Performance of the Neyman-Pearson Receiver for a Binary Thermal. Coherent State Signal

Transcription

1 ISSN Performance of the Neyman-Pearson Receiver for a Binary Thermal Coherent State Signal Kentaro Kato Quantum ICT Research Institute, Tamagawa University 6-- Tamagawa-gakuen, Machida, Tokyo , Japan Tamagawa University Quantum ICT Research Institute Bulletin, Vol5, No, -8, 05 Tamagawa University Quantum ICT Research Institute 05 All rights reserved No part of this publication may be reproduced in any form or by any means electrically, mechanically, by photocopying or otherwise, without prior permission of the copy right owner

2 Performance of the Neyman-Pearson Receiver for a Binary Thermal Coherent State Signal Kentaro Kato Quantum Communication Research Center, Quantum ICT Research Institute, Tamagawa University 6-- Tamagawa-gakuen, Machida, Tokyo Japan kkatop@labtamagawaacjp Abstract The performance of the quantum Neyman- Pearson receiver for a binary coherent state signal ie presence of thermal noise is numerically investigated To see the detection sensitivity of the quantum Neyman-Pearson receiver, its receiver operating characteristics (ROC) are numerically obtained The performance limit of the quantum Neyman-Pearson receiver is directly computed when the false alarm probability is very small, and the simulation result of the binary thermal coherent state signal is compared to that of the case of random phase signal I INTRODUCTION The Neyman-Pearson (NP) criterion was originally developed for binary hypothesis testing [, [ The idea of the NP criterion was applied to optimal signal detection problem ie early days of radar technologies [3, [4 It is nowadays considered as one of fundamental signal processing techniques for radar systems [5 The first application of the NP criterion to quantum signal detection was considered by Helstrom [6 Ie literatures [7, [8 the performance of the quantum NP receiver for a binary coherent state signal of random phase is well analyzed ie presence of thermal noise An analysis for the corresponding non-random phase case was discussed by Yoshitani [9 through a third-order perturbative calculation However, the analysis is valid only for the case of very small thermal noise due to the perturbative calculation, and hence the performance evaluation for the thermal coherent state signal having nonrandom phase in wider range of thermal noise exceeding the small noise case is remaining Therefore, the aim of this paper is to expand beyond the preceding works by focusing oe numerical evaluation of the receiver II NEYMAN-PEARSON CRITERION IN QUANTUM DETECTION Here, we introduce a general approach of the quantum NP criterion in accordance with Helstrom [0 Suppose that there are two quantum hypotheses, H 0 and H, that are associated with two distinct quantum-states ˆρ 0 and ˆρ, respectively, and let Π (ˆΠ 0, ˆΠ )(ˆ ˆΠ, ˆΠ ) denote a positive operator-valued measure (POVM) that describes the mathematical model of a detector The false alarm of the detection is characterized by Q 0 Pr{H H 0 } TrˆΠ ˆρ 0, () and the detection probability is defined as Q d Pr{H 0 H } Pr{H H } TrˆΠ ˆρ () The problem is to maximize the detection probability Q d under the constraint Q 0 α 0, where α 0 is a preassigned constant Using the undetermined multiplier method, the function F to be maximized is written as F Q d ζ(q 0 α 0 ) α 0 ζ +TrˆΠ (ˆρ ζ ˆρ 0 ) (3) with an undetermined multiplier ζ 0 If the operator ˆρ ζ ˆρ 0 can be decomposed into the form ˆρ ζ ˆρ 0 λ i (ζ) λ i (ζ) λ i (ζ) (4) i with its eigenvalues λ i (ζ) and eigenvectors λ i (ζ), then the detection operator ˆΠ (ζ) λ i (ζ) λ i (ζ), (5) i:λ i (ζ)>0 maximizes F for a fixed ζ when zero eigenvalues occur from a set of measure zero Letting α(ζ) λ i (ζ) ˆρ 0 λ i (ζ) (6) and i:λ i (ζ)>0 β(ζ) i:λ i (ζ)>0 λ i (ζ) ˆρ λ i (ζ), (7) the problem is reduced to finding the optimal ζ that maximizes β(ζ) under the constraint α(ζ) α 0 For this problem, one may use the bisection search (or its variant such as the golden section search) to find the optimal ζ, because the function α(ζ) is a monotonically decreasing function of ζ due to the convexity of the set of all possible points (Q 0,Q d ) (eg Fig4 See also the region D in Fig4 of the textbook [0 ) Therefore, a rough sketch of the calculation procedure for finding the best detection probability Q d β(ζ ) is given as follows: ) [α 0 is given ) Do the bisection search to find the optimal ζ under the constraint α(ζ) α 0 In each stage of the bisection search, compute α(ζ) by using Eq(6)

3 3) Once the optimal ζ is found, compute Q d β(ζ ) by using Eq(7) This procedure is used to find the best detection probability Q d for our problem in Section IV III MATRIX REPRESENTATION OF THERMAL COHERENT STATES Suppose that a binary coherent state signal { 0, γ } is ie pressure of thermal noise, where 0 is the vacuum state and γ the coherent state of complex amplitude γ Iis situation, we assume that the received signals are given by the thermal vacuum and thermal coherent states The density operator of the thermal vacuum is expressed ie following P -representation ˆρ 0 ˆρ th (0) π exp[ γ γ γ d γ, (8) where is the average photon number of thermal noise given by (e ω/kbt ) with the angular frequency ω of light, Boltzmann constant k B, and the absolute temperature T This state is also written as ˆρ 0 ( ν) ν n n n, (9) n0 where the real parameter ν is defined by ν /( + ),or ν/( ν) Oe other hand, the thermal coherent state with complex amplitude γ is given by ˆρ ˆρ th (γ) π exp[ γ γ γ γ d γ (0) The matrix representation of ˆρ ie basis { n } is given as follows [: m! m ˆρ n ( ν)νn n! ( ) ( ν)γ n m ν exp[ ( ν) γ L (n m) m [ ( ν) γ, ν if m n, () m ˆρ n ( n ˆρ m ), if m>n, () where L (m) n [x is the associated Laguerre polynomial ([, [4) By using Eq(9) and Eqs()-(), one can obtain a matrix representation of ˆρ ζ ˆρ 0 ie ordered basis { 0,,,} Since a computer can handle only a finite matrix size, one must employ an appropriate cut-off rule for the matrix size Iis study, we have used the same cut-off rule for the matrix size as that of the literature [3: For a finite size matrix representation ρ i [ m ˆρ i n n0,,,d m0,,,d of the state ˆρ i (i 0, ), (i) compute Trρ i and Trρ i, (ii) compare them with and ( ν)/( + ν), respectively, and (iii) check whether the errors are within a preassigned accuracy ɛ matrix IV PERFORMANCE OF THE NEYMAN-PEARSON RECEIVER FOR BINARY THERMAL COHERENT STATE SIGNAL Ie detectioeory, the receiver operating characteristic (ROC), which indicates the performance of Q d against α 0, is often used to evaluate the detection sensitivity of a receiver The ROC of the quantum NP receiver for decision between two pure coherent states 0 and γ is shown in (a) of Fig, where we have assumed γ is real and have set to γ 00, 0, 05,, and (See also APPENDIX) The effect of thermal noise can be observed in (b)-(b5) of Fig, where we have used the following conditions for calculation: γ is real; ˆρ 0 ˆρ th (0) and ˆρ ˆρ th (γ); γ 00, 0, 05,, and ; 0, 00, 0, and 5, orν 0, 00099, 009, 05 and 083 The calculation program is based oe procedure mentioned in Section II The source code of the calculation program was written in C++ and was compiled by Intel compiler on CentOS 5 (x86 64) To compute some special functions and to find the eigenvalues and eigenvectors of a matrix, GNU Scientific Library (GSL) [4 was used The accuracy ɛ matrix for determining the matrix size was set to 0 3, so that the number D of the rows (which equals to that of the columns) of the matrix representation ρ i (i 0, ) was determined according to satisfying this accuracy In each stage of the optimization process for parameter ζ, the eigenvalue problem of (/( + ζ))ρ (ζ/( + ζ))ρ 0 was solved instead of that of ρ ζρ 0 The initial values for the bisection search were set to ζ 0) lft /( + ζ0) lft ) 00 (or ζ0) lft 0) and ζ 0) rht /( + ζ0) rht ) (or ζ0) rht ), which was chosen heuristically to cover all the case under consideration Moreover, the accuracy for the bisection search ɛ search was set to ɛ search 0, that is, the iteration halted when ζ s) rht ζs) lft < ɛ search is satisfied at an sth stage In our simulation, we observed that the resulting ζ yields α(ζ )α 0 within accuracy 0 0 Finally, we numerically checked the completeness of the resulting eigenvectors with accuracy 0 It means that every entry of the matrix consisting of the sum of the dyadic product of the resulting eivenvectors is identical to the corresponding entry of the identity matrix within the accuracy 0 As shown in (b-) to (b-5), we observe the performance degradation of the quantum NP receiver due to the thermal noise in each γ From the comparison of the cases of γ 00 and γ, we can find the difference of the speed of the performance degradation against the

4 3 increase of the average number of photons of thermal noise Iis comparison, the latter might seem to be more degraded against the increase of thae former But, whee average number of photons of thermal noise is fixed, the ROC curves are lifted upwards as γ increases Hence the improvement of the signal-tonoise ratio is essential for the detection sensitivity of the quantum NP receiver In radar/lidar applications, the case of small α 0 is of much interest Fig shows the detection probability Q d versus γ when α 0 is small The simulation condition is as follows: α 0 0, 0 4, 0 6 ; γ is real, and 0 γ 8; 0, 005, 0, 05,, and 5, orν 0, 0048, 009, 033, 05, and 083; where the computing environment (including the initial set-ups, the verification level of the completeness of the eigenvectors, and so on) is the same as ie case of Fig From these figures, we see that the behavior of the curves in each figure is similar except its scaling of the probabilities It can be seen as a reflection of the trade-off betweee detection probability Q d and the false-alarm level α 0 Here we consider the case that the phase of the amplitude γ is uniformly distributed for comparison In this case the signal state ˆρ is replaced to ˆρ ˆρ th ( γ e ϕ ) π [ π 0 π exp[ γ γ e iϕ γ γ d γ dϕ exp[ γ + γ π I 0 [ γ γ γ γ d γ, (3) where I 0 [x is the modified Bessel function of order zero ([, [4, [5) The detection probability Q d for this random phase signal is given [7 by z Q d P n ( q)p z, (4) n0 where the fraction q is determined by α 0 q( ν)ν z + ν z+ and the decision level z (ln α 0 )/(ln ν), and P n0 ( ν)ν n, (5) P n ( ν)ν n exp[ ( ν) γ L n [ ( ν) γ /ν, (6) with the Laguerre polynomial L n [x ([, [4) The detection probability Q d for the case of random phase signal is shown in Fig 3, together with that for the nonrandom phase signal The solid lines are identical to the corresponding Q d shown in Fig, and the dashed lines are obtained from Eq(4) In each figure, the difference betweee detection probabilities of two cases becomes larger as γ increases The parameter γ can be regarded as a measure of closeness of the two signals The effect of randomness of the phase clearly appears whee two signals are apart from each other in each case of Conversely, it seems to be limited whee signals are closer V CONCLUSION We considered the quantum Neyman-Pearson receiver for a binary coherent state signal ie presence of thermal noise The receiver operating characteristic (ROC) curves of the quantum Neyman-Pearson receiver for binary thermal coherent state signal were numerically obtained The performance limit of the quantum Neyman- Pearson receiver was obtained via direct calculation of the eigenvalues and eigenvectors of the operator ˆρ ζ ˆρ 0 whee false alarm level α 0 0, 0 4, and 0 6 Further, the simulation result for the binary thermal coherent state signal is compared with the case of the signal with random phase This complements the preceding works done by Yoshitani [9 and Helstrom [8 in terms of the numerical evaluation of the quantum Neyman-Pearson receiver for a binary thermal coherent state signal ACKNOWLEDGMENTS The author is grateful to Professor Osamu Hirota of Tamagawa University for his helpful discussions This work was supported in part by JSPS KAKENHI Grant Number 5K0608 APPENDIX The case of binary pure state detection was analytically solved by Helstrom [6 This appendix is a short summary of his result To begin with, let us consider the following two pure states: H 0 :ˆρ 0 ψ 0 ψ 0 and H :ˆρ ψ ψ, (7) where ψ 0 ψ κe iθ, 0 <κ< and 0 θ<π If α 0 0, one can employ the Kennedy receiver [6 to construct the quantum NP receiver It is given by ˆΠ ˆ ψ 0 ψ 0 (8) and hence Q d κ (9) Here we let μ 0 r 00 ψ 0 + r 0 ψ, (0) μ r 0 ψ 0 + r ψ, () where r 00 ( ) + r, () +κ κ r 0 ( ) e iθ r 0 (3) +κ κ

5 noiseless (a) (b) (b) (b3) (b4) (b5) Fig Receiver operating characteristics (ROC) for binary coherent states ie presence of thermal noise (a) noiseless cases (γ 00, 0, 05, and ) (b) case for γ 00 with thermal noise 0, 00, 0, and 5 (b) γ 0 (b3) γ 05 (b4) γ (b5) γ

6 5 (a) (b) (c) Fig Detection probability Q d [% versus γ (a) α 0 0 (b) α (c) α These vectors form an ordered orthonormal basis B { μ 0, μ } Thee states ψ 0 and ψ are rewritten as where s 00 ( ) +κ + κ s, (6) s 0 ( ) +κ κ e iθ s 0 (7) ψ 0 s 00 μ 0 + s 0 μ [ s00 [ ψ 0 B, (4) s 0 ψ s 0 μ 0 + s μ [ s0 [ ψ B s, (5) Further, the matrix representations of ˆρ 0 and ˆρ ie basis B are respectively given as follows ˆρ 0 [ˆρ0 [ B ( + κ ) κeiθ κe iθ ( κ ), (8)

7 6 (a) (b) (c) Fig 3 Detection probability Q d [% for binary thermal coherent state signal of known phase and for that of unknown phase Solid line: known phase cases Dashed line: unknown phase cases (a) α 0 0 (b) α (c) α 0 0 6

8 7 and ˆρ [ˆρ [ B ( κ ) κeiθ κe iθ ( + κ ) Hence we have [ A00 A ˆρ ζ ˆρ 0 [ˆρ ζ ˆρ 0 B 0 A 0 A with (9) (30) A 00 ( κ ) ( + κ )ζ, (3) A 0 κeiθ κζeiθ, (3) A 0 κe iθ κζe iθ A 0, (33) A ( + κ ) ( κ )ζ (34) From this one can find the eigenvalues of ˆρ ζ ˆρ 0 as follows λ ( ζ Λ) < 0, (35) λ + ( ζ + Λ) > 0, (36) where Λ ( + ζ) 4ζκ The corresponding eigenvectors are then given by λ Û μ 0 u 00 μ 0 + u 0 μ [ λ B [ u00 u 0 λ + Û μ u 0 μ 0 + u μ where we have defined with u 00 u 0 [ λ + B [ u0 Û u 00 μ 0 μ 0 + u 0 μ 0 μ +u 0 μ μ 0 + u μ μ, [ [Û u00 u B 0 u 0 u, (37) u, (38) (39) Λ +(+ζ) κ Λ +(+ζ)λ κ u, (40) κ( ζ)e iθ Λ +(+ζ)λ κ u 0 (4) Letting ˆΠ 0 λ λ Û μ 0 μ 0 Û, ˆΠ 0 [ B Ω ( ζ) κ Λ ( ζ)κeiθ Λ ( ζ)κe iθ Ω ( ζ) κ (4) and ˆΠ λ + λ + Û μ μ Û ˆΠ [ B Ω ( ζ) κ Λ ( ζ)κeiθ Λ ( ζ)κe iθ Ω ( ζ) κ (43) with Ω Λ { Λ +(+ζ) κ }, the type-i error probability α(ζ) P ( 0) and the type-ii error probability β(ζ) P (0 ) are respectively given by α(ζ) ( + ζ) κ, Λ (44) β(ζ) ( + ζ) ζκ Λ (45) A typical behavior of α(ζ) and β(ζ) is illustrated in Fig 4 α(ζ) and β(ζ) Fig 4 Form this we observe there is a trade-off relation between α(ζ) and β(ζ) for the choice of ζ The remaining task for the the NP criterion is to find the optimal ζ that maximizes the detection probability β(ζ) under the constraint α(ζ) α 0 When α 0 is chosen as κ <α 0, the constraint α(ζ) α 0 can be replaced to α(ζ) κ since the maximum value of α(ζ) is κ and there is a trade off relation between α(ζ) and β(ζ) Solving the equation α(ζ) κ, the optimal ζ is 0 When α 0 is chosen as 0 <α 0 κ, thee constraint

9 8 α(ζ) α 0 can be replaced to α(ζ) α 0 Solving the equation α(ζ) α 0,wehave ζ ( κ ( κ )+( α 0 ) )κ, ( α 0 )α 0 where we have used the condition ζ>0 Substituting the optimal ζ into Eq(45), we have Q d β(ζ ) κ, if α 0 0; ( κ α 0 + ( α 0 )( κ )), (46) if 0 <α 0 κ ;, if κ <α 0 Thus we could reach to the result shown in Eq(33) of the textbook [0 (or Eq(8) of the literature [8) At the tail of this appendix, we must mentioat the analysis above is mainly based oe analysis used ie literature [7 which treats the problem of the minimax criterion Iis analysis we have used a technique of the square-root measurement [6 R S Kennedy, A near-optimum receiver for the binary coherent state quantum channel, MIT Res Lab Electron Quart Progr Rep, vol0, pp4-46, 973 [7 K Kato, Necessary and Sufficient Conditions for Minimax Strategy in Quantum Signal Detection, Proc IEEE ISIT 0, pp077-08, 0 REFERENCES [ J Neyman and E S Pearson, Oe Problem of the most Efficient Tests of Statistical Hypotheses, Phil Trans R Soc Lond A, vola3, no9, pp89-337, 933 [ J Neyman and E S Pearson, The testing of statistical hypotheses in relation to probabilities a priori, Proc Camb Philos Soc, vol9, no4, pp49-50, 933 [3 H V Hance, The optimization and analysis of systems for the detection of pulse signals in random noise, ScD dissertation, MIT, Jan, 95 [4 D Middleton, Statistical Criteria for the Detection of Pulsed Carriers in Noise I, J Appl Phys, vol4, no4, pp37-378, 953; D Middleton, Statistical Criteria for the Detection of Pulsed Carriers in Noise II, J Appl Phys, vol4, no4, pp379-39, 953 [5 M A Richards, Fundamentals of Radar Signal Processing, McGraw-Hill, New York, 005 [6 C W Helstrom, Detection Theory and Quantum Mechanics, Inform Contr, vol0, no3, pp54-9, 967; C W Helstrom, Detection Theory and Quantum Mechanics (II), Inform Contr, vol3, no, pp56-7, 968 [7 C W Helstrom, Performance of an Ideal Quantum Receiver of a Coherent Signal of Random Phase, IEEE Trans Aerosp Electron Syst, volaes-5, no3, pp56-564, 969 [8 C W Helstrom, Quantum Detection Theory, Progress in Optics, pp89-369, 97 [9 R Yoshitani, Oe Detectability Limit of Coherent Optical Signals in Thermal Radiation, J Stat Phys, vol, no4, pp , 970 [0 C W Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York, 976 [ B R Mollow and R J Glauber, Quantum Theory of Parametric Amplification I, Phys Rev, vol60, no5, pp , 967 [ M Abramowitz and I A Stegun, Ed, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55, National Bureau of Standards, United States Department of Commerce, Washington, DC, 964 [3 C W Helstrom, Bayes-Cost Reduction Algorithm in Quantum Hypothesis Testing, IEEE Trans Inform Theory, volit-8, no, 98 [4 GSL - GNU Scientific Library [5 G Lachs, Theoretical Aspects of Mixtures of Thermal and Coherent Radiation, Phys Rev, vol38, no4b, ppb0-b06, 965