Wiener Filtering for Passive Linear Quantum Systems

Wiener Filtering f or Passiv e Lin ear Quantum Systems V . Ugrinovskii M. R. James Abstract — This paper considers a versio n of the Wiener filtering problem fo r equalization of passiv e quan t um linear quantum systems. W e demonstrate that taking into c onsidera- tion the quantum nature of th e signals inv olved leads t o features typically not encoun tered i n classical equalization problems. Most significantly , findi n g a mean-square optimal quan t u m equalizing filter amounts to solv ing a nonconv ex constrained optimization problem. W e discuss two approaches to solving this problem, both i nvo lving a relaxation of the constraint. In both cases, unlike classical equalization, there is a t hreshold o n the variance of the noise below which an improv ement of the mean-square err or cannot be guaranteed. I . I N T R O D U C T I O N The task o f transferrin g quantu m inform ation differs sig- nificantly from its classical ( n on-q uantum) coun terpart, since the laws o f q uantum mech anics limit the accu racy of in- formation transfer th roug h qu antum cha nnels. Sp e cifically , the signal-to-noise ratio of possible qu antum measurements on th e tr ansmission lin e is lim ited [4], reflecting the well known fact that a q uantum state canno t be cloned at the remote locatio n. This motiv ates a great intere st in devel- oping systematic methodolog ies for the de sig n of optim a lly perfor ming quantum communication systems. In the classical communication th eory , op timization plays an instru mental role in balan cing various trade-offs in the design of classical communication systems. The most cele- brated example of u sing op timization in signal p rocessing are due to N. W iener [1 5] who developed a general method for reducing the effects of n o ise a nd chann el distor tion th rough minimization of the mean square error (M SE) between the signal an d its estimate over a class o f line ar filters. This paper highlights conceptual challenges that ar ise when the W iener o ptimization p a r adigm is applied in the deriv ation of coh erent quan tu m filters, i.e., filters which them selves are quantum systems. T o b e concr ete, we restrict attention to one type of the co herent filtering problem concer ned with equalizing distortions of quan tum sign als transmitted via a quan tum commun ication channel. Owing to the an alogy with classical chan nel equ a lization, we call this prob lem th e quantu m eq u alization pr oblem . The pap er shows that the requirem ent fo r the filter to be physically realizable translates This work was supp orted by the Australi an Researc h Council and the ARC Centre for Quant um Computation and Communication T echnology . V . Ugrino vskii is with the School of Enginee ring and Infor - mation T echnology , Unive rsity of N ew South W ales at the Aus- tralia n Defence Force Academy , Canberra, ACT 2600, Australia, v.ougrinovski @adfa.edu.au M. R. James are with the ARC Centre for Quantum Computa- tion and Communication T echnology , Research School of Engineer - ing, The Austra lian na tional Univ ersity , Canb erra, ACT 2601, Aust ralia, matthew.james @anu.edu.au. into add itional constraints wh ich rend er the pro blem of o pti- mizing the me a n square of the eq u alization error non conv ex. The pa per is centered aro und the so-called passive quan- tum equalizers. Mathematically , dynamics of a passi ve quan- tum system in the Heisenb erg pictur e ar e described by complex quantu m stochastic differential equation s expressed in term s of annihilation op erators only [7]. Such systems are simple to implement exper imentally by cascading co n- ventional quantu m optics co mpon ents such as beam splitters and optical cavities [ 1 1]. Fu r thermor e, in a gen eral qu antum system, passi v ity ensures that the sy stem dissipates energy in the input. A strik ing o bservation that emerges from o ur analysis is that passivity appear s to be a r ather restrictive proper ty in the context of equalization , in that an optim al passiv e coher ent equ alizer is not always a b le to imp rove the MSE. It turns out that the achiev able improvement dep ends on the v ariance of the quantum noise in the filter input signal. W e give examples which r ev eal a thresho ld on this variance above w h ich th e op timal passive coheren t equ a lize r delivers an improved MSE. The p aper is organ ized as f ollows. In th e n ext section we present the basics of p assi ve linear quantu m systems. The quantum passiv e equ alization pro blem is posed in Section II I. A relaxation of the pro b lem is proposed in Section IV. Next, in Section V, the pr oblem is particularized to dem onstrate the dependency b etween th e power spectrum density of the equalization err or and the variance of the system noise. T wo examples of t he qua n tum coherent filter design are presented in that section, r eflecting two approac hes to optimizatio n of the equalizatio n erro r , the first one is via dir ect op timization of the power spectrum density , and th e second o ne is using the W iener-Hopf f actorizatio n techn ique [8]. Finally , conclud in g remar ks are gi ven in Section VI. Notation: For an operato r a in a Hilbert sp ace H , a ∗ denotes the Hermitian adjoint operator , and if a is a complex number, a ∗ is its comp lex con jugate. Let a = ( a 1 , . . . , a n ) be a column vector co m prised of n operators (i.e., a is an operato r H → H n ); th en a # = ( a ∗ 1 , . . . , a ∗ n ) , a T = ( a T 1 . . . a T n ) (i.e, the row of operators), and a † = ( a # ) T . The notation col( a, b ) de n otes the colum n vector obtained by co ncatenating vectors a and b . For a comp lex matrix A = ( A ij ) , A # , A T , A † denote, respec tively , the matr ix of co mplex conjug ates ( A ∗ ij ) , th e transp ose matrix and the Hermitian ad joint matrix. [ · , · ] denotes the comm u tator o f two operators in H . tr[ · ] deno tes the trace of a matrix. I is the iden tity ma trix. The quantu m exp ectation of a n op e rator V with respec t to a state ρ , is den oted h V i = tr[ ρV ] [12]. I I . O P E N L I N E A R PA S S I V E Q U A N T U M S Y S T E M S An op en q u antum annih ilatio n-only system represents a linear system ˙ a = A a + B u , a ( t 0 ) = a , y = C a + D u ; (1) where A , B , C , D are complex m × m , m × n , n × m , n × n matr ices, and u is a (column) vector of n qu a ntum input pro cesses. T h e input is assumed to b e of the fo rm u ( t ) = u 0 ( t ) + b ( t ) , (2) where b is a (co lumn) vector o f n quantum no ise pr ocesses, b = ( b 1 , . . . , b n ) , a n d u 0 ( t ) is an adapted process [5]. The noise p rocesses can be rep r esented as a n nihilation o perator s on an ap p ropria te F ock space [5], but fro m the system theory viewpoint they can be trea te d as quantum Gaussian white noise processes with z e r o mean, an d the covariance  b ( t ) b # ( t )   b † ( t ′ ) b T ( t ′ )  =  I + Σ T b Π b Π † b Σ b  δ ( t − t ′ ) , (3) where Σ b , Π b are complex matrices with the properties that Σ b = Σ † b , Π T b = Π b . Alon g with their adjoint (creation) oper a to rs b ∗ j ( t ) , the noise o perators satisfy can on- ical commutatio n relation s [ b j ( t ) , b ∗ k ( t ′ )] = δ j k δ ( t − t ′ ) , [ b j ( t ) , b k ( t ′ )] = 0 . Here, δ j k = 0 when j 6 = k , and is the identity ope rator when j = k ; δ ( t − t ′ ) is the δ -fun ction. The colu mn vector a ( t ) = ( a 1 ( t ) , . . . , a m ( t )) repr esents the system modes an d consists of annihilation o perator s on a certain Hilb ert space H . A discussion abou t open linear quantum systems can be found in [7], [2], [6]. From now on, it will b e assumed th a t th e pair ( A, B ) is controllab le . For a system o f the form (1) to correspond to quantum physical d y namics, it mu st pre serve the c anonical commuta- tion relations d uring its ev olutio n [ 13], [6]. A c cording to [9], for th e system (1) this is guar anteed if an d on ly if th ere exists a Hermitian co mplex matrix Θ such th a t A Θ + Θ A † + B B † = 0 , B = − Θ C † , D = I . (4) W ithout lo ss of gener a lity we will assume f r om now on that the c ondition s (4) are satisfied for th e systems under consideratio n with Θ = I ; this can always be achie ved by an app ropria te choic e of c oordin ates [9]. Furth e rmore, we will assume th a t th e matrix A is Hurwitz. From (1), th e output of th e system can be represented as y ( t ) = C e A ( t − t 0 ) a + Z t t 0 g ( t − τ ) u 0 ( τ ) dτ + Z t t 0 g ( t − τ ) b ( τ ) dτ . (5) Here we intr oduced the no tation f or the impulse respo nse, associated with the system [16], g ( t ) = ( C e At B + δ ( t ) I , t ≥ 0 , 0 , t < 0 . (6) Let us in troduce the transfer fun ction of the system (1), G ( s ) = C ( sI − A ) − 1 B + I . Since B = − C † , the transfer function G ( s ) is square. This o b servation h olds for all passi ve systems consider ed hencefo rth. Furthermo re, if f ollows from the prop erties of the p hysical realizability [13] that for the passive system (1), G ( s )[ G ( − s ∗ )] † = I . (7) In the sequ el we will b e interested in stationar y behaviours of the systems und er consideration . Since the m atrix A is assumed to be stable and assuming that u 0 ( t ) is stationary , the stationary compon ent of the system outpu t is ob tained from (5) by letting t 0 → −∞ : y ( t ) = Z + ∞ −∞ g ( t − τ ) u 0 ( τ ) dτ + Z + ∞ −∞ g ( t − τ ) b ( τ ) dτ . (8) Also, f or conv enien ce the upp er limit of integration has been changed to + ∞ since g ( t ) is cau sal. Consider the correlation function o f stationary quantum operator pro cesses x j ( t ) , x k ( t ) associated with th e system, R x j , x k ( t ) = h ( x j (0) − h x j (0) i )( x k ∗ ( t ) − h x k ∗ ( t ) i ) i . The corresp onding power spectrum density is th en P x j , x k ( iω ) = Z + ∞ −∞ e − iωt R x j , x k ( t ) dt. (9) The Fourier tran sform is understo od in the sense of temp ered distributions when R x j , x k is not integrab le. Als o, consider the e xtension o f P x j , x k ( iω ) to the complex pla n e, given b y the bilater a l Lap lace tra n sform of R x j , x k , P x j , x k ( s ) = Z + ∞ −∞ e − st R x j , x k ( t ) dt. (10) Often, P x j , x k ( s ) is also referred to as the po wer sp ectrum density fun c tion [8], althou gh in general it may no t be real. Since th e matrix A is Hurwitz, P x j , x k ( s ) is well defined on s = iω and P x j , x k ( s ) | s = iω = P x j , x k ( iω ) , where the expression on th e left-han d side refers to the power-spectrum density d efined in (10), an d the expression on righ t-hand side is defined in (9). I t is easy to o b tain that the power spectr um density m atrix of the outpu t y ( t ) , P y , y ( s ) = ( P y j , y k ( s )) is related to the power spectrum de n sity m a tr ix o f th e noise b , P b , b ( s ) = ( P b j , b k ( s )) , in the standard ma n ner: P y , y ( s ) = G ( s ) P b , b ( iω )[ G ( − s ∗ )] † . (11) I I I . E Q UA L I Z A T I O N P R O B L E M F O R A N N I H I L AT I O N - O N LY C O M M U N I C AT I O N S Y S T E M S In this sectio n , a gener al eq ualization scheme fo r a passive commun ication system is outlined. Consider a system in Fig. 1 consisting of a quantum channel and an equa lize r . The inp ut si gn a l u plays the role of a me ssage signal to be transmitted thr ough the chan nel, of the fo rm u ( t ) = u 0 ( t ) + b ( t ) , (12) and w denotes the vector comprised of additional quantum noises. It include s the n oise in puts that are n ecessarily present in the p h ysically realizable system G ( s ) [6], [14], as well as noises introduc e d by measurem ent devices. In P S f r a g r e p l a c e m e n t s G ( s ) H ( s ) u w y w z ˆ u ˆ z y u Fig. 1. A general quantum communica tion system. T he transfer function G ( s ) represents the channel, and H ( s ) represents an equali zing filter . terms of the notation adopted in the previous sectio n, we have u 0 = col( u 0 , 0) , an d b = col( b, w ) . The comb ined input u = col( u, w ) drives an annihilation - only (passiv e) quantum system G ( s ) , as described in the previous section , to p roduce the outp ut y = co l( y u , y w ) , altho ugh f or filtering purpo ses, we are on ly interested in the o utput comp onent y u which corr e sp onds to the inpu t channel u . The objective: In the classical fil tering theo ry [8], the equalizer is to com pensate for signal distortions in the output y u ( t ) , by min imizing the eq ualization error e ( t ) = ˆ u ( t ) − u ( t ) between classical signals ˆ u ( t ) , u ( t ) in the mea n - square sense. The classical power sp e ctrum density P e,e ( iω ) is usually L 2 -integrable an d is related to the c o rrelation function of the er r or e ( t ) via the inverse Fourie r tr ansform , R e,e ( t ) = 1 2 π Z + ∞ −∞ P e,e ( iω ) e iωt dω . In this case, minim izing the mean -square error covariance measure tr R e,e (0) is equiv alent to minimizing tr P e,e ( iω ) pointwise in ω . Alternativ ely , the optimal causal filter can be sought to satisfy the W iener-Hopf equation [8], R u,y u ( t ) = Z + ∞ 0 h ( t − τ ) R y u ,y u ( τ ) dτ , t > 0; (13) here h ( t ) is the un ilateral in verse Laplace transf o rm o f a causal transfe r fun c tion H ( s ) . The equation (13) reflects the p rojection pro perty of classical least-square estimates, E ( e ( t ) y u ( τ ) † ) = 0 f or − ∞ < τ < t . Th e so lu tion to equation (13) is obtained usin g spectral factorization. Analogou s to the classical mean-squ are equalization, we wish to obtain a quantum system H ( s ) whose outpu t ˆ u matches th e inpu t u op timally , in th e sense that the equal- ization err or e ( t ) = ˆ u ( t ) − u ( t ) mu st ha ve a minimu m covariance. Owing to the physical realizability requirem ent reflected in the identity (7), quantu m channels ar e n ot gu ar- anteed to generate L 2 -integrable power spectrum densities. For this reaso n, we will pose the prob le m dire c tly as opti- mization o f the p ower spectrum density , to either m inimize tr P e,e ( iω ) p ointwise for every ω , or obtain a causal H ( s ) by solving the corre sp onding spe c tr al factorization prob lem. Both app roaches will b e discussed in Section V. Admissible equ alizing fi lters: The key distinction of the proble m und er conside ration fro m classical coun terparts is that the system H ( s ) must be ph ysically realizab le as a q uantum sy stem. This mand ates imposing additiona l r e - quiremen ts o n the filter . Firstly , to ensure tha t the L TI filter system obtain e d fro m th e optimizatio n pro blem ( 1 6) or fr om spectral factorization can be made physically realizable, it may need to be equipped with additional noise inputs z — it was observed in [ 6], [14] that any L TI system can b e made phy sically re alizable b y addin g no ise. W ith out loss of generality , we will assume that the ad ded noise z is in a Gaussian v acuum state, i.e. , the corresp o nding mean and covariance o f z are h z ( t ) i = 0 ,  z ( t ) z # ( t )   z † ( t ′ ) z T ( t ′ )  =  I 0 0 0  δ ( t − t ′ ) . (14) Secondly , to facilitate implem entation of the resu ltin g quantum filter [ 11], we restrict attention to passiv e equalizer systems. In this case, the requ irement f or ph ysical realizab il- ity of the filter leads to a fo r mal constraint of the form (7) on the tr a nsfer f unction H ( s ) : H ( s )[ H ( − s ∗ )] † = I . (15) Let us deno te the set o f passiv e phy sically realizab le equal- izers s atisfying (15) as H r . T he pointwise optimizatio n of tr P e,e ( iω ) in th e class of phy sically realizab le filters is th us a constrain ed o ptimization prob lem, min H ∈ H r tr P e,e ( iω ) . (16) The con straint (15) precludes th e d irect app lication of standard filtering tech n iques to ob ta in an optimal q uantum W iener equalizer . In the next section we o utline a relaxation technique which h e lp s to overcome this prob lem. I V . C O N S T R A I N T R E L A X A T I O N Let us define th e par titions of the transf e r function s G ( s ) and H ( s ) c o mpatible with the par titio ns of u = col( u, w ) , y = c o l( y u , y w ) , and co l( y u , z ) , col( ˆ u, ˆ z ) , respectively: G ( s ) =  G 11 ( s ) G 12 ( s ) G 21 ( s ) G 22 ( s )  , H ( s ) =  H 11 ( s ) H 12 ( s ) H 21 ( s ) H 22 ( s )  . (17) W ith this no tation, we have that P e,e ( s ) = ( H 11 ( s ) G 11 ( s ) − I )( I + Σ T b )( G 11 ( − s ∗ ) † H 11 ( − s ∗ ) † − I ) + H 11 ( s ) G 12 ( s )( I + Σ T w ) G 12 ( − s ∗ ) † H 11 ( − s ∗ ) † + H 12 ( s ) H 12 ( − s ∗ ) † . (18) Also, the con straint (15) is equivalent to H 11 ( s ) H 11 ( − s ∗ ) † + H 12 ( s ) H 12 ( − s ∗ ) † = I , (19) H 11 ( s ) H 21 ( − s ∗ ) † + H 12 ( s ) H 22 ( − s ∗ ) † = 0 , (20) H 21 ( s ) H 21 ( − s ∗ ) † + H 22 ( s ) H 22 ( − s ∗ ) † = I . (21) From ( 18), we o bserve that the spectral d ensity fu nction P e,e ( s ) depends on th e v ariables H 11 , H 12 only . Therefo re one possible app roach to solving the equalizer design p rob- lem is to employ a two-step procedure whose first step is to optimize th e equ alization erro r with respect to H 11 ( s ) , H 12 ( s ) , sub ject to the constra int (19), followed by the seco nd step during which the remaining tran sfer functions H 21 ( s ) , H 22 ( s ) are computed to fulfill th e remain ing p hysical real- izability constrain ts (2 0), (21). Of co urse, there is no guaran tee th at with H 11 ( s ) , H 12 ( s ) found d uring th e first step, the remaining transf e r fun ctions H 21 ( s ) , H 22 ( s ) e xist and satisfy the conditions (20), (21). Nev ertheless, this appr o ach is attr a c ti ve in that it allows us to obtain tractable relaxation s of th e o r iginal quan tum equ alizer design p roblem . Indeed , using (19), H 12 ( s ) can b e elimina ted from the expression (18): P e,e ( s ) = ( H 11 ( s ) G 11 ( s ) − I )( I + Σ T b ) × ( G 11 ( − s ∗ ) † H 11 ( − s ∗ ) † − I ) + H 11 ( s ) G 12 ( s )( I + Σ T w ) G 12 ( − s ∗ ) † H 11 ( − s ∗ ) † − H 11 ( s ) H 11 ( − s ∗ ) † + I . (22) It also f ollows from (19) that H 11 ( iω ) H 11 ( iω ) † ≤ I ∀ ω ∈ R 1 . (23) This allows us to replace the o riginal prob lem of findin g an op tim al passi ve equ alizer H ( s ) with the p roblem of optimizing the equalization er ror in the class of causal transfer function s H 11 ( s ) subject to the quadra tic constraint (23). W e will give a precise me a ning to this statemen t in the next section, where we d iscuss two relaxed quan tum W iener filter pro b lem f ormulatio ns. V . T W O A P P RO AC H E S T O Q UA N T U M W I E N E R E Q UA L I Z AT I O N In this section w e apply the relax ation tec h nique discussed in the pr evious section to two problem s which demonstrate features of the qu antum Wiener filtering . Our aim is to highligh t ne w fea tu res of the problem of co h erent W iener equalization owing to the phy sical realizability co n straint (15), r a ther th a n ob tain a general solution to this problem. All signals in this section are a ssumed to b e scalar un less specified otherwise. A. Equalizatio n via o ptimization of power spectru m density: An optical bea m splitter In th is section, we focus on the problem (1 6). The c o n- straint relax ation proposed in the pr evious section allows to r eplace this pro blem with the pr oblem inv olving the constraint (23). In the case of scalar signals u , y u and ˆ u , P ee ( s ) and Σ b are scalars, an d this pr o blems simp lifies significantly: min | H 11 ( iω ) | ≤ 1 P e,e ( iω ) , (24) P e,e ( iω ) = (1 + Σ b ) | H 11 ( iω ) G 11 ( iω ) − 1 | 2 + | H 11 ( iω ) | 2 G 12 ( iω )( I + Σ T w ) G 12 ( iω ) † −| H 11 ( iω ) | 2 + 1 . (25) In (24), th e m inimum is taken over the set of c a usal tran sfer function s H 11 ( s ) subje c t to the scalar version of the condi- tion (23). Obviously , we have in this ca se min | H 11 ( iω ) | ≤ 1 P e,e ( iω ) ≤ min H ∈ H r P e,e ( iω ); (26) i.e., the problem (2 4) delivers a lower bou nd o n the o ptimal power spectrum density . The requir e ment for H 11 ( s ) to b e causal is also no ntrivial — while the fre quency pointwise optimization is ea sy to perfo rm over complex H 11 , th e P S f r a g r e p l a c e m e n t s u w z ˆ u ˆ z y w H ( s ) y u Fig. 2. A beam splitter and a quantum equalizer system. pointwise optimal H 11 ,ω obtained this way mu st admit a causal extension into th e complex plane. I n general, th is issue can be addressed nu merically [1], using the standar d Matlab sof tware [10]. There f ore in the rema inder of th is section, we will be c o ncerne d with equalization of a static quantum system for which the cau sality co ndition is satisfied automatically . This simplified analysis aims to demonstrate that the p r oposed relaxation can lead to ph ysically realizab le equalizers which ar e optimal in the sense of ( 16). As an example of a static qu antum system consider a quantum -mechan ical bea m splitter, which is a two-input two- output quan tum system; see Fig. 2. In Fig. 2, the input u represents the signal we would like to split, and th e second input w is an a u xiliary no ise inpu t. The bea m splitter mixes the signals u and w , its outputs and inp uts are related via a unitary transfor mation:  y u y w  = G  u w  , G ( s ) =  √ η √ 1 − η − √ 1 − η √ η  ; (27 ) η ∈ (0 , 1) is a real par a meter k n own as transmittance. Th at is, G ( s ) is static in th is case, and y u = √ η u + p 1 − η w . The equalization problem is to estimate the sign al u from the outp ut y u of this d evice usin g a coherent equ alizer, i. e., a device which pre serves the cano n ical c o mmutation r elations. T o demo nstrate the applica tio n of a quantu m Wiener filter in this problem , suppose that the in put n oise b in (12) is in Gaussian vacuum state, and Σ b = 0 , Π b = 0 , wherea s the beamsplitter n oise w is in a Gaussian thermal state, so that Σ w = σ 2 w > 0 , Π w = 0 . With these assumptions, the expression f or the objective functio n in (25) bec omes P e,e ( iω ) = (1 − η ) σ 2 w | H 11 ( iω ) | 2 − 2 √ η Re H 11 ( iω ) + 2 . (28) The constrain t co ndition (23) red u ces in this case to | H 11 ( iω ) | 2 ≤ 1 . (29) Since all c o efficients in (28) ar e constants, th e optimal value and the op timal equalizer should also be constant. The prob lem (24) is thu s a regular con strained op tim ization problem , wh ich can be solved u sin g th e Lag range mu ltip lier technique . Pr op osition 1: 1. If σ 2 w ≤ √ η (1 − η ) , then the optimal equal- izer which attains m inimum in (1 6) is H ( s ) = I . 2. On the other h a nd, when σ 2 w > √ η (1 − η ) , an op timal equalizer is g iven by H 11 ( s ) = √ η σ 2 w (1 − η ) , H 12 ( s ) = r 1 − η σ 4 w (1 − η ) 2 , H 21 ( s ) = − H 12 ( s ) , H 22 ( s ) = H 11 ( s ) . (30) P S f r a g r e p l a c e m e n t s u v α β w z ˆ u y w v out u out ˆ z y u H ( s ) Fig. 3. A ca vity , beam splitters and an equaliz er system. Such an eq ualizer atten uates th e in put y u , an d must include an additio nal noise input z , to en sure that it is physical realizable. The co r respond ing expressions f or the optimal error power spectrum density are min H ∈ H r P ee = ( σ 2 w (1 − η ) − 2 √ η + 2 , if σ 2 w ≤ √ η (1 − η ) ; 2 − η σ 2 (1 − η ) , if σ 2 w > √ η (1 − η ) . (31) Comparing the power spe c tr um density of the err or at the input o f th e filter, P ( y u − u ) , ( y u − u ) ( iω ) , with P e,e ( iω ) in (31), we ob serve that P ( y u − u ) , ( y u − u ) ( iω ) = P e,e ( iω ) if σ 2 w ≤ √ η (1 − η ) , a nd P ( y u − u ) , ( y u − u ) ( iω ) > P e,e ( iω ) if σ 2 w > √ η (1 − η ) . Thus, Proposition 1 shows that the requirem e nt f o r p hysical realizability r estricts the capacity of an optimal coher e nt equalizer to respon d to noise in the input s ignal. It is still possible to red uce th e MSE by means of a co herent equalizer, howe ver this is on ly possible p rovided the covariance of the th ermal no ise in the inpu t signal is sufficiently large. This situation differs strikingly from the classical W iener equalization theor y . B. The W ien er -Hop f technique for q uantu m equa lization: A n equalizer for an o ptical cavity Let us modify the system in Fig. 2 to includ e an optical cavity and two ad ditional beam splitter s of tran smittance α and β ; see Fig. 3. W ith these mo dification the system becomes d ynamical. In Fig. 3, v denotes an ad ditional thermal Gaussian noise input into the system, with zer o mean and covariance  v ( t ) v ∗ ( t )   v ∗ ( t ′ ) v ( t ′ )  =  1 + σ 2 v 0 0 σ 2 v  δ ( t − t ′ ) . Correspon d ingly , the relation between the chan n el outpu t col( y u , y w ) and its in put col( u, v ) is fo und fro m th e r elations  y u y w  =  √ η √ 1 − η − √ 1 − η √ η   u out w  ,  u out v out  = ¯ G ( s )  u v  , ¯ G ( s ) =  ¯ G 11 ( s ) ¯ G 12 ( s ) ¯ G 21 ( s ) ¯ G 22 ( s )  = p αβ  G c − √ α ′ β ′ √ α ′ G c + √ β ′ − √ β ′ G c − √ α ′ − √ α ′ β ′ G c + 1  ; ( 32) G c ( s ) den otes the transfer fu nction of the o ptical cavity G c ( s ) = s − γ 2 + i Ω s + γ 2 + i Ω ; (33) γ , Ω a r e real co nstants, and α ′ = 1 − α α , β ′ = 1 − β β . Note th at G c ( s )[ G c ( − s ∗ )] ∗ = I . After these m o difications, the p ower sp ectrum density of the equ alization erro r in equation (22) is expressed as P e,e ( s ) = 2 + ( η σ 2 v ¯ G 12 ( s )[ ¯ G 12 ( − s ∗ )] ∗ + (1 − η ) σ 2 w ) × H 11 ( s )[ H 11 ( − s ∗ )] ∗ − √ η  H 11 ( s ) ¯ G 11 ( s ) + [ H 11 ( − s ∗ )] ∗ [ ¯ G 11 ( − s ∗ )] ∗  . (34) The au xiliary optimization problem considered in the p revi- ous sections is therefore to obtain a causal transfer function H 11 ( s ) which optimizes (3 4) su b ject to the constraint (29). Unlike the previous sectio n, the system con tains dynam- ics and the correspo nding optima l filter is expected to b e dynamica l. Therefo re, we cannot e xpect that the pointwise optimization in (24) will produce a causal tra n sfer fu nction H 11 ( s ) . In the classical case, th is issue is resolved using the W iener-Hopf spe ctral factorization method [8]. Therefo re, here we pro ceed as f ollows. First, we ap p ly the W iener-Hopf spectral factorization metho d [8] to obtain a cau sal op timal H 11 ( s ) that minimize s tr P e,e ( iω ) for P e,e ( s ) in (34); this step does not in volve th e physical realizability constraints. Next, we sho w that in fact the found H 11 ( s ) v alidates the required co nstraint (29), provided the variance of the system noise excee d s a cer ta in thr eshold. T h en we show that in this case a co mplete phy sically realizab le filter transfer f unction H ( s ) wh ich satisfies (19)–(21) can be constructed f rom th e found H 11 ( s ) . Since P e,e ( s ) in (34) de p ends o n H 11 ( s ) only , we c an minimize tr P e,e ( iω ) b y treating P e,e ( s ) as a power spectrum density of a class ical sy stem . Define ζ = 1 − η η σ 2 w αβ , ρ = α ′ + β ′ + ζ σ 2 v 2 √ α ′ β ′ ≥ 1 . (35) Letting M ( s ) be the follo wing causal transfer f unction, M ( s ) = q 2 η σ 2 v p αβ (1 − α )(1 − β )( ρ + 1 ) × s + γ 2 q ρ − 1 ρ +1 + i Ω s + γ 2 + i Ω , (36) we obtain the identity M ( s )[ M ( − s ∗ )] ∗ = η σ 2 v ¯ G 12 ( s )[ ¯ G 12 ( − s ∗ )] ∗ + (1 − η ) σ 2 w . Therefo re, P e,e ( s ) = (( M ( s ) H 11 ( s ) − √ η Q ( s )) × (( M ( − s ) ∗ H 11 ( − s ) ∗ − √ η [ Q ( − s ∗ )] ∗ ) − η Q ( s )[ Q ( − s ∗ )] ∗ + 2 , (37) where Q ( s ) ,  ¯ G 11 ( − s ∗ ) M ( − s ∗ )  ∗ = 1 − √ α ′ β ′ p 2 η σ 2 v √ α ′ β ′ ( ρ + 1)   1 + γ 2 q ρ − 1 ρ +1 + 1+ √ α ′ β ′ 1 − √ α ′ β ′  s + i Ω − γ 2 q ρ − 1 ρ +1   . (38) Now co nsider a classical filtering problem of minimizing the M SE between the filter output ˆ u = H ( s ) y u , wh ere y u = M ( s ) u , and the sign al ¯ u = √ η Q ( s ) u . Let [ Q ( s )] + denote the causal pa rt of Q ( s ) . Acc o rding to the W iener- Hopf metho d [ 8], th e causal solution to th is problem is H 11 ( s ) = √ η M ( s ) [ Q ( s )] + . (39) This filter en sures that the erro r u − ¯ u and the filter input y u are ortho gonal. Since the expression for the power spe c tr um density of the error in th is pr oblem is exactly equa l to th e first term in (37), we con clude th at the filter (3 9) minimizes P e,e ( iω ) in the class o f ca u sal transfer fu n ctions. Th is yields the explicit expression for the op timal filter which is causal by way of co n struction: H 11 ( s ) = (1 − √ α ′ β ′ ) / √ η σ 2 v ( √ α ′ + √ β ′ ) 2 + ζ × s + γ 2 + i Ω s + γ 2 q ρ − 1 ρ +1 + i Ω . (40) Pr op osition 2: Under th e condition σ 2 v > − ζ ( α ′ + β ′ ) + q 4 ζ 2 α ′ β ′ + (1 − √ α ′ β ′ ) 2 ηαβ ( α ′ − β ′ ) 2 ( α ′ − β ′ ) 2 (41) the tran sfe r fu nction H 11 ( s ) in (4 0) satisfies (2 9). It can b e shown using Proposition 2 th at th e f ollowing constants are real under (4 1): α 11 = 1 − √ α ′ β ′ 2 √ η σ 2 v ( ρ + 1) p (1 − α )(1 − β ) , α 12 = q 1 − α 2 11 , β 12 = γ 2 r ρ − 1 ρ + 1 − α 2 11 . Pr op osition 3: Suppo se (41) hold s. Th en the op timal causal equalizer for the system un der consideration in this section is g iv en by the following tr ansfer fun ctions H 11 ( s ) = α 11 s + γ 2 + i Ω s + γ 2 q ρ − 1 ρ +1 + i Ω , (42) H 12 ( s ) = α 12 s + β 12 + iα 12 Ω s + γ 2 q ρ − 1 ρ +1 + i Ω , (43) H 21 ( s ) = − α 12 s − β 12 + iα 12 Ω s + γ 2 q ρ − 1 ρ +1 + i Ω , (44) H 22 ( s ) = α 11 s − γ 2 + i Ω s + γ 2 q ρ − 1 ρ +1 + i Ω . (45) As we see, the condition ( 41) plays a critical r ole in the above analysis. Th e expr ession on the rig ht-hand sid e of ( 41) depend s on σ 2 w . If ζ ( α ′ + β ′ ) > s 4 ζ 2 α ′ β ′ + (1 − √ α ′ β ′ ) 2 η αβ ( α ′ − β ′ ) 2 , (46) then th is expre ssion is n egative, and (41) h olds trivially . It can b e sh own that if σ 2 w > | 1 − √ α ′ β ′ | √ 1 − η then ( 46) holds, and hence (41) is trivially satisfied. T hus we h ave arri ved at a conclusion similar to that made in th e previous section: If the variance of the thermal n oise in th e system is sufficiently large, then there exists a filter which attenuates the therm a l noise compo nent of y u while injectin g a small amo unt of noise throu g h th e z channel. V I . C O N C L U S I O N S The paper h a s discussed a qu antum counter p art of th e clas- sical W iener filtering problem fo r equalization of quantum systems. The requirem e nt to ob tain a physically realizable passiv e causal equalizer imposes no nconve x constrain ts o n the filter transfer function. W e ha ve d iscussed one f orm of relax ation of th e se con straints, and h ave shown, via examples, th at the relaxation does not pre clude find ing a physically r ealizable coheren t filter able to re d uce the sig n al distortion caused by th e noisy quantum chan n el. R E F E R E N C E S [1] Z. Drmac, S. Gugercin, and C. Beat tie. 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