Quantum Signal Processing and Non-Orthogonal State Detection

Transcription

1 Sesors & Trasducers Vol. 60 Issue December 03 pp Sesors & Trasducers 03 by IFSA Quatum Sigal Processig ad o-orthogoal State Detectio Webi YU Baoyu ZHEG Shegmei ZHAO Istitute of Sigal Processig ad Trasmissio ajig Uiversity of Posts ad Telecommuicatios ajig 0003 Chia Tel.: fax: Received: September 03 /Accepted: 5 ovember 03 /Published: 3 December 03 Abstract: Quatum state sigal for most time is o-orthogoal sigal; it s ot fit to applyig the classical sigal processig methods to their sigal aalyzig ad estimatig. I accordace with this we discussed a basic model for pure quatum sigal processig. By combiig the least-squares estimatio ad Positive-Operator- Valued Measure we furthermore developed a ovel quatum detectio method for o-orthogoal state sigal. The simulatio test result i additio verified its effectiveess. Copyright 03 IFSA. Keywords: Sigal processig Quatum detectio o-orthogoal state Least-square estimatio POVM.. Itroductio The quatum sigal processig (QSP) cocept is itroduced by Eldar i article [ ] yet he barely discussed this framewor of sigal processig with a view o the algorithm of classic sigal processig (CSP). The classical sigals are orthogoal states as a special case of quatum sigal while most quatum state sigals are o-orthogoal states [9-]. The detectig method of o-orthogoal states is show i paper [3 4]. Paper [5] itroduced the Positive- Operator-Valued Measure (POVM) of quatum states. To detectig quatum sigal effectively we focus o the problem of pure quatum sigal processig. I this paper Sectio describes this model sectio 3 discusses the least-squares estimatio for quatum sigals. Sectio 4 surveys the detectio of oorthogoal state sigal usig the least-square estimatio ad POVM. Sectio 5 presets the umerical results.. Quatum Sigal Processig.. The Model of Quatum Sigal Processig I order to build a framewor of QSP we redefie the quatum sigal little differetly from the classic sigal. The descriptio of classic sigal (here refers to discrete time sigal) is ow as the time series a i i the Hilbert space l while i the case of quatum mechaics the descriptio of quatum sigal series is i () where i is the desity operator of quatum bits (-qubits) ad is the ifiite dimesio space which is a tesor product of etire the -dimesioal uitary spaces. Therefore i ca Article umber P_654 54

2 Sesors & Trasducers Vol. 60 Issue December 03 pp be cosidered as a quatum sigal series i this tesor space. The defiitio of the quatum sigal eables us to propose the framewor of QSP. As is illustrated i Fig. both the iput ad the output sigal are the classical sigals. But the erel of the QSP procedure is the quatum system. Cosequetly the sigals eed to be trasformed ito quatum sigals. Classic Sigal Quatum Sigal Quatum Sigal Classic Sigal Modulus System Quatum System Measuremet Fig.. The QSP model. Meawhile to process classic sigal of macroworld system with quatum mechaics a approach of sigal modulatio is eeded to trasform sigal from classic to quatum. Ad a quatum measuremet must be tae o the last step of the QSP procedure so that outcome iformatio ca be read by the macro world. I case that both iput ad output are classic sigal QSP ca provide the advatages of parallel computig which is exactly the purpose of quatum computatio. As a example the quatum algorithm of large umber factorizatio proposed by Shor has wo more improvemet o the calculatio efficiecy tha the covetioal algorithm of classic does [5-8]... Trasform for Quatum System To stress out our study of quatum sigal trasform the modulatio of classic sigal ad the quatum measuremet are ot icluded i the followig discussio. The the trasform which describes the quatum system i mathematics is showed up. For a iput series i of a quatum system assume the output series to be i. If for each i the quatum system does t cause the correlatio ' betwee i ad j i j that is the output i ' ad j still remai idepedet with each other the we ca give a equatio of the relatioship betwee iput ad output sigal accordig to the quatum operatios formalism [5]. tr U U ' i i ev i ev. () Yet a geeral quatum system will cause the associatio of the iput sigal elemets which maes the states of every elemet etagled. I this case the quatum system should be preseted by U which is the operator o the discrete ifiite dimesioal space. Let matrix is i be the iput series so the desity 0 (3) The the desity matrix of output series is ' tr ev U ev U. As ca be see the states of every elemet of output series have already etagled..3. Frequecy Domai Trasform of Quatum Sigal As we have metioed above the quatum sigal essetially is the tesor i Hilbert space. There is still a problem that the choice of differet basis results i differet represetatio of the tesor. Ad QSP also has the similar trasform as frequecy domai (FD) trasform i the field of CSP. Cocerig the quatum discrete Fourier trasform of pure states j / j e 0 (4) as j refers to the basis state ad for the geeral states we have where x j y j j0 0 y j/ xje j0. 54

3 Sesors & Trasducers Vol. 60 Issue December 03 pp Quatum discrete Fourier trasform is a ey part of quatum algorithm such as large umber factorizatio ad may other iterestig quatum algorithms. Ad oe of its importat roles is phase estimatio that ca approximate the eigevalues of uitary operator o some specified occasios [7]. But here trasformig quatum sigal from time domai (TD) to the FD is its aother importat applicatio. I order to get the formalism of this trasform we eed to exted the cocept of pure states to mixed states. For the quatum series the tesor i Hilbert space is expressed as ad its basis is R The let R i i i. i 0 decomposed upo the tesor basis we ca get j0 x R j j. Cosequetly the correspodig quatum discrete Fourier trasform is here x R y R j j j0 0 y j/ xje j0 (5) ad yr is the outcome of the trasform. With 0 this quatum FD trasform the quatum liear filter that is able to separate quatum sigals which is mixed i TD but idepedet i FD ca be possibly realized. 3. Least-Squares Estimatio of Quatum Sigal I the field of CSP there is a problem of the least-squares (LS) estimatio of Wieer filter which ca estimate the sigal that eeds detectio. oe the less i the field of QSP the same problem still exists: how to subtract the idispesable sigal we eed from the iput series. As respects we ll give a optimal estimatio of the quatum sigal i the LS sese below. X S V Fig.. Optimal LS estimatio of quatum sigal. H As is show i Fig. assume X S V S is the iput quatum sigal ad S V 0 l0 here ad l stad for the desity matrices of oe qubit. We hope that the quatum sigal become S after it goes through the filter H ad its elemet S is the correct estimatio of S o the criterio of LS. It meas to miimize e S S as much as possible. ote that S S are the th elemets of tesor S ad S respectively ad. Therefore it s a evidet result that E e mi. By the priciple of orthogoal projectio it meas that etire e are orthogoal with ad S HX HX h X i i i0 l (6) where HX is the liear combiatio of X i. So etire e must be orthogoal with all the X i i which case we have E ex i =0 i 0. (7) Accordig to equatio (3.) there are a group of equatios i0 where RXX i j h R i j R ( j) i XX SX j 0 (8) is the autocorrelated fuctio of X ad RSX ( j) is the cross-correlated fuctio for S ad X. I this way ca we get the last elemets of H. Ad repeatig this step the all elemets of H ca be obtaied. Obviously the matrix H we obtaied usig this method complies to the axiomatic stadards for quatum operatios [5]. 543

4 Sesors & Trasducers Vol. 60 Issue December 03 pp ) Let H ormalized the tr H ay. 0 for p ) It s easy to prove that for ay probability i by the liear property of tesor H H p ph i i i i i i. 3) H is completely positive-defiite. Due to the basic priciple of quatum operatios either H is a uitary operator itself or the joit operator is uitary after the additioal expasio o the H system. Therefore the quatum trasform of LS estimatio ca be physically realized. Usig the result of quatum LS estimatio it s easy to fid that whe V is the quatum oise of a system as log as the iformatio for autocorrelatio ad cross-correlatio be available we will be able to costruct the optimal quatum filter i the LS sese so as to resist the quatum oise pollutio. Ad this method of QSP provides us with a distict approach to deal with the quatum oise i cotrast with the quatum error-correctig code which is vital for the realizatio of quatum computer ad reliable quatum commuicatio system. It should be metioed that i terms of sigle qubit its probability amplitude which is a complexvalued umber cotais its etire iformatio. So LS criterio ca be applied to the estimatio of probability amplitude. But the liear correlated fuctio is supposed to reveal the part of the etire iformatio for the qubit s probability amplitude. Therefore high-level correlatio should be more sigificat o the estimatio of complex-valued sigals tha the real-valued. Meawhile i terms of QSP itroducig high-level correlated fuctio may possibly improve the accuracy of the quatum sigal estimatio. 4. The Optimum Detectio for o- Orthogoal State Sigal 4.. Use Least-Square Filter to Estimate Uow o-orthogoal State Sigals We too the biary o-orthogoal sigal as the sample the OOK (O-Off Keyig) sigal is referred here geerally. H0 : 0 H : H 0 ad H deotig absece ad presece of sigal. They tae the state 0 ad the state as the sigal carrier i tur. The the POVM operator for OOK sigal ca be described as follows [0]: E E / / e 0 0 e / e e / / 0 e 0 e / e e E3 I E E. After the measuremet if the cosequeces tur out to be E the the source sigal is cofirmed to be H if it tur out to be E the source should be H 0 ad if the cosequeces is as E 3 the origi state of the sigal is uow. For the case of E 3 as the measuremet result the origi state of the sigal will be affirmed by the least-square estimatio. Simultaeously the error probability (the measuremet outcome is E 3) is Pe e / (9) where stads for the average eergy of sigals. Let L bits sigal sequece passig by the POVM assume m bits get the correct outcome ad L-m elemets get the false results. We ca give the estimatio of the L-m elemets by the m correct bits usig least-square filterig. The filter is desiged with FIR (Fiite Impulse Respose) its impulse respose vector is Let h [ h 0 h h m ] T. T P E X s R E XX s is the uow sigal that is to be estimated X s s s i i im plays for a vector whose elemets are m ow sigals R is the auto-correlatio fuctio of the source which is a m* m matrix. Accordig to equatio P Rh h R P - so the uow sigal estimatio is ad its least-square error is mi T s hx T E s PR P. T 544

5 Sesors & Trasducers Vol. 60 Issue December 03 pp The Boud of Bit Error Probability for Combiig POVM ad Least-Square Estimatio (CPLSE) The estimatio error is primarily decided by the auto-correlatio fuctio of the sigal sequece ad the positio of the elemets which to be estimated i the sequece. Cocerig the auto-correlatio fuctio of the geeral sigal is a atteuatio fuctio we ca cosequetly give the upper ad lower boud of the estimatio error. For this istace let s assume s s s s s m m L be a sigal sequece s to s m are the ow sigals sm to s L are the uow sigals. Of all the estimatio cosequeces s L has a maximum value of the least-square error. Ad i the sigal sequece s sb sb sbm/ sbm/ s s s bm/ bm L where sb to sb m/ ad sb m/ to sb m sigifies the ow sigal the sb m/ sigifies the midst uow sigal ad the rests symbolize the remaiig uow sigals. Sigal sb m/ therefore will have the miimum value of least-square error. Obviously whe the sigal to be estimated lies i the most distat place with the ow sigal it has the maximum value of least-square estimatio error while the oe stads the midst of the ow sigals has the miimum value of estimatio error withstadig that the ow sigals are all adjacet with each other. ext we ll derive the maximum ad miimum bit error probability. As the bit error probability ca be derived from least-square error approximately we have: mi s s P s s s s s is the real value of the sigal s is the estimatio value P represets for the error probability of s s s s. From equatio 0.5 / P 00.5 / P mi ss0.5 ss0.5 we get 0.65P 0.5P P s s 0.5 s 0.5 s ss0.5 mi 0.5 taig the priciple of the least-square error ito cosideratio the error probability is P s s. 0.5 The the bit error probability is betwee the maximum value of least-square error ad the miimum value of least-square error that is mi_ lower 0.5 Pe _ bit mi_ upper 0.5. (0) 5. umerical Results I this sectio we mae umerical aalysis to verify above results. Firstly assume the average eergy to be 5. Secodly the auto-correlatio fuctio of source sigals is set to be R j j e j 4 which is show i Fig. 3. Fially the error probability of the POVM detectio as well as the error probability of the CPLSE are calculated with (9) ad (0). R(j-) j- Fig. 3. Auto-correlatio fuctio of the sigal sequece. Fig. 4 is the bit error probability varyig with the iput sigal sequece legth we set 5 L varies from 50 to 500 the outcome tur to be straight lie the POVM detectio get the error probability of more tha 0.08 while the CPLSE detectio gets the error probability betwee 0.0 ad 0.0. Bit Error Probability CPLSE Lower Boud CPLSE Upper Boud POVM Sigal Sequece Legth Fig. 4. Bit Error Probability of sigal varyig with the sequece legth L. 545

6 Sesors & Trasducers Vol. 60 Issue December 03 pp I the case as i Fig. 5 we set L 00 ad varies from 3 to 0. From the results we ca see that with the sigal power icreasig the bit error probability decrease promptly which tae o a expoetial decay. Log Value of Bit Error Probability CPLSE Upper Boud CPLSE Lower Boud POVM Average Eegy of Sigals Fig. 5. Bit Error Probability of sigal sequece varyig with. It s easy to see from all the above figures ad results that the CPLSE method is able to reduce the bit error probability efficietly ad thus improves the performace of coheret-state sigal detectio. 6. Coclusios I this paper we preset the mathematical model of quatum sigal processig. These basic wors eable us to get the optimal LS estimatio of quatum sigal. O this basis the POVM operators ad the average bit error probability for oorthogoal state sigal are deducted. Combiig the least-square error theory we proposed the CPLSE method which reduced the bit error probability ad improved the performace of detectio. Quatum trasform ad the realizatio of quatum filter alog with the quatum system aalysis will be the future wor. Refereces []. Y. C. Eldar A. V. Oppeheim Quatum sigal processig Sigal Processig Vol. 9 Issue 6 00 pp. -3. []. Y. C. Eldar Quatum sigal processig Dissertatio Mass Istitute Techology 00. [3]. V. A. Vilrotter ad C.-W. Lau Quatum detectio of biary ad terary sigals i the presece of thermal oise fields The Iterplaetary etwor Progress Report IP PR 4-5 Jet Propulsio Laboratory Califoria Istitute of Techology October-December 00 pp. -3. [4]. Victor A. Vilrotter Quatum receiver for distiguishig betwee biary coheret-state sigals with partitioed-iterval detectio ad costatitesity local lasers The Iterplaetary etwor Progress Report Vol Jet Propulsio Laboratory Califoria Istitute of Techology May 5 0 pp.. [5]. M. A. ielse I. L. Chuag Quatum computatio ad quatum iformatio Cambridge Uiversity Press Cambridge 000. [6]. Guagca Guo Tao Guo Tie Zheg Quatum computer Quatum Optics Trasactio Vol. 3 Issue 997 pp. -4. [7]. M. Broos Quatum computig ad commuicatio Spriger-Verlag Berli Heidelberg 999. [8]. P. W. Shor Polyomial time algorithms for prime factorizatio ad discrete logarithms o a quatum computer SIAM Joural of Computer Vol. 6 Issue pp [9]. R. P. Feyma R. B. Leighto M. Sads The Feyma lectures o physics Addiso-Wesley 965. [0]. R. P. Feyma Simulatig physics with computers Iteratioal Joural of Theoretical Physics Vol. Issue 6/7 98 pp []. P. A. M. Dirac The priciples of quatum mechaics 4 th editio Oxford Uiversity Press Oxford 958. []. Y. C. Eldar A. V. Oppeheim D. Egor Orthogoal ad projected orthogoal matched filter detectio Sigal Processig Vol. 84 Issue pp Copyright Iteratioal Frequecy Sesor Associatio (IFSA). All rights reserved. ( 546