\alpha e^{\frac{2\pi i}{N}x},$ (22)

with a non-negative integer $\alpha$, then the ensemble $\mathcal{E}_{gcs}$ is
called $N$-ary generalized PSK ($N$-GPSK) signal.

Moreover, $N$-GPSK signal is called $N$-ary standard PSK ($N$-SPSK) signal
g.cariolaro  if every coherent state in Eq. (21) is S-CS, and $N$-PSK signal
is called $N$-ary non-standard PSK ($N$-NSPSK) signal if every coherent state
in Eq. (21) is NS-CS. The following theorem shows that the generalized
coherent states in Definition 3 are symmetric.

###### Theorem 1

For given distinct generalized coherent states
$|\alpha_{1},\vec{h}\rangle,\cdots,|\alpha_{N},\vec{h}\rangle$, there exists a
unitary operator $U$ such that

$\displaystyle|\alpha_{x},\vec{h}\rangle=U^{x-1}|\alpha_{1},\vec{h}\rangle,\ \
\forall x\in\\{1,2,\cdots,N\\},$ (23)

for $x=1,2,\cdots,N$ and

$U^{N}=\mathbb{I},$ (24)

where $\mathbb{I}$ is an identity operator on a subspace spanned by
$\\{|\alpha_{1},\vec{h}\rangle,\cdots,|\alpha_{N},\vec{h}\rangle\\}$.

Proof. Consider a unitary operator

$\displaystyle U=e^{\frac{2\pi i}{N}a^{\dagger}a},$ (25)

where $a$ and $a^{\dagger}$ are the annihilation and creation operators
satisfying

$\displaystyle a|n\rangle=\sqrt{n}|n-1\rangle,\ \ \forall n\in\mathbb{Z}^{+},$
(26) $\displaystyle a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle,\ \ \forall
n\in\mathbb{Z}^{+}\cup\\{0\\},$ (27)

respectively. It is straightforward to show that the unitary operator $U$ in
Eq. (25) satisfies Eq. (24).

We also note that

$\displaystyle U|n\rangle=e^{\frac{2\pi i}{N}n}|n\rangle,$ (28)

for any non-negative integer $n$, therefore we have that

$\displaystyle U|\alpha_{x},\vec{h}\rangle$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\alpha_{x}^{n}h_{n}(|\alpha_{x}|^{2})e^{\frac{2\pi
i}{N}a^{\dagger}a}|n\rangle$ (29) $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\alpha_{x}^{n}h_{n}(|\alpha_{x}|^{2})e^{\frac{2\pi
i}{N}n}|n\rangle$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}(\alpha_{x}e^{\frac{2\pi
i}{N}})^{n}h_{n}(|\alpha_{x}|^{2})|n\rangle,$

for every $x\in\\{1,2,\cdots,N-1\\}$. Moreover, Eq. (22) leads us to

$\displaystyle\alpha_{x}e^{\frac{2\pi i}{N}}=\alpha_{x+1},$ (30)

for $x\in\\{1,2,\cdots,N-1\\}$ and

$\displaystyle|\alpha_{x}|=\alpha,$ (31)

for $x\in\\{1,2,\cdots,N\\}$. From Eqs. (29), (30) and (31), we have

$\displaystyle
U|\alpha_{x},\vec{h}\rangle=\sum_{n=0}^{\infty}(\alpha_{x+1})^{n}h_{n}(|\alpha_{x+1}|^{2})|n\rangle=|\alpha_{x+1},\vec{h}\rangle.$
(32)

Eq. (23) can be shown by an inductive use of Eq. (32), which completes the
proof.

Theorem 1 means that the Helstrom bound of quantum communication with $N$-GPSK
signal is given by Eq. (9) in Proposition 1, which is encapsulated in the
following theorem.

###### Theorem 2

The Helstrom bound of $N$-GPSK signal is given by

$\displaystyle
P_{hel}(\mathcal{E}_{gcs})=\frac{1}{N^{2}}\left(\sum_{p=1}^{N}\sqrt{\lambda_{p}^{(G)}}\right)^{2},$
(33)

where $\lambda_{p}^{(G)}$ takes the form of

$\displaystyle\lambda_{p}^{(G)}=\sum_{k=0}^{M-1}\left[\sum_{n=0}^{\infty}\alpha^{2n}\cos\left\\{\frac{2\pi}{N}k(n+p-1)\right\\}\left\\{h_{n}(\alpha^{2})\right\\}^{2}\right].$
(34)

for every $p\in\\{1,2,\cdots,N\\}$.

Proof. For every $j,k\in\\{1,2,\cdots,N\\}$, the inner product
$\langle\alpha_{j},\vec{h}|\alpha_{k},\vec{h}\rangle$ is

$\displaystyle\langle\alpha_{j},\vec{h}|\alpha_{k},\vec{h}\rangle=\sum_{n=0}^{\infty}\left\\{\alpha^{2}e^{i\frac{2\pi}{N}(k-j)}\right\\}^{n}\left\\{h_{n}(\alpha^{2})\right\\}^{2}.$
(35)

From Eq. (35) together with Eq. (8), $\lambda_{p}^{(G)}$ is also obtained by

$\displaystyle\lambda_{p}^{(G)}$
$\displaystyle=\sum_{k=1}^{N}\left[\sum_{n=0}^{\infty}\left\\{\alpha^{2}e^{i\frac{2\pi}{N}(k-j)}\right\\}^{n}\left\\{h_{n}(\alpha^{2})\right\\}^{2}\right]e^{-\frac{2\pi
i(p-1)(j-k)}{N}}$
$\displaystyle=\sum_{k=1}^{N}\left[\sum_{n=0}^{\infty}\alpha^{2n}e^{i\frac{2\pi}{N}(k-j)(n+p-1)}\left\\{h_{n}(\alpha^{2})\right\\}^{2}\right].$
(36)

As mentioned before, the set $\\{\lambda_{p}^{(G)}\\}_{p=1}^{N}$ is invariant
under the choice of $j\in\\{1,2,\cdots,N\\}$. By choosing $j=1$ and
substituting $k$ to $k-1$, $\lambda_{p}^{(G)}$ can be rewritten by

$\displaystyle\lambda_{p}^{(G)}=\sum_{k=0}^{N-1}\left[\sum_{n=0}^{\infty}\alpha^{2n}e^{i\frac{2\pi}{N}k(n+p-1)}\left\\{h_{n}(\alpha^{2})\right\\}^{2}\right].$
(37)

Since the Gram matrix is Hermitian, $\lambda_{p}^{(G)}$ is a real number.
Thus, by using the relation

$\lambda_{p}^{(G)}=\frac{\lambda_{p}^{(G)}+\lambda_{p}^{(G)*}}{2},$ (38)

together with Eq. (37), we have Eq. (34). Due to Theorem 1, every generalized
coherent state in $N$-GPSK signal is symmetric. Thus, Proposition 1 and Eq.
(34) lead us to the Helstrom bound in Eq. (33).

##  IV Sub-Poissonianity of NS-CS and the Helstrom bound

For $N=3,4$ and 8, we provide illustrative results of the Helstrom bound of
$N$-NSPSK signal of Eq. (33) in case of OS-CS, P-CS, BG-CS and mSG-CS. We also
compare these results with the case of $N$-SPSK signal.

###  IV.1 Optical Spin Coherent States (OS-CS)

Figure 1: The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK
signal composed of OS-CS , where (a), (b) and (c) show the case of $N=$3, 4
and 8, respectively. In these figures, purple, red, blue and green lines show
the case of $N$-NSPSK signal with $\widetilde{n}=3$, $\widetilde{n}=5$,
$\widetilde{n}=7$ and $\widetilde{n}=11$, respectively. Black lines in the
figures show the case of $N$-SPSK signal.  Figure 2: The minimum error
probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of BG-CS, where
(a), (b) and (c) shows the case of $N=$3, 4 and 8, respectively. In these
figures, blue and red lines show the case of $N$-NSPSK signal with
$\varsigma=0.5$ and $\varsigma=1.5$, respectively. Black lines show the case
of $N$-SPSK.

The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal
composed of OS-CS are illustrated in Fig. 1, where Fig. 1(a), (b) and (c) show
the case of $N=$3, 4 and 8, respectively. In these figures, purple, red, blue
and green lines show the case of $N$-NSPSK signal with $\widetilde{n}=3$,
$\widetilde{n}=5$, $\widetilde{n}=7$ and $\widetilde{n}=11$, respectively.
Black lines in the figures show the case of $N$-SPSK signal.

In Fig. 1(a), the minimum error probabilities of 3-NSPSK signal composed of
OS-CS is smaller than that of 3-SPSK signal when mean photon number is large
($\langle n\rangle>0.45$, $\langle n\rangle>0.42$, $\langle n\rangle>0.38$ and
$\langle n\rangle>0.37$ in case of $\widetilde{n}=3$, $\widetilde{n}=5$,
$\widetilde{n}=7$ and $\widetilde{n}=11$, respectively). In other words, 3-PSK
quantum communication can be enhanced by a non-standard coherent state using
OS-CS. However, in Fig. 1(b), each minimum error probability of 4-NSPSK signal
is larger than that of 4-SPSK signal for arbitrary mean photon number. This
aspect repeatedly happens in Fig. 1(c) where 8-NSPSK signal is considered.
These results imply that 4-PSK and 8-PSK quantum communication cannot be
enhanced by OS-CS.

###  IV.2 Barut-Girardello Coherent States (BG-CS)

The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal
composed of BG-CS are illustrated in Fig. 2, where Fig. 2(a), (b) and (c)
shows the case of $N=$3, 4 and 8, respectively. In these figures, blue and red
lines show the case of $N$-NSPSK signal with $\varsigma=0.5$ and
$\varsigma=1.5$, respectively. Black lines show the case of $N$-SPSK.

In Fig. 2(a), each minimum error probabilty of 3-NSPSK signal with
$\varsigma=1.5$ is smaller than that of 3-SPSK signal when mean photon number
is larger than 0.48. Meanwhile, each minimum error probabilty of 3-NSPSK
signal with $\varsigma=0.5$ is larger than that of 3-SPSK signal for arbitrary
mean photon number. Thus, enhancing 3-PSK quantum communication by non-
standard coherent state using BG-CS depends on the parameter $\varsigma$.
However, in Fig. 2(b), each minimum error probability of $4$-NSPSK signal is
larger than that of $4$-SPSK signal for arbitrary mean photon number. This
aspect repeatedly happens in Fig. 2(c) where 8-NSPSK signal is considered.
These results imply that 4-PSK and 8-PSK quantum communication cannot be
enhanced by BG-CS.

###  IV.3 Modified Susskind-Glogower Coherent States (mSG-CS)

Figure 3: The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK
signal composed of mSG-CS, where (a), (b) and (c) shows the case of $N=$3, 4
and 8, respectively. In these figures, red lines show the case of $N$-NSPSK
signal and black lines show the case of $N$-SPSK.  Figure 4: The minimum error
probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of P-CS, where
(a), (b) and (c) shows the case of $N=$3, 4 and 8, respectively. In these
figures, blue and red lines show the case of P-CS with $\varsigma=0.5$ and
$\varsigma=1.5$, respectively. Black lines show the case of S-CS.

The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal
composed of mSG-CS are illustrated in Fig. 3, where Fig. 3(a), (b) and (c)
shows the case of $N=$3, 4 and 8, respectively. In these figures, red lines
show the case of $N$-NSPSK signal and black lines show the case of $N$-SPSK.

In Fig. 3, each minimum error probability of $N$-NSPSK signal is larger than
that of $N$-SPSK signal for any $N=$3, 4 and 8 and any mean photon number.
This result implies that 3-PSK, 4-PSK, and 8-PSK quantum communication cannot
be enhanced by mSG-CS. We also compare this result with the previous work
about on-off keying signal e.m.f.curado ; it is known that the minimum error
probability of on-off keying signal composed of mSG-CS has a singular point
where the logarithm of the minimum error probability diverges to $-\infty$.
This implies that the minimum error probability can achieve to zero. Unlike
the result in the previous work e.m.f.curado , the minimum error probabilities
of 3, 4 and 8-NSPSK signal in Fig. 3 do not have such singular points.

###  IV.4 Perelomov Coherent States (P-CS)

The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal
composed of P-CS are illustrated in Fig. 4, where Fig. 4(a), (b) and (c) shows
the case of $N=$3, 4 and 8, respectively. In these figures, blue and red lines
show the case of P-CS with $\varsigma=0.5$ and $\varsigma=1.5$, respectively.
Black lines show the case of S-CS.

In Fig. 4(a), each minimum error probabilty of 3-NSPSK signal composed of P-CS
with $\varsigma=$0.5 and 1.5 is larger than that of 3-SPSK signal for
arbitrary mean photon number. In other words, $3$-PSK quantum communication
cannot be enhanced by non-standard coherent state using P-CS. However, in Fig.
4(b), each minimum error probabilty of 4-NSPSK signal composed of P-CS is
smaller than that of 4-SPSK signal when mean photon number is small ($\langle
n\rangle<0.585$ and $\langle n\rangle<0.786$ in case of $\varsigma=0.5$ and
$\varsigma=1.5$, respectively). This implies that 4-PSK quantum communication
can be enhanced by P-CS. In Fig. 4(c), each minimum error probabilty of
8-NSPSK signal composed of P-CS is smaller than that of 8-SPSK signal for
arbitrary mean photon number. This result is rather surprising since even
super-Poissonianity in P-CS can enhance the 4-PSK and 8-PSK quantum
communication unlike the binary case m.namkung . We discuss the details in the
next section.

###  IV.5 Mandel Parameter and $N$-NSPSK Quantum Communication

It is known that sub-Poissonianity of non-classical light is one of the
important statistical properties for improving Helstrom bound of binary
quantum optical communication e.m.f.curado . For this reason, we consider the
following Mandel parameter,

$Q_{M}^{(NS)}=\frac{(\Delta n)^{2}}{\langle n\rangle}-1,$ (39)

where $\langle n\rangle$ is mean photon number and $\Delta n$ is standard
deviation of the number of photons. It is known that if $Q_{M}^{(NS)}>0(<0)$,
then the generalized coherent state is super-Poissonian(sub-Poissonian)
l.mandel ; r.short . If $Q_{M}^{(NS)}=0$ (for example, S-CS), then the
generalized coherent state is Poissonian. Here, we consider the relation
between the performance of the $N$-PSK quantum communication and the Mandel
parameter.

  1. 1.

In case of OS-CS, the Mandel parameter is analytically driven as e.m.f.curado

$\displaystyle Q_{M}^{(OS)}=-\frac{\langle n\rangle}{\widetilde{n}},$ (40)

which means that OS-CS is always sub-Poissonian. According to Fig. 1, we note
that sub-Poissonianity of OS-CS does not always guarantee the enhancement of
the $N$-PSK quantum communication.

  2. 2.

In case of BG-CS, the Mandel parameter is analytically driven in terms of the
Modified Bessel function of the first kind as e.m.f.curado

$\displaystyle
Q_{M}^{(BG)}=\alpha\left[\frac{I_{2\varsigma+1}(2\alpha)}{I_{2\varsigma}(2\alpha)}-\frac{I_{2\varsigma}(2\alpha)}{I_{2\varsigma-1}(2\alpha)}\right].$
(41)

Since the inequality $\left\\{I_{\nu+1}(x)\right\\}^{2}\geq
I_{\nu}(x)I_{\nu+2}(x)$ holds for every $x\geq 0$, $Q_{M}^{(BG)}$ is negative
semidefinite. Therefore, BG-CS is always sub-Poissonian or Poissonian.
Moreover, $Q_{M}^{(BG)}$ is known to be strictly negative for non-zero mean
photon number e.m.f.curado . Nevertheless, Fig. 2 shows that sub-Poissonianity
of BG-CS does not always guarantee the enhancement of the $N$-PSK quantum
communication.

  3. 3.

Since the analytic form of the Mandel parameter of the mSG-CS Mandel parameter
is too complex e.m.f.curado , we do not introduce the analytic form here.
According to the result of e.m.f.curado , the Mandel parameter of mSG-CS is
negative when the mean photon number is not too large. Nevertheless, Fig. 3
shows that the sub-Poissonianity of the mSG-CS cannot provide any advantage on
the $N$-PSK quantum communication.

  4. 4.

In case of P-CS, the Mandel parameter is analytically driven as e.m.f.curado

$\displaystyle Q_{M}^{(P)}=\frac{\langle n\rangle}{2\varsigma},$ (42)

which means that P-CS is super-Poissonian. However, Fig. 4 shows that P-CS can
enhance the $N$-PSK quantum communication for $N=$3, 4 or 8. It is surprising
since the super-Poissonianity of NS-CS can even enhance $N$-PSK quantum
communication unlike the binary case.

##  V Conclusion

In the present article, we have considered the quantum communication with
$N$-ary phase shift keying ($N$-PSK) signal for arbitrary an arbitrary
positive integer $N>1$. By using NS-CS, we have analytically provided the
Helstrom bound of the quantum communication with $N$-PSK. Unlike the binary
case e.m.f.curado ; m.namkung , we have shown that even super-Poissonianity of
NS-CS can improve the Helstrom bound of $N$-PSK quantum communication: The
Helstrom bound can be improved by considering the sub-Poissonian NS-CS for
$N=3$, meanwhile the super-Poissonian NS-CS can improve the Helstrom bound for
$N=4$ and $N=8$.

Using $N$-PSK signal with $N>2$, we can achieve a better transmission rate per
a signal pulse than that of binary-BPSK even if the receiver’s measurement is
slow i.a.burenkov . On the other hands, the maximal success probability of
discriminating a message encoded in $N$-PSK signal generally decreases as $N$
is getting large. Thus our results about the possible enhancement of the
maximal success probability in $N$-PSK quantum communication by NS-CS is
important and even necessary to design efficient quantum communication
schemes.

In the present article, we have only considered PSK signal with equal prior
probabilities, which is composed of symmetric pure states. However, it is
interesting and even important to consider a non-equiprobable or asymmetric
ensemble of NS-CS for several reasons: First, it is practically difficult to
implement the PSK signal having perfect symmetry or equal prior probabilities.
Moreover, in discriminating three non-equiprobable and asymmetric pure states,
there is possibility that sub-Poissonianity of non-classical light can enhance
the Helstrom bound. We note that it is also interesting to consider
unambiguous discrimination i.d.ivanovic ; d.dieks ; a.peres ; g.jaeger ;
s.pang ; j.a.bergou2  of NS-CS since this strategy can provide us with better
confidence than the minimum error discrimination.

##  VI Acknowledgement

This work was supported by Quantum Computing Technology Development Program
(NRF2020M3E4A1080088) through the National Research Foundation of Korea (NRF)
grant funded by the Korea government (Ministry of Science and ICT).

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# IACT event analysis with the MAGIC telescopes using deep convolutional
neural networks with CTLearn

T. Miener1, R. López-Coto2, J. L. Contreras1, J. G. Green3, D. Green4 for the
MAGIC Collaboration, E. Mariotti2, D. Nieto1, L. Romanato2, and S. Yadav5

###### Abstract

The Major Atmospheric Gamma Imaging Cherenkov (MAGIC) telescope system
consists of two imaging atmospheric Cherenkov telescopes (IACTs) and is
located on the Canary island of La Palma. IACTs are excellent tools to inspect
the very-high-energy (few tens of GeV and above) gamma-ray sky by capturing
images of the air showers, originated by the absorption of gamma rays and
cosmic rays by the atmosphere, through the detection of Cherenkov photons
emitted in the shower. One of the main factors determining the sensitivity of
IACTs to gamma-ray sources, in general, is how well reconstructed the
properties (type, energy, and incoming direction) of the primary particle
triggering the air shower are. We present how deep convolutional neural
networks (CNNs) are being explored as a promising method for IACT full-event
reconstruction. The performance of the method is evaluated on observational
data using the standard MAGIC Analysis and Reconstruction Software, MARS, and
CTLearn, a package for IACT event reconstruction through deep learning.

1Instituto de Física de Partículas y del Cosmos and Departamento de EMFTEL,
Universidad Complutense de Madrid, Spain<EMAIL_ADDRESS>
2 Dipartimento di Fisica e Astronomia dell?Università and Sezione INFN,
Padova,Italy

3INAF - National Institute for Astrophysics, Roma, Italy

4Max-Planck-Institut für Physik, München, Germany

5Birla Institute of Technology and Science, Pilani, India

##  1 Introduction

In this contribution, we show how deep learning (DL) algorithms like CNNs are
incorporated into the analysis workflow of the MAGIC telescopes to perform
full-event reconstruction. We also explore the robustness of CNN-based
methods, when applying them to real observational data and compare the
sensitivity to the standard analysis of MARS (Zanin et al. 2013; Aleksić et
al. 2016). The DL workflow consists of three main building bricks (see Fig.
1). The Monte Carlo (MC) simulations and observational data are processed by
the MARS software. A complementary macro translate crucial information into
uproot111https://github.com/scikit-hep/uproot4-readable branches (Pivarski et
al. 2021). Then, the DL1-Data-Handler222https://github.com/cta-
observatory/dl1-data-handler (Kim et al. 2020) (DL1DH) assembles several data
levels from the standard approach and unifies them in a common data format in
HDF5 designed for DL purposes. The training of the CNN-based models and their
prediction, the actual full-event reconstruction, are performed with
CTLearn333https://github.com/ctlearn-project/ctlearn (Nieto et al. 2019a;
Brill et al. 2019), a backend for IACT analysis using TensorFlow.

Figure 1.: Diagram depicting the main analysis steps of the MAGIC DL analysis
with CTLearn.

##  2 DL analysis with the MAGIC telescopes

### Model selection

For this work, CTLearn’s Thin-ResNet (TRN) was selected based on previous
studies (Grespan et al. 2021; Miener et al. 2021). Stereoscopic information
are explored by concatenating the images (integrated pixel charges and signal
arrival times) of MAGIC1 and MAGIC2 channel-wise before feeding the network.
We explored two different analysis schemes, where we trained the same TRN
model with raw images, containing besides the Cherekov light of the shower
also the fluctuations of the Night Sky Background (NSB), and cleaned images,
where pixels dominated by noise rather than Cherenkov light emission are set
to zero. The cleaning mask are obtained with the standard MARS analysis
cleaning. Since the pixel layout of the MAGIC cameras is a hexagonal lattice,
we mapped them to a Cartesian lattice using bilinear interpolation to directly
apply CNNs (Nieto et al. 2019b).

### Validation on simulations

To evaluate the performance common metrics like ROC curves, energy and angular
resolution curves are used, applying the same quality cuts (see Fig. 2). The
reconstruction performance is obtained using MC gamma simulations coming
uniformly from a $0.4^{\circ}$ offset of the telescope pointing (ringwobble).
To demonstrate the robustness of CNNs trained with cleaned images, we tested
all methods for the background rejection against MC proton simulations and
observational off data, where we do not expect any gamma-ray signal.

Figure 2.: The performance parameters are obtained using the MC gamma
simulations (ringwobble). _Left)_ ROC curves tested against MC proton
simulations and observational off data. _Center)_ Angular resolution vs.
reconstructed energy. _Right)_ Energy resolution vs. simulated energy.

### Results on observational data

In this work, 2.93h observation of the standard gamma-ray candle Crab Nebula,
taken on four different nights in 2018 under good weather condition at low
zenith (zd < $35^{\circ}$), are considered. The data have been analyzed with
latest MARS software using the standard settings for the analysis focusing of
the medium energy (ME) - and low energy (LE) range. For CTLearn, we strictly
adopted the quality cuts from the MARS analysis. ME analysis (> $250$ GeV)
apply the cuts: valid stereo reconstruction, $\theta^{2}$ < 0.009
$\text{deg}^{2}$, hadronness < 0.16 and both hillas intensity sizes > 300 phe,
while the LE analysis (> $100$ GeV) apply the cuts: valid stereo
reconstruction, $\theta^{2}$ < 0.02 $\text{deg}^{2}$, hadronness < 0.28 and
both hillas intensity sizes > 60 phe. To fairly compare the results, obtained
with CNN-based models, with the standard approach (random forest (RF) for the
background rejection, Look-Up tables (LUTs) for the energy estimation and RF
for bidimensional direction reconstruction), the hadronness cut is adjusted in
the CTLearn analysis to equalize the background (bkg) rates for the
corresponding standard MARS analyses (ME or LE). A source detection is
determined using a $\theta^{2}$ plot (see Fig. 3 for the CTLearn ME analysis
with cleaned images) and the significance (Sig. in Tab. 1) calculation (Eq. 17
in (Li & Ma 1983)). The main properties of all analyses are summarized in Tab.
1. The sensitivity (Sen. in Tab. 1) is computed as the strength of the source
that gives excess/sqrt(background) = 5 after 50h.

Analysis | $N_{on}$ | $N_{off}$ | $N_{ex}$ |  $\gamma$ rate [/min] | bkg rate [/min] | Sen. [% Crab] | Sig. (Li&Ma)
---|---|---|---|---|---|---|---
MARS – ME | $819$ | $21.0\pm 2.6$ | $798.0\pm 28.7$ | $4.54\pm 0.16$ | $0.119\pm 0.015$ | $0.70\pm 0.05$ | $43.0\sigma$
CTLearn – ME (raw) | $629$ | $23.3\pm 3.1$ | $605.7\pm 25.3$ | $3.45\pm 0.14$ | $0.133\pm 0.018$ | $0.97\pm 0.08$ | $36.5\sigma$
CTLearn – ME (cleaned) | $844$ | $22.0\pm 2.7$ | $822.0\pm 29.2$ | $4.68\pm 0.17$ | $0.125\pm 0.015$ | $0.69\pm 0.05$ | $43.6\sigma$
MARS – LE | $3579$ | $679.0\pm 15.0$ | $2900.0\pm 61.7$ | $16.49\pm 0.35$ | $3.861\pm 0.086$ | $1.09\pm 0.03$ | $61.1\sigma$
CTLearn – LE (raw) | $2730$ | $673.7\pm 20.0$ | $2056.3\pm 56.0$ | $11.70\pm 0.32$ | $3.832\pm 0.114$ | $1.53\pm 0.05$ | $47.5\sigma$
CTLearn – LE (cleaned) | $3536$ | $680.7\pm 15.1$ | $2855.3\pm 61.3$ | $16.24\pm 0.35$ | $3.872\pm 0.086$ | $1.11\pm 0.03$ | $60.4\sigma$

Table 1.: Summary of all performed analyses of the same Crab Nebula sample.
Figure 3.: $\theta^{2}$ plot for the CTLearn ME analysis with cleaned images.

##  3 Conclusions and outlook

This contribution shows for the first time that CNN-based full-event
reconstruction works for the MAGIC telescopes and that CTLearn analyses are
capable of detecting the Crab Nebula with a clear signal. We demonstrate that
CNNs trained with cleaned images rather than raw images show a stronger
robustness, when applying them to observational data, and the performance
already matches the sensitivity of detection of the conventional analysis on
real data. The performance of CNNs trained with raw images can be optimized by
pixel-wise tuning of the NSB noise of the MCs (Vuillaume et al. 2021) to match
the NSB level of each observation. The selected TRN model is relatively
shallow and further performance enhancements are foreseen by increasing the
model depth/complexity. Future work is planned, where the full performance of
CNNs under various observation conditions are evaluated.

### Acknowledgments

We would like to thank the Instituto de Astrofísica de Canarias for the
excellent working conditions at the Observatorio del Roque de los Muchachos in
La Palma. The financial support of the German BMBF, MPG and HGF; the Italian
INFN and INAF; the Swiss National Fund SNF; the ERDF under the Spanish
Ministerio de Ciencia e Innovación (MICINN) (FPA2017-87859-P, FPA2017-
85668-P, FPA2017-82729-C6-5-R, FPA2017-90566-REDC, PID2019-104114RB-C31,
PID2019-104114RB-C32, PID2019- 105510GB-C31C42, PID2019- 107847RB-C44,
PID2019-107988GB-C22); the Indian Department of Atomic Energy; the Japanese
ICRR, the University of Tokyo, JSPS, and MEXT; the Bulgarian Ministry of
Education and Science, National RI Roadmap Project DO1-268/16.12.2019 and the
Academy of Finland grant nr. 317637 and 320045 are gratefully acknowledged.
This work was also supported by the Spanish Centro de Excelencia “Severo
Ochoa” SEV-2016- 0588, SEV-2017-0709 and CEX2019-000920-S, and "María de
Maeztu” CEX2019-000918-M, the Unidad de Excelencia “María de Maeztu”
MDM-2015-0509-18-2 and the "la Caixa" Foundation (fellowship
LCF/BQ/PI18/11630012), by the Croatian Science Foundation (HrZZ) Project
IP-2016-06-9782 and the University of Rĳeka Project 13.12.1.3.02, by the DFG
Collaborative Research Centers SFB823/C4 and SFB876/C3, the Polish National
Research Centre grant UMO-2016/22/M/ST9/00382 and by the Brazilian MCTIC, CNPq
and FAPERJ. TM acknowledges support from PID2019-104114RB-C32. JLC and DN
acknowledges partial support from The European Science Cluster of Astronomy &
Particle Physics ESFRI Research Infrastructures funded by the European Union’s
Horizon 2020 research and innovation program under Grant Agreement no. 824064.
SY acknowledges financial support from Google LLC through the Google Summer of
Code 2020 program. We acknowledge the support of NVIDIA Corporation with the
donation of a Titan X Pascal GPU used for part of this research.

This paper has gone through internal review by the MAGIC Collaboration.

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