Abstract—This is a note on quantum detection of quater-nary amplitude-shift keying (QASK) coherent state signal.The closed-form expression of the square-root measurement(SRM) of QASK coherent state signal is derived by solvingthe eigenvalue problem of the Gram matrix consisting ofthe signal. The Bayes-optimal detection and the minimaxdetection for QASK coherent state signal are respectivelyanalyzed by using novel iterative calculation algorithms ofNakahira et al. [15]. Toward derivation of the closed-formexpressions of the Bayes-optimal detection and the minimaxdetection, mathematical structure of the corresponding op-timal detection vectors is discussed based on the numericalcalculation results.
I. INTRODUCTION
The main role of quantum signal detection theory[1], [2], [3], [4] is to examine the performance limitof quantum state signals and to clarify the mathemati-cal structure of optimal quantum detection. Hence it isexpected to be a guiding theory that provides designmethods for optimal quantum detection that enables ahighly functional communication system which close toquantum limits beyond classical ones. In this article, weattempt to apply quantum signal detection theory to aparticular coherent state signal.
A coherent state signal is characterized by its mod-ulation format. For example, phase-shift keying (PSK)and quadrature amplitude modulation (QAM) are majorformats widely used in advanced digital coherent opticalcommunication systems [5]. Here let us recall somepreceding studies about coherent state signals based onquantum signal detection theory. The early work on theoptimal detection problem of PSK coherent state signalcan be found in the literatures [6], [7] in which theso-called Belavkin weighted square-root measurement(BWSRM) was introduced. Since PSK coherent statesignal is a kind of symmetric quantum state signals,analysis of symmetric signals may involve the case ofPSK coherent state signal. In this context, the analysis ofoptimal detection of symmetric quantum states by Ban etal. [8] is remarkable. Based on the analyses of Belavkinand Ban et al., it can be understood that the square-root measurement (SRM) for PSK coherent state signalis not only the Bayes-optimal detection at the uniformsignal distribution but also the minimax detection. Thenumerical comparison of the error rate performance of
quantum detection for M -ary PSK and M -ary QAMcoherent state signals was done under the assumption thatevery signal state is pure and the SRM is employed as areceiver [9]. Extending this result, numerical simulationsfor 4PSK, 8PSK and 16QAM coherent state signals in thepresence of thermal noise were performed by Cariolaroand Pierobon [10]. Since the error rate performance ofBayes-optimal detection depends on a priori probabilitydistribution of signal elements in general, the minimaxstrategy would be preferable than Bayes-optimal strategyin some cases. The error rate performance of the minimaxdetection for 16QAM coherent state signal was numeri-cally investigated by the author [11].
As for another type of coherent state signals, someresults with respect to amplitude-shift keying (ASK)coherent state signal can be found. In the literature [12]by Helstrom, the error rates of Bayes-optimal detectionfor ternary amplitude-shift keying (3ASK) and quaternaryamplitude-shift keying (QASK) coherent state signalswere computed using his Bayes-cost reduction algorithm[12]. In contrast, the closed-form expressions of the SRMand the minimax detection for 3ASK coherent state signalwere derived by the author [13], [14]. Taking accountof recent progress in the applications of quantum signaldetection theory, the analysis of M -ary ASK coherentstate signal is of importance. However, no comprehensiveanalysis of M -ary ASK coherent state signal has beendone. Towards a future study on M -ary ASK coherentstate signal, we focus on quaternary amplitude-shift key-ing (QASK) coherent state signal in this article as a firststep.
As mentioned above, the error rate performance ofthe Bayes-optimal detection for QASK coherent statesignal at the uniform signal distribution has been alreadyshown in the literature [12]. Hence the central problemin our analysis is to derive closed-form expressions ofoptimal quantum detection in each detection strategy.However, as we will see later, derivation of the closed-form expressions of optimal quantum detection for QASKcoherent state signal was restricted only to the case ofSRM. Aiming to break through this deadlock, we performnumerical analysis of QASK coherent state signal withthe help of the calculation algorithms of Nakahira et al.[15], because the results to be obtained in the numerical
Tamagawa University Quantum ICT Research Institute BulletinVol.6 No.1 : 9―24（2016）
10
calculation will be helpful to figure out the structure ofoptimal quantum detection.
The remaining part of this article is organized asfollows. In Section II, the error rate performance andthe optimal detection operators of QASK coherent statesignal are investigated for the SRM, the Bayes-optimaldetection at the uniform signal distribution, and the min-imax detection, respectively. In the first half of SectionII, the closed-form expression of the SRM for QASKcoherent state signal is derived by solving the eigenvalueproblem of the corresponding Gram matrix. In the re-maining part of Section II, the Bayes-optimal detectionand the minimax detection cases are numerically investi-gated. In each case, the optimal detection vectors and thecorresponding minimal average probability of error arenumerically shown. In Section III, some discussions onthe properties of QASK coherent state signal are given,and we summarize the results in Section IV.
II. ERROR RATE PERFORMANCE OF QASK COHERENTSTATE SIGNAL
A. QASK coherent state signal
Let |ψi〉 denote a quantum state that corresponds to asignal element. We define QASK coherent state signal asfollows (See also Appendix A).
where |α〉 is a coherent state defined by a|α〉 = α|α〉with the photon annihilation operator a. In this article,we assume α > 0 for simplicity.
The signal constellation of QASK coherent sate signalis shown in Fig. 1, where xc = (a + a†)/2, xs = (a −a†)/2i, and i =
√−1. Let p = (p1, p2, p3, p4) denote
xs
xc
−3α −α α(> 0) 3α
ψ1 ψ2 ψ3 ψ4
0
Fig. 1. Signal constellation of QASK.
a probability distribution of the signal elements. If thedistribution is uniform, p = u = (1/4, 1/4, 1/4, 1/4),then the average number of signal photons for QASKcoherent state signal is given as ns = 5|α|2.
In general, the Gram operator and the Gram matrix ofpure states |ψi〉 are respectively defined by
where M is the number of states. Suppose that |ψi〉 arelinearly independent. Then the Gram operator is strictlypositive-definite, G > 0. The SRM vectors are definedby
|μ•i 〉 ≡ G−1/2|ψi〉, 1 ≤ i ≤M. (4)
Conversely,
|ψi〉 = G1/2|μ•i 〉, 1 ≤ i ≤M. (5)
Letting γ = {|μ•i 〉 : 1 ≤ i ≤ M}, the set γ is
an orthonormal basis for the space spanned by linearlyindependent states {|ψi〉}. In fact, it satisfies
〈μ•i |μ•
j 〉 = δij ∀(i, j), and
M∑k=1
|μ•k〉〈μ•
k| = 1, (6)
where δij is the Kronecker delta and 1 is the identity.Note that matrix representation of G in the basis γ is theGram matrix G:
[G]γ =[〈μ•
i |G|μ•j 〉]
=[〈μ•
i |G1/2G1/2|μ•j 〉]
= [〈ψi|ψj〉] = G. (7)
Applying the completeness relation of Eq.(6) to Eq.(5),we have
|ψi〉 =
(M∑k=1
|μ•k〉〈μ•
k|)G1/2|μ•
i 〉
=M∑k=1
[〈μ•
k|G1/2|μ•i 〉]|μ•
k〉
=M∑k=1
(g1/2
)ki|μ•
k〉, (8)
where(g1/2
)ki
is the (k, i)-entry of G1/2. From this weobserve that the ith column of G1/2 corresponds to thecoefficients in the expansion by the basis γ. Conversely,the relation G1/2G−1/2 = 1 and the completeness rela-tion provide the following expression.
|μ•i 〉 = G1/2G−1/2|μ•
i 〉
= G1/2
(M∑k=1
|μ•k〉〈μ•
k|)G−1/2|μ•
i 〉
=M∑k=1
[〈μ•
k|G−1/2|μ•i 〉] (
G1/2|μ•k〉)
=M∑k=1
(g−1/2
)ki|ψk〉, (9)
11
where(g−1/2
)ki
is the (k, i)-entry of G−1/2.From the analysis mentioned above, the column vector
representation of each signal element of QASK coherentstate signal in the basis γ is given as follows.
|ψi〉 .= [|ψi〉]γ =
⎡⎢⎢⎣
(g1/2)1i(g1/2)2i(g1/2)3i(g1/2)4i
⎤⎥⎥⎦ , i = 1, 2, 3, 4, (10)
where the symbol .= means the left-hand side is rep-
resented by the right-hand side. Similarly, the columnvector representation of each SRM vector for QASKcoherent state signal in the basis γ is given by
|μ•i 〉 .
= [|μ•i 〉]γ =
⎡⎢⎢⎣
δ1iδ2iδ3iδ4i
⎤⎥⎥⎦ , i = 1, 2, 3, 4. (11)
B. Square-root of Gram matrix of QASK
The Gram matrix of QASK coherent state signal isgiven by
G =
⎡⎢⎢⎣
1 κ κ4 κ9
κ 1 κ κ4
κ4 κ 1 κκ9 κ4 κ 1
⎤⎥⎥⎦ , (12)
where κ = exp[−2|α|2]. The eigenvalues and the cor-responding eigenvectors of G are respectively given asfollows:
and where the eigenvectors have been normalized. Sincethe signal elements are linearly independent, the Grammatrix G of Eq.(12) is a positive definite matrix. There-fore every eigenvalue is positive, λi > 0, i = 1, 2, 3, 4.
Further, the projectors Pi = �λit�λi, where t�λi means the
Some specific examples of the square-root of the Grammatrix for QASK coherent state signal are shown inAppendix B. Further, we show some examples of thecolumn vector representation of the signal elements inAppendix C for the later analysis done in Section III.
13
C. SRM for QASK
The detection vectors of the SRM are defined by|μ•
i 〉 = G−1/2|ψi〉. The column vector representation of|μ•
i 〉 for QASK coherent state signal in the basis γ wasalready shown in Eq.(11). In accordance with Theorem 5of the literature [16], we set p• = (p•1, p
•2, p
•3, p
•4) with
p•1 = p•4 =(g1/2)22
2{(g1/2)11 + (g1/2)22
} , (47)
p•2 = p•3 =(g1/2)11
2{(g1/2)11 + (g1/2)22
} . (48)
It satisfies
|μ•i 〉〈μ•
i |(p•i |ψi〉〈ψi| − p•j |ψj〉〈ψj |
)|μ•
j 〉〈μ•j | = 0 (49)
for every (i, j). Therefore, Π• = {|μ•i 〉〈μ•
i | : i =1, 2, 3, 4} becomes the Bayes-optimal detection strategyat the signal distribution p•, and the closed-form expres-sion of the minimal average probability of error at p• isgiven by
P •e = min
ΠPe(Π,p•)
= Pe(Π•,p•) = 1− (g1/2)11(g
1/2)22. (50)
This minimal error probability P •e at p• is illustrated
in (a) of Fig. 2. The associated signal probabilities,p•1 = p•4 and p•2 = p•3, are plotted in (b) and (c) ofFig. 2, respectively. In each figure, the parameter κ wastaken from 0.001 to 0.99. Some specific examples ofthe optimal distribution p•, the channel matrix obtainedby the SRM, and the minimal error probability P •
e forκ = 0.1, 0.3, 0.5, 0.7, and 0.9 are shown in Appendix D.
D. Bayes-optimal detection of QASK at the uniform input
Numerical analysis of error rate performance of theBayes-optimal detection for QASK coherent state signalat the uniform signal distribution can be found in Fig.1 ofthe literature [12] by Helstrom. In this section, we redoa numerical simulation in the same problem settings asthat but by using another algorithm.
The problem is to find the minimal error probabilityP bayese (u) such that
P bayese (u) = min
ΠPe(Π,u), (51)
where Π stands for a positive operator-valued measure(POVM). For this type of optimization problem, severalnumerical calculation algorithms have been developed[12], [17], [18], [15]. In this article, we use Nakahira’siterative algorithm for finding the Bayes-optimal errorprobability (Section IV.A of the literature [15]). Calcula-tion program was implemented by Mathematica, and theconstant for stopping criteria in Nakahira’s algorithm wasset to be δPC = 10−12. The optimality of the simulationresults has been verified with the condition (15) of theliterature [15] which is equivalent to the Holevo’s originalcondition for Bayes-optimality [2].
1.0E-1
1.0E-2
1.0E-3
1.0E-4
1.0E-5
1.0E-6
1.0E-7
1.0E-0
10.10.010.001
(a)
10.10.010.001
(b)
10.10.010.001
(c)
SRM
SRM
SRM
0.21
0.22
0.23
0.24
0.25
0.26
0.24
0.25
0.26
0.27
0.28
0.29
aver
age
pro
bab
ilit
y o
f er
ror
pro
bab
ilit
y o
f si
gnal
pro
bab
ilit
y o
f si
gnal
Fig. 2. SRM: (a) P •e , (b) p•1 = p•4 , and (c) p•2 = p•3 .
The simulation results are shown in Fig. 3, whichis essentially the same as Fig.1 of the literature [12]although the parameters of horizontal axis are different.
14
The parameter κ in Fig. 3 was taken from 0.001to 0.99. Some specific examples of the Bayes-optimaldetection vectors |μbayes
i (u)〉, the channel matrix obtainedby the Bayes-optimal detection, and the minimal errorprobability P bayes
e (u) for κ = 0.1, 0.3, 0.5, 0.7, and 0.9are shown in Appendix E.
1.0E-1
1.0E-2
1.0E-3
1.0E-4
1.0E-5
1.0E-6
1.0E-7
1.0E-0
10.10.010.001
Bayes, uniform
aver
age
pro
bab
ilit
y o
f er
ror
Fig. 3. Bayes-optimal detection at u: Pbayese (u).
E. Minimax detection of QASK
The minimax strategy is one of fundamental strategiesin quantum signal detection theory [4], [19], [20]. Theminimax detection problem is formulated as follows:
P ◦e = Pe(Π
◦,p◦)= min
Πmaxp
Pe(Π,p)
= maxp
minΠ
Pe(Π,p). (52)
To solve this problem, Nakahira’s iterative calculationalgorithm for the minimax detection problem (SectionV.A of the literature [15]) is used. Like in the Bayes-optimal detection case, calculation program was imple-mented by Mathematica and the stopping constant wasset to δPC = 10−12. The simulation results have beenverified with the condition (22) of the literature [15]which is equivalent to the optimality conditions of Hirotaand Ikehara [4]. The simulation results are shown in Fig.4. Note that numerically obtained result on the optimal
probability p◦1 (or p◦2) is equal to that of p◦4 (p◦3). Thisis a reflection of its own symmetric structure of QASKcoherent state signal. Some specific examples of theminimax distribution p◦, the minimax detection vectors|μ◦
i 〉, the channel matrix, and the minimax value P ◦e for
κ = 0.1, 0.3, 0.5, 0.7, and 0.9 are shown in Appendix F.
III. SOME DISCUSSIONS ON QASK COHERENT STATESIGNAL
A. Comparison of SRM, Bayes-optimal, minimax
Here let us compare the three cases considered in thepreceding section. Basically, the following discussion isparallel to Section IV of the literature [14].
To begin with, we define the following factors.• The rate of difference for Pe:
ε(Pe) =P ◦e − P •
e
P •e
and ε′(Pe) =P bayese (u)− P •
e
P •e
.
• The rate of difference for p1:
ε(p1) =p◦1 − p•1
p•1and ε′(p1) =
1/4− p•1p•1
.
• The rate of difference for p2:
ε(p2) =p◦2 − p•2
p•2and ε′(p2) =
1/4− p•2p•2
.
The behavior of these factors is shown in Fig.5. In eachcomparison, the parameter κ was taken from 0.001 to0.99. Fig. 5 (a) shows ε(Pe) > 0.0 and ε′(Pe) < 0.0.That is,
P ◦e > P •
e > P bayese (u) (53)
holds for 0.001 ≤ κ ≤ 0.99. The similar relation wasobserved also in the case of 3ASK [14]. Similarly, weobserved ε(p1) < 0.0 and ε′(p1) > 0.0 in (b) of Fig. 5.Hence we can say that
p◦1 < p•1 < 0.25 (54)
holds for 0.001 ≤ κ ≤ 0.99. Alternatively,
p◦2 > p•2 > 0.25 (55)
holds for 0.001 ≤ κ ≤ 0.99, because ε(p2) > 0.0 andε′(p2) < 0.0 are observed in (c) of Fig. 5. Overviewing(a), (b), and (c) of Fig. 5 and the specific examples forκ = 0.1, we observed that the structure of the detectionvectors of the SRM is similar to that of the Bayes-optimal detection at the uniform distribution of signalelements when κ < 0.1. On the other hand, all the optimalprobability of signal converges to 1/4 when κ is close to1, but this is just a reflection of pure guessing situationcaused by the preparation of almost identical quantumstates.
A typical schematic of the relationships (53)-(55) isshown in Fig. 6, where we have assumed p1 = p4
15
1.0E-1
1.0E-2
1.0E-3
1.0E-4
1.0E-5
1.0E-6
1.0E-7
1.0E-0
10.10.010.001
(a)
10.10.010.001
(b)
10.10.010.001
(c)
minimax
minimax
minimax
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
0.25
0.26
0.27
0.28
0.29
0.30
0.31
0.32
aver
age
pro
bab
ilit
y o
f er
ror
pro
bab
ilit
y o
f si
gnal
pro
bab
ilit
y o
f si
gnal
Fig. 4. Minimax detection: (a) P ◦e , (b) p◦1 = p◦4 , and (c) p◦2 = p◦3 .
and p2 = p3 for simplicity. In this figure, the con-cave curve stands for the all of Bayes-optimal detectioncases, P bayes
the case of SRM, Point B(0.18750, 0.24621) the caseof minimax detection, and Point C(0.25000, 0.23348)the case of Bayes-optimum detection at the uniformdistribution of signal elements. The straight lines (dashed,dotted and dot-dashed) are tangent lines that touch tothe concave curve at Points A, B, and C, respectively.Each line represents the error rate performance of thecorresponding detection strategy when the probabilitydistribution of signal elements varies from the optimalone. The error rate performance of the minimax detectionis stable even if the distribution varies. This is a basicfeature of the minimax detection.
B. Toward derivation of the optimal detection vectors
In the analysis of error rate performance for the Bayes-optimal detection and the minimax detection, we per-formed a numerical calculation. Hence the closed-formsolutions are still open. Toward analytical derivation ofthe optimal detection vectors in each case, we analyze thestructure of the vectors based on the examples shown in
16
Sections II.D and II.E. From observations of the results,a template of the optimal detection vectors is conjecturedto the following form.
[|μbayes
1 (u)〉]γ, or [|μ◦
1〉]γ →
⎡⎢⎢⎣
ABCD
⎤⎥⎥⎦ ; (56)
[|μbayes
2 (u)〉]γ, or [|μ◦
2〉]γ →
⎡⎢⎢⎣−BAD−C
⎤⎥⎥⎦ ; (57)
[|μbayes
3 (u)〉]γ, or [|μ◦
3〉]γ →
⎡⎢⎢⎣−CDA
−B
⎤⎥⎥⎦ ; (58)
[|μbayes
4 (u)〉]γ, or [|μ◦
4〉]γ →
⎡⎢⎢⎣
DCBA
⎤⎥⎥⎦ . (59)
As demonstrated in the literatures [4], [21], this type ofconjecture might be helpful to derive the closed-formexpressions. If these are the optimal detection vectors,then the parameters A,B,C, and D must satisfy theorthonormality condition,
A2 +B2 + C2 +D2 = 1, AD +BC = 0. (60)
In the case of Bayes-optimal detection at the uniformdistribution of signal elements, its optimality conditionyields the following equations.
(60), (61), and (62) form a system of equations forunknowns A, B, C, and D. Unfortunately, it is stilldifficult to solve this system of equations analytically.However, it has been verified that the same results as theexamples shown in Section II.D are numerically obtainedfrom the system of equations. Although it is just onlya numerical verification, we expect the set of equationsmentioned above correctly captures an algebraic structureof the optimal detection. .
In the case of minimax detection, the following condi-tions are enforced:
• Symmetry of the minimax distribution.
p◦1 = p◦4 and p◦2 = p◦3, (63)
and p◦1 + p◦2 + p◦3 + p◦4 = 1 and p◦i ≥ 0 for every i.• Bayes-optimality at the minimax distribution.
We expect the conditions, (60) and (63)-(66), will behelpful for finding the analytical solution to the minimaxdetection problem of QASK.
IV. SUMMARY
Quantum detection of quaternary amplitude-shift key-ing (QASK) coherent state signal was investigated. Theclosed-form expression of the square-root measurement(SRM) for QASK coherent state signal was derived bysolving the eigenvalue problem of the correspondingGram matrix. The optimal detection vectors of the Bayes-optimal detection at the uniform distribution of signalelements and the minimax detection were respectivelycalculated by Nakahira’s iterative calculation algorithms[15]. Toward derivation of the closed-form expressions ofthe Bayes-optimal detection and the minimax detection,the structure of optimal detection vectors was discussedbased on the calculation results. From this, templates ofthe optimal detection vectors for the Bayes-optimal detec-tion at the uniform signal distribution and for the minimaxdetection were conjectured. Based on this conjecture,the systems of equations to determine the correspond-ing optimal detection vectors for QASK coherent statesignal were proposed. Thus the derivation of the closed-form expressions for the Bayes-optimal detection at theuniform signal distribution and for the minimax detectionfor QASK coherent state signal are still remaining. Thisproblem will be discussed else where.
ACKNOWLEDGMENT
The author is grateful to Professor Osamu Hirota ofTamagawa University for his encouraging comments.This work was supported in part by JSPS KAKENHIGrant Number JP15K06082.
17
0.0 0.1 0.2 0.3 0.4 0.50.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
B A C
aver
age
pro
bab
ilit
y o
f er
ror
QASK
condi�ons
Fig. 6. SRM, minimax, and Bayes-optimal cases for κ = 0.7
REFERENCES
[1] C. W. Helstrom, QUANTUM DETECTION AND ESTIMATION THE-ORY, Academic Press, New York, 1976.
[2] A. S. Holevo, “Statistical decision theory for quantum systems,”J. Multivar. Analys., vol.3, pp.337-394, 1973.
[3] H. P. Yuen, R. S. Kennedy, and M. Lax, “Optimum testing ofmultiple hypotheses in quantum detection theory,” IEEE Trans.Inform. Theory, vol.IT-21, no.2, pp.125-134, 1975.
[4] O. Hirota and S. Ikehara, “Minimax strategy in the quantumdetection theory and its application to optical communication,”Trans. IECE. Japan, vol.E65, pp.627-633, 1982.
[5] K. Kikuchi, “Digital coherent optical communication sys-tems:Digital coherent optical communication systems,” IEICEElectronics Express, vol.8, no.20, pp.1642-1662, 2011.
[6] V. P. Belavkin, “Optimal multiple quantum statistical hypothesistesting,” Stochastics, vol.1, pp.315-345, 1975.
[7] V. P. Belavkin, “Optimal distinction of non-orthogonal quantumsignals,” Radio Engineering and Electronic Physics, vol.20, pp.39-47, 1975.
[8] M. Ban, K. Kurokawa, R. Momose, and O. Hirota, “Optimummeasurements for discrimination among symmetric quantum statesand parameter estimation,” Int. J. Theor. Phys., vol.36, no.6,pp.1269-1288, 1997.
[9] K. Kato, M. Osaki, M. Sasaki, and O. Hirota, “Quantum detectionand mutual information for QAM and PSK signals,” IEEE Trans.Commun., vol.47, no.2, pp.248-254, 1999.
[10] G. Cariolaro, and G. Pierobon, “Performance of quantum datatransmission systems in the presence of thermal noise,” IEEETrans. Commun., vol.58, no.2, pp.623-630, 2010.
[11] K. Kato, “Error performance of quantum minimax receiver for16QAM coherent state signal,” Tamagawa University QuantumICT Research Institute Bulletin, vol.2, no.1, pp.19-24, 2013.
[12] C. W. Helstrom, “Bayes-cost reduction algorithm in quantumhypothesis testing,” IEEE Trans. Inform. Theory, vol.IT-28, no.2,1982.
[13] K. Kato, “Quantum minimax receiver for ternary coherent statesignal in the presence of thermal noise,” J. Phys.: Confer. Series,vol.414, 012039, 2013.
[14] K. Kato, “Square-root measurement for ternary coherent statesignal,” Tamagawa University Quantum ICT Research InstituteBulletin, vol.3, no.1, pp.29-33, 2014.
[15] K. Nakahira, K. Kato, and T. S. Usuda, “Iterative methods forfinding optimal quantum measurements under minimum-error andminimax criteria,” Phys. Rev. A, vol.91, 012318, 2015.
[16] C. Mochon, “Family of generalized “pretty good” measurementsand the minimal-error pure-state discrimination problems forwhich they are optimal,” Phys. Rev. A, vol.73, 032328, 2006.
[17] M. Jezek, J. Rehacek, and J. Fiurasek, “Finding optimal strategiesfor minimum-error quantum-state discrimination,” Phys. Rev. A,vol.65, no.6, 06301(R), 2002.
[18] Y. C. Eldar, A. Megretski, and G. C. Verghese, “Designing optimalquantum detectors via semidefinite programming,” IEEE Trans.Inform. Theory, vol.49, no.4, 2003.
[19] K. Kato, “Necessary and sufficient conditions for minimax strat-egy in quantum signal detection,” 2012 IEEE Int. Symp. In-form. Theory Proc., pp.1082-1086, 2012.
[20] K. Nakahira, K. Kato, and T. S. Usuda, “Minimax strategy inquantum signal detection with inconclusive results,” Phys. Rev. A,vol.88, 032314, 2013.
[21] M. Osaki, M. Ban, and O. Hirota, “Derivation and physicalinterpretation of the optimum detection operators for coherent-state signals,” Phys. Rev. A, vol.54, no.2, pp.1691-1701, 1996.
APPENDIX
A. Definition of QASK
We have defined QASK coherent state signal by Eq.(1).However, QASK coherent state signal may be defined asfollows.
S ′ = {|0〉, |2α〉, |4α〉, |6α〉}.When this type of definition is employed, the averagenumber of signal photons is given as ns = 14|α|2. Evenin this case, the Gram matrix is given by Eq.(12).
B. Numerical example of the square-root of Gram matrix
The following examples are obtained from Eqs.(13)-(16) and (17)-(20).
The quadrature phase-shift keying (QPSK) coherentstate signal is defined as
S = {|α〉, |iα〉, |−α〉, |−iα〉}.The average number of signal photons for QPSK isgiven as ns = |α|2. The optimality of the SRM for thesymmetric pure state signal (including PSK coherent statesignal) was proved by Belavkin [6] and by Ban et al. [8],independently.
The Gram matrix of QPSK is given as follows [21].
G =
⎡⎢⎢⎣
1 Zc + iZs ζ2 Zc − iZs
Zc − iZs 1 Zc + iZs ζ2
ζ2 Zc − iZs 1 Zc + iZs
Zc + iZs ζ2 Zc − iZs 1
⎤⎥⎥⎦ ,
where ζ = exp[−|α|2], Zc = ζ cos[|α|2], and Zs =ζ sin[|α|2]. Since this matrix is a circular matrix, itseigenvalues and eigenvectors are respectively given asfollows.
λQPSK1 = 1 + ζ2 − 2Zc, �λQPSK
1 =1
2
⎡⎢⎢⎣−11−11
⎤⎥⎥⎦ ;
λQPSK2 = 1 + ζ2 + 2Zc, �λQPSK
2 =1
2
⎡⎢⎢⎣
1111
⎤⎥⎥⎦ ;
λQPSK3 = 1− ζ2 − 2Zs, �λQPSK
3 =1
2
⎡⎢⎢⎣
i−1−i1
⎤⎥⎥⎦ ;
λQPSK4 = 1− ζ2 + 2Zs, �λQPSK
4 =1
2
⎡⎢⎢⎣−i−1i1
⎤⎥⎥⎦ .
From these, the minimal average probability of error atthe uniform distribution of signal elements is given asfollows (See also [9]).
Pe(QPSK) = 1− 1
16
(4∑
i=1
√λQPSKi
)2
. (67)
Here let us compare the error rate performance ofQASK coherent state signal with that of QPSK. SinceQPSK coherent state signal is a symmetric pure statesignal, its SRM is identical to the Bayes-optimal detectionat the uniform distribution of signal elements and to theminimax detection. The simulation results are shown inFig. 7. When the designed error probability is set toPe = 10−5, QASK requires 13.2 photons, while QPSKrequires 5.4 photons. Thus, 3.9dB power budget of QPSK
is expected than QASK at Pe = 10−5. In an ordinaryusage of the optical communications system, QPSK co-herent state signal is clearly better than QASK. However,QASK might have a potential than QPSK in some casesof unusual usage of an optical communications system.Such applications of QASK coherent state signal will bediscussed else where.
