Quantum Filtering for Systems Driven by Fields in Single Photon States and Superposition of Coherent States using Non-Markovian Embeddings

Quan tum Filtering for Systems Driv en b y Fields in Single Photo n States and Sup erp osition of Coheren t States using Non-Mark o vian Em b eddings ∗ John E. Gough † Matthew R. James ‡ Hendra I. Nurdin § No vem b er 19, 2018 Abstract The purp ose of this pap er is to determine q u an tum master and filter equ ations f or systems coupled to fields in certain non-classical conti nuous-mo d e states. Sp ecifically , w e consider t w o t yp es of field states (i) single photon states, and (ii) su p erp ositions of coheren t sta tes. The system a nd field are describ ed using a qu an tum stochastic unitary model. Master equatio ns are deriv ed from this mo del a nd are giv en in terms of systems of coupled equations. The output fi eld carries information ab out the system, and is con tin uously mon itored. The quantum filters are determined with the aid of an embedd ing of the system into a larger n on-Mark o vian system, and are give n b y a system of coupled sto c hastic d ifferen tial equations. Keyw ords: q uantum filtering, con tin uous-mo de single photon states, con tin uous-mo de sup erp ositions o f coheren t states, quan tum sto c hastic pr o cesses 1 In tro duction In recen t y ears single photon states of ligh t and sup erp ositions of coheren t states hav e b ecome increasingly imp o rtan t due to applications in quan tum technology , in particular, quan tum computing and quantum information systems, [25], [27], [23], [18], [30]. F or instance, the ligh t ma y inte ract with a system, say an atom, quan tum dot, o r cav ity , and ∗ This work was supp orted by the Austra lian Res e arch Council a nd the UK Engineering and Physical Sciences Research Council g r ant EP/G03 9275 /1. † Institute for Ma thematics and Physics, Abery stwyth University , SY23 3BZ, W ale s , United Kingdo m. Email: jug@ab er.a c .uk ‡ AR C Centre for Qua ntu m Co mputation and Communication T echnology , Research Sc ho ol of Enginee r- ing, Austr a lian National University , Can b erra, A CT 0200, Australia . Ema il: Matthew.J a mes@anu.edu.au § Research School of E ngineering, Australian National University , Canberra , ACT 0200 , Austra lia. Email: Hendra.Nurdin@anu.edu.au 1 Single Photon Quantum Filter 2 this system ma y b e used as a quantum memory , [2 5], or to con trol the pulse shap e of the single photon state [27]. When light interacts with a quantum system, information ab out the system is con tained in the scattered light. This informat ion ma y b e useful for monitoring the b ehav ior of the system, or fo r con trolling it. The topic of this pap er concerns the extraction of information fr o m the scattered light when the incoming ligh t is placed in a single photon state | Ψ i = | 1 ξ i , or a sup erp osition of coheren t states | Ψ i = P j α j | f j i , as illustrated in Figure 1 . filter HD measurement signal estimates system detector input output Figure 1 : A system initialized in a state | η i coupled to a field in a state | Ψ i (single photon or sup erp osition of coheren t states). The output field is contin uously monitored by homo dyne detection (a ssumed p erfect) to pro duce a classical measuremen t signal Y ( t ). The o utput Y ( t ) is filtered to pro duce estimates ˆ X ( t ) = π t ( X ) of system op erators X at time t . The problem of extracting info rmation from con tinuous measuremen t of the scattered ligh t is a problem of quantum fi ltering , [4], [5], [6], [11], [3 1], [3], [9], [32]. The curren t state of the a rt for quantum filtering considers incoming light in a v acuum or other Gaussian state, with quadrature or coun ting measuremen ts. Both single photon states of ligh t, and sup erp ositions of coheren t states of light, are highly non-classical, and are fundamentally differen t from Ga ussian states. In view of the increasing imp ort ance of these non-Gaussian states of light, the purp ose of this paper is to solve a quan tum filtering problem for systems driv en by fields in single photon states a nd sup erp ositions of coherent states. In the case of single photon fields, the master equation describing unconditional dy- namics w as shown to b e a system of coupled equations in [17 ], a feature of non-Mark ovian c haracter. Mark ovian em b eddings w ere used in [10] to deriv e quan tum tra jectory equa- tions (quan tum filtering equations) for a class of non-Marko vian master equations. In recen t w o rk, the authors hav e sho wn ho w to construct ancilla syste ms to com bine with t he system of in terest to fo rm a Mark o vian extended system drive n b y v acuum from whic h quan tum filtering results ma y b e obtained for single photon states a nd sup erp o sitions of coheren t states from the standard filter fo r the extended system, [19], [20]. Ho w ev er, dep ending on the complexit y of the non-classical stat e, it may p o ssibly b e difficult to determine suitable ancilla systems, and indeed the sup erp osition case w as not straig h tforw ard. In this pa p er w e presen t an alternativ e approa c h to the embedding that also a llo ws for the deriv ation of the quantum filter. The extended system forms a non- Mar ko vian system, with the ancilla, system and field initialized in a sup erp osition state. While standard filtering results do not apply , the quantum sto chas tic metho ds can neve rtheless b e applied to determine t he quan tum filters. In this w ay , w e expand the range of metho ds that may b e a pplied to deriv e quan tum filters for non- classical states. Single Photon Quantum Filter 3 The pap er is o r g anized as follo ws. In sec tion 2 the idealized filtering problem t o b e solv ed in this pap er is form ulated. The con tin uous mo de single photon states are defined in Section 3.1 . The master equation for the single photon field state is deriv ed in Section 3.2 using the mo del presen ted in Section 2. This leads natura lly to Section 3.3, where the system is em b edded in a larger mo del, inspired b y the approac h used in [10] but differing in the details. This extended system pro vides a compact and transparent description of the problem, a nd may readily b e generalized to n -photo n states, and indeed, multiple c hannels of n -photon states. The quan tum filter for the extended system is presen ted in Section 3.4 , with a deriv ation extending the reference metho d app earing in Section 3.6. The filtering results for the extended system are used to find the filtering equations for the original problem in v olving a single photon field in Section 3.5. The sup erp osition of coherent field states is defined in Section 4.1, and a suitable embedded system for this case is describ ed in Section 4.2. The corresp onding master and filtering equations are presen ted in Sections 4.3 and 4.4, resp ectiv ely . Some concluding remarks are made in Section 5. In this pap er w e are not concerned with technic al issues concerning domains of un- b ounded op erators and relat ed matters, and indeed, w e assume that the system op erators are b ounded, and t hat all quan tum sto c ha stic in tegrals a re w ell- defined in the sens e of Hudson-P a rthasarath y , [22]. Notation: W e use the standard Dirac notatio n | ψ i to denote state v ectors (v ectors in a Hilb ert space) [26], [1]. The superscript ∗ indicates Hilb ert space adjoin t or complex conjugate. The inner pro duct of state v ectors | ψ 1 i and | ψ 2 i is denoted h ψ 1 | ψ 2 i . The exp ected v alue of an op erator X when the system is in state | ψ i is denoted E ψ [ X ] = h ψ | X | ψ i . F or op erators A and B w e write h A, B i = tr[ A ∗ B ] . 2 Problem F orm ulation W e consider a quan tum system S coupled to a quan tum field B , as sho wn in Figure 1. The field B has t w o comp onen ts, the input field B in and the output field, B out . In this pap er w e consider t w o non- classical cases for the state | Ψ i of the input field (i) a single photon state | Ψ i = | 1 ξ i , where ξ is a complex v alued function such t hat R ∞ 0 | ξ ( s ) | 2 ds = 1 (represen t ing the w a v e pac k et shap e), or (ii) a sup erp o sition of coheren t states | Ψ i = P j α j | f j i , where | f j i are coherent states and the complex num b ers α j ( j = 1 , . . . , n ) are normalized w eights . As illustrated in Figure 1, the field in teracts with the quan tum system S , and the results of this in teraction pro vide infor ma t io n ab out the system that ma y b e obtained through con tin uous measuremen t of an o bserv able Y ( t ) of the output field B out ( t ). The filtering pro blem of in terest in this pa p er is to determine the conditio nal state fro m whic h estimates ˆ X ( t ) of system o p erators X may b e determined at time t based on know ledge of the observ ables { Y ( s ), 0 ≤ s ≤ t } . In what f ollo ws the system S is assumed to b e defined o n a Hilb ert space H S , with a n initial state denoted | η i ∈ H S . The input field B in is described in terms of annihilation B ( ξ ) and creation B ∗ ( ξ ) op erators defined on a F o ck space F , [28, Chapter I I], [9, Section 4]. Quan tum exp ectatio n will b e denoted by the sym b o l E , and when we wish to displa y Single Photon Quantum Filter 4 the underlying state, w e emplo y subscripts; fo r example, E η Ψ denotes quantum exp ectation with resp ect to the stat e | η i ⊗ | Ψ i . The dynamics of the system will b e describ ed using the quantum sto c hastic calculus, [22], [15], [28 ], [16 ], [9]. Quantum sto c hastic integrals a re defined in terms of fundamen tal field o p erators B ( t ), B ∗ ( t ) and Λ( t ), [28, Chapter I I], [9, Section 4]. 1 The non- zero Ito pro ducts for the field op erato r s are dB ( t ) dB ∗ ( t ) = dt, dB ( t ) d Λ( t ) = dB ( t ) , d Λ( t ) d Λ( t ) = d Λ( t ) , d Λ ( t ) dB ∗ ( t ) = dB ∗ ( t ) . (1) The dynamics of the comp osite system is describ ed by a unitary U ( t ) solving the Sc hr¨ odinger equation, or quan tum sto c hastic differen tial equation (QSDE), dU ( t ) = { ( S − I ) d Λ( t ) + LdB ∗ ( t ) − L ∗ S dB ( t ) − ( 1 2 L ∗ L + iH ) d t } U ( t ) , (2) with init ia l condition U (0) = I . Here, H is a fixed self-adjoint o p erator r epresen ting the free Hamiltonian of the system, a nd L and S are system op erators determining the coupling of the system to the field, with S unitary . In this pa p er, for simplicit y w e assume that the parameters S, L, H are b ounded op erators on the system Hilb ert space H S . Ho w ev er, w e r emark under some suitable additional conditions the results and equations obtained in this pap er should a lso b e extendable to some sp ecial classes of QSD Es with unbounded parameters, exploiting the results in [13, 14]. A system op erat o r X a t time t is giv en in t he Heisen b erg picture b y X ( t ) = j t ( X ) = U ( t ) ∗ ( X ⊗ I ) U ( t ) and it follo ws from t he quan tum Ito calculus that d j t ( X ) = j t ( S ∗ X S − X ) d Λ( t ) + j t ( S ∗ [ X , L ]) dB ( t ) ∗ + j t ([ L ∗ , X ] S ) d B ( t ) + j t ( L ( X )) dt, (3) where L ( X ) = 1 2 L ∗ [ X , L ] + 1 2 [ L ∗ , X ] L − i [ X , H ] . (4) The map X 7→ L ( X ) is kno wn a s the Lindblad ge n er ator , while the quartet of maps X 7→ L ( X ) , S ∗ X S − X , S ∗ [ X , L ] , [ L ∗ , X ] S are kno wn as Evans-Hudson maps . The output field is defined by B out ( t ) = U ( t ) ∗ B ( t ) U ( t ). 2 In this pap er w e consider the output field observ able Y ( t ) defined by Y ( t ) = U ( t ) ∗ Z ( t ) U ( t ) , (5) where Z ( t ) = B ( t ) + B ∗ ( t ) , (6) 1 In ter ms o f a nnihilation and cre ation white noise o per ators b ( t ) , b ∗ ( t ) that s atisfy sing ula r commutation relations [ b ( s ) , b ∗ ( t )] = δ ( t − s ), the fundamental field op er a tors ar e given by B ( t ) = R t 0 b ( s ) ds , B ∗ ( t ) = R t 0 b ∗ ( s ) ds , and Λ( t ) = R t 0 b ∗ ( s ) b ( s ) ds . Also, we may write B ( ξ ) = R ∞ 0 ξ ∗ ( s ) dB ( s ). 2 Recall B ( t ) = B in ( t ) is the input field. Single Photon Quantum Filter 5 is a quadrature observ a ble of the input field ( t he counting case Z ( t ) = Λ( t ) is discussed briefly in Section 5). Note that b ot h Z ( t ) and Y ( t ) are self-adj o in t and self-commutativ e: [ Z ( t ) , Z ( s )] = 0 a nd [ Y ( t ) , Y ( s )] = 0. W e write Z t and Y t for the subspaces of comm uting op erators generated b y the observ ables Z ( s ), Y ( s ), 0 ≤ s ≤ t , resp ectiv ely . 3 They are related by the unitary rotation Y t = U ( t ) ∗ Z t U ( t ). Ph ysically , Y ( t ) ma y represen t the in tegrated photo curren t arising in an idealized (p erfect) homo dyne photo detection sc heme, as in Figure 1. F o r further information on homo dyne detection, we refer the reader to the literature; for example, [2], [3], [32]. The primary goa l of this pap er is to determine t he quantum fil ter for the quan tum conditional exp ectation (see, e.g. [9, Definition 3.13]) ˆ X ( t ) = E η Ψ [ X ( t ) | Y t ] . (7) This conditional exp ectation is w ell defined, since X ( t ) comm utes with the subspace Y t (non-demolition condition). The conditional estimate ˆ X ( t ) is affiliated to Y t (written in abbreviated fashion a s ˆ X ( t ) ∈ Y t ) and is characterized by the requiremen t that E η Ψ [ ˆ X ( t ) K ] = E η Ψ [ X ( t ) K ] (8) for all K ∈ Y t . 3 Single Pho ton Inp ut Fields 3.1 Single Photon F ields S tates In this section w e consider the contin uous-mo de single photon state | Ψ i = | 1 ξ i defined by [24, sec. 6.3], [27 , eq. (9)] | 1 ξ i = B ∗ ( ξ ) | 0 i , (9) where ξ is a complex v alued function suc h that R ∞ 0 | ξ ( s ) | 2 ds = 1, and | 0 i is the v a cuum state of the field. Expression (9) say s t ha t the single photon wa ve pac k et with t emp oral shap e ξ is created from the v acuum using the field op erator B ∗ ( ξ ). The Hilb ert space fo r the comp osite system is H = H S ⊗ F = H S ⊗ F t ] ⊗ F ( t , where here w e ha v e exhibite d the contin uous temp oral tensor pro duct decomp o sition of the F o c k space F = F t ] ⊗ F ( t in to past a nd future comp onents , whic h is of basic imp ortance in what follows. W rite E 11 [ X ⊗ F ] = h η 1 ξ | ( X ⊗ F ) | η 1 ξ i = h η | X | η ih 1 ξ | F | 1 ξ i (10) for the exp ectatio n with resp ect to the pro duct state | η 1 ξ i , where t he field is in the single photon state. Here and in what follows X is a b ounded system op erator acting on H S , and 3 Z t and Y t are commutativ e von Neumann algebra s. They a re als o filtrations, e .g . Z t 1 ⊂ Z t 2 whenever t 1 < t 2 . Single Photon Quantum Filter 6 F is a field op erator acting on t he F o c k space F . Similarly , we ma y define the exp ectation when the field is in the v acuum state, E 00 [ X ⊗ F ] = h η 0 | ( X ⊗ F ) | η 0 i = h η | X | η ih 0 | F | 0 i . (11) W e will also hav e need for the cross-expectations E 10 [ X ⊗ F ] = h η 1 ξ | ( X ⊗ F ) | η 0 i , and E 01 [ X ⊗ F ] = h η 0 | ( X ⊗ F ) | η 1 ξ i . (12) A crucial difference b etw een the single photon state and the v acuum state is that the later state factorizes | 0 i = | 0 t ] i ⊗| 0 ( t i with resp ect t o the temp oral factorization F = F t ] ⊗ F ( t of the F o c k space, with | 0 t ] i ∈ F t ] and | 0 ( t i ∈ F ( t , while the f o rmer do es not. Rather, w e ha v e | 1 ξ i = B ∗ ( ξ ) | 0 i = | 1 ξ t ] i ⊗ | 0 ( t i + | 0 t ] i ⊗ | 1 ξ ( t i , (13) where | 1 ξ t ] i = B −∗ t ( ξ ) | 0 t ] i , and | 1 ξ ( t i = B + ∗ t ( ξ ) | 0 ( t i , (14) and B − t ( ξ ) = B ( ξ χ [0 ,t ] ) , B + t ( ξ ) = B ( ξ χ ( t, ∞ ] ) , B ( ξ ) = B − t ( ξ ) + B + t ( ξ ) . (15) Here, χ [0 ,t ] is the indicator function for the time interv al [0 , t ]. No te that while | 1 ξ i has unit norm, we ha v e k | 1 ξ t ] i k 2 = Z t 0 | ξ ( s ) | 2 ds, and k | 1 ξ ( t i k 2 = Z ∞ t | ξ ( s ) | 2 ds. (16) A consequenc e of the additiv e decomp osition (13) and the definitions (1 5) is the follow - ing. Let K ( t ) b e a b ounded op erator acting on the full Hilb ert space H t ha t is adapted, i.e. K ( t ) a cts trivially on F ( t , the field in the future. Then the exp ectation with resp ect to the single photon field may b e expressed in terms of the v acuum state as follow s: E 11 [ K ( t )] = E 00 [ B − t ( ξ ) K ( t ) B −∗ t ( ξ ) + r ( t ) K ( t )] (17) where r ( t ) = R ∞ t | ξ ( s ) | 2 ds . 3.2 Master Equation Before deriving the quan t um filter, w e w ork out dynamical equations for the unconditioned single photo n exp ectation, [17 ]. T o assist us in ev alua t ing this exp ectation, w e mak e use of the followin g lemma. Lemma 3.1 L et K ( t ) b e a b ounde d quantum sto chas tic pr o c ess define d b y K ( t ) = Z t 0 M 0 ( s ) ds + Z t 0 M − ( s ) dB ( s ) + Z t 0 M + ( s ) dB ∗ ( s ) + Z t 0 M 1 ( s ) d Λ( s ) , (18) Single Photon Quantum Filter 7 wher e M 0 , M ± and M 1 ar e b ounde d and adapte d. Then we have E 11 [ K ( t )] = E 11 [ Z t 0 M 0 ( s ) ds ] + E 10 [ Z t 0 M − ( s ) ξ ( s ) ds ] + E 01 [ Z t 0 M + ( s ) ξ ∗ ( s ) ds ] + E 00 [ Z t 0 M 1 ( s ) | ξ ( s ) | 2 ds ] , (19) E 10 [ K ( t )] = E 10 [ Z t 0 M 0 ( s ) ds ] + E 00 [ Z t 0 M + ( s ) ξ ∗ ( s ) ds ] , (20) E 01 [ K ( t )] = E 01 [ Z t 0 M 0 ( s ) ds ] + E 00 [ Z t 0 M − ( s ) ξ ( s ) ds ] , (21) E 00 [ K ( t )] = E 00 [ Z t 0 M 0 ( s ) ds ] . (22) Proo f. Using (17), the expressions B − t ( ξ ) = R t 0 ξ ∗ ( s ) dB ( s ), B −∗ t ( ξ ) = R t 0 ξ ( s ) dB ∗ ( s ), and the Ito rule we hav e E 11 [ dK ( t )] = E 00 [ d ( B − t ( ξ ) K ( t ) B −∗ t ( ξ ) + r ( t ) K ( t ))] = E 00 [ B − t ( ξ ) M 0 ( t ) B −∗ t ( ξ ) + r ( t ) M 0 ( t ) + M + ( t ) B −∗ t ( ξ ) ξ ∗ ( t ) + B − t ( ξ ) M − ( t ) ξ ( t ) + M 1 ( t ) | ξ ( t ) | 2 ] dt = E 11 [ M 0 ( t )] dt + E 00 [ M + ( t ) B ∗ ( ξ ) ξ ∗ ( t ) + B ( ξ ) M − ( t ) ξ ( t ) + M 1 | ξ ( t ) | 2 ] dt. (23) This last line is j ustified since M ± are a dapted and E 00 [ B + t ( ξ )] = 0. That is, E 11 [ dK ( t )] = E 11 [ M 0 ( t )] dt + E 01 [ M + ( t )] ξ ∗ ( t ) dt + E 10 [ M − ( t )] ξ ( t ) dt + E 00 [ Z t 0 M 1 ( s ) | ξ ( s ) | 2 ds ] . (24) This prov es (19). The remaining expressions ar e pro v en in a similar manner.  W e will first express the master equation in Heisen b erg form using the exp ectations µ j k t ( X ) = E j k [ X ( t )] . (25) Note that for all t ≥ 0 we ha v e µ 00 t ( I ) = 1 = µ 11 t ( I ) , µ 01 t ( I ) = 0 = µ 10 t ( I ) . (26) Theorem 3.2 The mas ter e quation in Heisenb er g form f o r the system when the fie l d is in the single photon state | 1 ξ i i s given by the system o f e quations ˙ µ 11 t ( X ) = µ 11 t ( L ( X )) + µ 01 t ( S ∗ [ X , L ]) ξ ∗ ( t ) + µ 10 t ([ L ∗ , X ] S ) ξ ( t ) + µ 00 t ( S ∗ X S − S ) | ξ ( t ) | 2 , (27) ˙ µ 10 t ( X ) = µ 10 t ( L ( X )) + µ 00 t ( S ∗ [ X , L ]) ξ ∗ ( t ) , (28) ˙ µ 01 t ( X ) = µ 01 t ( L ( X )) + µ 00 t ([ L ∗ , X ] S ) ξ ( t ) , (29) ˙ µ 00 t ( X ) = µ 00 t ( L ( X )) . (30) The initial c ond i tion s ar e µ 11 0 ( X ) = µ 00 0 ( X ) = h η , X η i , µ 10 0 ( X ) = µ 01 0 ( X ) = 0 . (3 1) Single Photon Quantum Filter 8 Proo f. Equations (27)- (30) are obtained by applying Lemma 3.1 to t he Heisen b erg equation (3).  It is apparen t from Theorem 3.2 that the single photon exp ectation µ 11 t ( X ) = E 11 [ X ( t )] cannot b e determined b y a single differen tial equation, and that instead a system of coupled equations is required, equation (27)- ( 3 0). No t e that the unita ry matrix S app earing in t he Sc hr¨ odinger equation (2) do es app ear in the single photo n master equations (27)- (30), in con trast to the v acuum case (whic h corresp o nds to (30)). 3.3 Em b edding In this section w e construct a suitable em b edding fo r the sys tem and single photon field, and sho w how the sys tem of master equations from Section 3.2 can b e compactly represen ted as a single equation for a larger system. This em b edding will b e used in subseque n t sections to deriv e the quantum filter. W e should emphasize, how ev er, that our embedding is not the same as that used in [10], [19], [20]. The em b edding is illustrated in Figure 2. HD ancilla Figure 2 : System em b edded in the extended system. While the analysis do es not emplo y an y coupling b et w een the system and ancilla t w o-lev el system, the ancilla, system and field are assumed to b e initialized in a sup erp osition state | Σ i defined in equation (36). Recall tha t the system and field ar e defined on a Hilb ert space H = H S ⊗ F . W e define an extended space ˜ H = C 2 ⊗ H = H ⊕ H (32) whic h includes the system, field and an ancilla tw o-lev el system. L et | e 0 i and | e 1 i b e an orthonormal basis fo r C 2 , | e 0 i =  0 1  , | e 1 i =  1 0  , (33) and let A b e a n o p erator acting on C 2 , i.e. a complex 2 × 2 mat r ix A =  a 11 a 10 a 01 a 00  . (34) Single Photon Quantum Filter 9 It ma y b e helpful to think of op era t ors A ⊗ X ⊗ F on the extended space ˜ H represen ted in the Kronec k er pro duct form A ⊗ ( X ⊗ F ) =  a 11 ( X ⊗ F ) a 10 ( X ⊗ F ) a 01 ( X ⊗ F ) a 00 ( X ⊗ F )  . (35) W e allo w the extended system to evolv e unitarily according to I ⊗ U ( t ), where U ( t ) is the unitary op erator for the system a nd field, giv en b y the Sc hr¨ odinger equation (2). Note in pa rticular that the system is not coupled to the a ncilla C 2 , and observ a bles of this t w o-lev el system are static. W e initialize the extended system in the sup erp osition state | Σ i = α 1 | e 1 η 1 ξ i + α 0 | e 0 η 0 i , (36) where | α 0 | 2 + | α 1 | 2 = 1. This state evolv es according to | Σ( t ) i = ( I ⊗ U ( t )) | Σ i . (37) F or notational conv enience w e write w j k = α ∗ j α k (38) and note that w = P j k w j k | e j ih e k | is a densit y mat r ix for C 2 . The exp ectation with resp ect to t he sup erp osition state | Σ i is giv en b y ˜ µ t ( A ⊗ X ) = E ψ [ A ⊗ X ( t )] = h Σ | ( A ⊗ X ( t )) | Σ i = X j k w j k a j k µ j k t ( X ) . (39) This exp ectation is correctly nor malized, µ t ( I ⊗ I ) = 1 , and the exp ectations µ j k t ( X ) defined in Section 3.2 ar e scaled comp onents o f ˜ µ t ( A ⊗ X ): µ j k t ( X ) = ˜ µ t ( | e j ih e k | ⊗ X ) w j k , (40) for w j k 6 = 0, otherwise it can b e set to, say , 0 . W e also hav e µ j k t ( X ) = w 11 ˜ µ t ( | e j ih e k | ⊗ X ) w j k ˜ µ t ( | e 1 ih e 1 | ⊗ I ) . (41) Note that in the extended space the Schr¨ odinger and Heisen b erg pictures ar e related b y E Σ( t ) [ A ⊗ X ⊗ F ] = E Σ [ A ⊗ U ∗ ( t )( X ⊗ F ) U ( t )] . (42) In order to deriv e t he equation for exp ectations in the extended system, w e need t he follo wing lemma, whic h follow s from Lemma 3.1, and makes use of the matrices σ + = | e 1 ih e 0 | =  0 1 0 0  , σ − = | e 0 ih e 1 | =  0 0 1 0  . (43) Single Photon Quantum Filter 10 Lemma 3.3 Assume α 0 6 = 0 , and let M ( t ) b e b ounde d and adap te d. Th en E Σ [ Z t 0 A ⊗ M ( s ) dB ( s )] = ν E Σ [ Z t 0 ( Aσ + ) ⊗ M ( s ) ξ ( s ) ds ] , (44) E Σ [ Z t 0 A ⊗ M ( s ) dB ∗ ( s )] = ν ∗ E Σ [ Z t 0 ( σ − A ) ⊗ M ( s ) ξ ∗ ( s ) ds ] , (45) E Σ [ Z t 0 A ⊗ M ( s ) d Λ( s ) ] = | ν | 2 E Σ [ Z t 0 ( σ − Aσ + ) ⊗ M ( s ) | ξ ( s ) | 2 ds ] , (46) wher e ν = α 1 α 0 . (47) In Lemma 3.3, exp ectations of sto c hastic in tegrals with resp ect to the sup erp osition state | Σ i are expressed in terms of exp ectations o f non-sto c hastic integrals again with resp ect to | Σ i with the aid of the matrices σ ± acting on the ancilla system C 2 . The action of the field a nnihilation, creation and gauge op erators is t herefore captured algebraically and all exp ectations in these relations ar e with resp ect to the same state. W e no w hav e Theorem 3.4 Assume α 0 6 = 0 . Then the exp e ctation ˜ µ t ( A ⊗ X ) (de fi ne d by (39)) evolve s ac c or d i n g to ˙ ˜ µ t ( A ⊗ X ) = ˜ µ t ( G t ( A ⊗ X )) , (48) wher e G t ( A ⊗ X ) = A ⊗ L ( X ) + ( Aσ + ) ⊗ [ L ∗ , X ] S ν ξ ( t ) + ( σ − A ) ⊗ S ∗ [ X , L ] ν ∗ ξ ∗ ( t ) +( σ − Aσ + ) ⊗ ( S ∗ X S − X )) | ν ξ ( t ) | 2 . (49) The reader may easily v erify t ha t the system of master equations (27)- (30) for µ j k t ( X ), j, k = 1 , 0, follow s from equation (48) by setting A = | e j ih e k | . 3.4 Quan tum Filter for the Extended S ystem The extended system provides a con v enien t f r amew or k for quan tum filtering, since all exp ectatio ns can b e expressed in terms of the sup erp osition state | Σ i . Our immediate goal in this section is t o determine the equation for the quan tum conditional exp ectation ˜ π t ( A ⊗ X ) = E Σ [ A ⊗ X ( t ) | I ⊗ Y t ] , (50) and in Section 3.5 w e will explain ho w the quan tum filter for the single photo n field ma y b e obtained fr om t his equation. The con tin uously monitored field observ able that corresp onds to the conditional exp ec- tation (50) is I ⊗ Y ( t ), and fro m (5) w e ha ve the corresp onding output equation f o r the extended system: d ( I ⊗ Y ( t )) = I ⊗ ( L ( t ) + L ∗ ( t )) dt + I ⊗ ( S ( t ) dB ( t ) + S ∗ ( t ) dB ∗ ( t )) . (51) Single Photon Quantum Filter 11 In what follows w e will mak e use o f the following lemma concerning exp ectations of the pro cess V ( t ) = Z t 0 ( S ( s ) dB ( s ) + S ∗ ( s ) dB ∗ ( s )) (52) with resp ect to the single photon state. Lemma 3.5 F or any K ∈ Y s , we have E 11 [( V ( t ) − V ( s ) ) K ] = E 10 [ Z t s S ( r ) ξ ( r ) dr K ] + E 01 [ Z t s S ∗ ( r ) ξ ∗ ( r ) dr K ] . (53) Proo f. Equation (53) is obta ined using the additive decomp osition (13) and Lemma 3.1.  Note that an op erator K in the unital comm utativ e alg ebra I ⊗ Y t has the for m K = I ⊗ ˜ K , where ˜ K ∈ Y t . B y the spectral theorem, [9, Theorem 3.3], we ma y iden tify K and ˜ K , b oth of whic h are equiv alen t to a classical sto chastic pro cess K t ( s ), 0 ≤ s ≤ t . In the r emainder o f this pap er, w e use these iden tifications without f urt her comment. The quan tum conditio nal exp ectation ˜ π t ( A ⊗ X ) ∈ I ⊗ Y t is w ell defined b ecause A ⊗ X ( t ) is in the comm utan t I ⊗ Y ′ t of the algebra I ⊗ Y t , and is characterize d by the r equiremen t that E Σ [ ˜ π t ( A ⊗ X ) I ⊗ K ] = E Σ [( A ⊗ X ( t ))( I ⊗ K )] (54) for all K ∈ Y t , see, e.g. [9, Definition 3.13]. Theorem 3.6 Assume α 0 6 = 0 . T he c o n ditional exp e ctation ˜ π t ( A ⊗ X ) define d by (50) for the extende d system satisfie s d ˜ π t ( A ⊗ X ) = ˜ π t ( G t ( A ⊗ X )) dt + H t ( A ⊗ X ) d W ( t ) , (55) wher e H t ( A ⊗ X ) = ˜ π t ( A ⊗ ( X L + L ∗ X )) − ˜ π t ( A ⊗ X ) π t ( I ⊗ ( L + L ∗ )) + ˜ π t (( Aσ + ) ⊗ X S ) ν ξ ( t ) + ˜ π t (( σ − A ) ⊗ S ∗ X ) ν ∗ ξ ∗ ( t ) − ˜ π t ( A ⊗ X ) ˜ π t (( σ + ⊗ S ) ν ξ ( t ) + ( σ − ⊗ S ∗ ) ν ∗ ξ ∗ ( t )) (56) and dW ( t ) = d Y ( t ) − ˜ π t ( I ⊗ ( L + L ∗ ) + ( σ + ⊗ S ) ν ξ ( t ) + ( σ − ⊗ S ∗ ) ν ∗ ξ ∗ ( t )) dt. (57) The p r o c ess W ( t ) define d by (5 7 ) is a I ⊗ Y t Wiener pr o c ess with r e sp e ct to | Σ i a nd is c al le d the inno v ations pro cess . Proo f. W e follow the characteristic function metho d [29], [7], [5], whereb y w e p os- tulate that the filter ha s the form d ˜ π t ( A ⊗ X ) = F t ( A ⊗ X ) d t + H t ( A ⊗ X ) I ⊗ d Y ( t ) , (58) Single Photon Quantum Filter 12 where F t and H t are t o b e determined. Let f b e square in tegrable, and define a pro cess c f b y dc f ( t ) = f ( t ) c f ( t ) d Y ( t ) , c f (0) = 1 . (59) Then I ⊗ c f ( t ) is adapted to I ⊗ Y t , a nd the defining relation (54) implies that E Σ [ A ⊗ ( X ( t ) c f ( t ))] = E Σ [ ˜ π t ( A ⊗ X ) I ⊗ c f ( t ))] (60) holds for all f . By calculating the differen tials of b oth sides, taking exp ectatio ns and conditioning w e obtain E Σ [ A ⊗ ( d X ( t ) c f ( t ))] = E Σ [( I ⊗ c f ( t )) ˜ π t ( G ( A ⊗ X )) (61) +( I ⊗ f ( t ) c f ( t )) { ˜ π t ( A ⊗ ( X L + L ∗ X )) + ˜ π t ( Aσ + ⊗ X S ) ν ξ ( t ) + ˜ π t ( σ − A ⊗ S ∗ X ) ν ∗ ξ ∗ ( t ) } ] dt and E Σ [ A ⊗ ( d ˜ π t ( A ⊗ X ) c f ( t ))] (62) = E Σ [( I ⊗ c f ( t ) {F t ( A ⊗ X ) + H t ( A ⊗ X ) ˜ π t ( I ⊗ ( L + L ∗ )) + H t ( A ⊗ X ) π t (( σ + ⊗ S ) ν ξ ( t ) + ( σ − ⊗ S ∗ ) ν ∗ ξ ∗ ( t )) } +( I ⊗ f ( t ) c f ( t )) { ˜ π t ( A ⊗ X ) ˜ π t ( I ⊗ ( L + L ∗ )) + H t + ˜ π t ( A ⊗ X ) ˜ π t ( σ + ⊗ S ν ξ ( t ) + σ − ⊗ S ∗ ν ∗ ξ ∗ ( t )) } ] dt. No w equating co efficien ts of c f ( t ) and f ( t ) c f ( t ) we solv e for F t ( A ⊗ X ) and H t ( A ⊗ X ) to obtain the filter equation. W e no w prov e the martingale prop erty E Σ [ I ⊗ ( W ( t ) − W ( s )) | I ⊗ Y s ] = 0 , that is, E Σ [ I ⊗ ( W ( t ) − W ( s ))( I ⊗ K )] = 0 for all K ∈ Y t . Now E Σ [ I ⊗ ( W ( t ) − W ( s ) )( I ⊗ K )] = E Σ [ { I ⊗ ( Y ( t ) − Y ( s )) − Z t s π r ( I ⊗ ( L + L ∗ ) + σ + ⊗ S ξ ( r ) + σ − ⊗ S ∗ ξ ∗ ( r )) dr } I ⊗ K ] = E Σ [ { I ⊗ ( Y ( t ) − Y ( s )) − Z t s ( I ⊗ ( L ( r ) + L ∗ ( r )) + σ + ⊗ S ( r ) ξ ( r ) + σ − ⊗ S ∗ ( r ) ξ ∗ ( r )) dr } I ⊗ K ] = E Σ [ { I ⊗ ( V ( t ) − V ( s )) − Z t s ( σ + ⊗ S ( r ) ν ξ ( r ) + σ − ⊗ S ∗ ( r ) ν ∗ ξ ∗ ( r )) dr } I ⊗ K ] = 0 . T o see that this last expression is zero, we mak e use of Lemma 3.5, the m ultiplicative factorization of the v acuum state, the fact that V ( t ) has zero exp ectation in the v a cuum state, to find that E Σ [ I ⊗ ( V ( t ) − V ( s )) I ⊗ K ] = w 11 E 11 [( V ( t ) − V ( s )) K ] + w 00 E 00 [( V ( t ) − V ( s ) ) K ] = w 11 ( Z t s E 10 [ S ( r ) K ] ξ ( r ) dr + Z t s E 01 [ S ∗ ( r ) K ] ξ ∗ ( r ) dr ) , Single Photon Quantum Filter 13 and E Σ [ Z t s ( σ + ⊗ S ( r ) K ) ξ ( t ) + σ − ⊗ S ∗ ( r ) K ξ ∗ ( t )) dr ] = w 11 ( Z t s E 10 [ S ( r ) K ] ξ ( r ) dr + Z t s E 01 [ S ∗ ( r ) K ] ξ ∗ ( r ) dr ) . Finally , since dW ( t ) dW ( t ) = dt , Levy’s Theorem implies that W ( t ) is a Y t Wiener pro cess. This completes the pro of.  Notice the terms in v olving σ ± in the filter ( equation (56)) and in the innov ations pro cess (equation (57)). These terms arise from exp ectations inv olving the single photon state. Note that due to the martingale pro p ert y o f the inno v a tions pro cess W ( t ) we see that if w e ta ke the expected v alue of equation (56) we recov er equation (48), consisten t with E Σ [ ˜ π t ( A ⊗ X )] = ˜ µ t ( A ⊗ X ) and the definition of conditional exp ectatio n. 3.5 Single Photon Q uan tum Filter W e return now to t he main goal of the pap er, namely the determination of the quantum filter for the conditio na l state when the field is in the single photon state, a s stated in equation (7). As discussed earlier, our strategy is to make use of the filtering results obtained in Section 3.4 f o r the extended system. Lemma 3.7 Assume α 0 6 = 0 . D efine the c ondi tion a l quantities π j k t ( X ) by π j k t ( X ) = w 11 ˜ π t ( | e j ih e k | ⊗ X ) w j k ˜ π t ( | e 1 ih e 1 | ⊗ I ) . (63) wher e ˜ π t ( A ⊗ X ) is the c onditional state for the extende d system defin e d by (50). Then f o r al l K ∈ Y t we have E 11 [ π j k t ( X ) K ] = E j k [ j t ( X ) K ] . (64) Proo f. W e ha v e E 11 [ π j k t ( X ) K ] = 1 w 11 E Σ [ | e 1 ih e 1 | ⊗ ( π j k t ( X ) K )] = 1 w 11 E Σ [ π t ( | e 1 ih e 1 | ⊗ I )( I ⊗ π j k t ( X ) K )] = 1 w j k E Σ [ π t ( | e j ih e k | ⊗ X )( I ⊗ K )] = 1 w j k E Σ [( | e j ih e k | ⊗ j t ( X ))( I ⊗ K )] = E j k [ j t ( X ) K ] as required.  W e can no w presen t our main theorem for the quantum filter for the single photon field state. Single Photon Quantum Filter 14 Theorem 3.8 The quantum filter for the c ond itional exp e ctation with r esp e ct to the single photon field is given in the Heisenb er g pictur e by ˆ X ( t ) = E 11 [ X ( t ) | Y t ] = π 11 t ( X ) , (65) wher e π 11 t ( X ) i s define d by (63) (f o r j = k = 1 ) , and is given by the system of e quations dπ 11 t ( X ) = ( π 11 t ( L ( X )) + π 01 t ( S ∗ [ X , L ]) ξ ∗ ( t ) + π 10 t ([ L ∗ , X ] S ) ξ ( t ) + π 00 t ( S ∗ X S − X ) | ξ ( t ) | 2 ) dt +( π 11 t ( X L + L ∗ X ) + π 01 t ( S ∗ X ) ξ ∗ ( t ) + π 10 t ( X S ) ξ ( t ) − π 11 t ( X )( π 11 t ( L + L ∗ ) + π 01 t ( S ) ξ ( t ) + π 10 t ( S ∗ ) ξ ∗ ( t ))) dW ( t ) , (66) dπ 10 t ( X ) = ( π 10 t ( L ( X )) + π 00 t ( S ∗ [ X , L ]) ξ ∗ ( t )) dt +( π 10 t ( X L + L ∗ X ) + π 00 t ( S ∗ X ) ξ ∗ ( t ) − π 10 t ( X )( π 11 t ( L + L ∗ ) + π 01 t ( S ) ξ ( t ) + π 10 t ( S ∗ ) ξ ∗ ( t ))) dW ( t ) , (67) dπ 01 t ( X ) = ( π 01 t ( L ( X )) + π 00 t ([ L ∗ , X ] S ) ξ ( t )) dt +( π 01 t ( X L + L ∗ X ) + π 00 t ( X S ) ξ ( t ) − π 01 t ( X )( π 11 t ( L + L ∗ ) + π 01 t ( S ) ξ ( t ) + π 10 t ( S ∗ ) ξ ∗ ( t ))) dW ( t ) , (68) dπ 00 t ( X ) = π 00 t ( L ( X )) dt + ( π 00 t ( X L + L ∗ X ) − π 00 t ( X )( π 11 t ( L + L ∗ ) + π 01 t ( S ) ξ ( t ) + π 10 t ( S ∗ ) ξ ∗ ( t ))) dW ( t ) . (69) Her e, the inno v a tions pr o c ess W ( t ) is a Y t Wiener pr o c ess with r esp e ct to the single photon state and is define d b y dW ( t ) = d Y ( t ) − ( π 11 t ( L + L ∗ ) + π 10 t ( S ) ξ ( t ) + π 01 t ( S ∗ ) ξ ∗ ( t )) dt. (70) The initial c ond i tion s ar e π 11 0 ( X ) = π 00 0 ( X ) = h η , X η i , π 10 0 ( X ) = π 01 0 ( X ) = 0 . (7 1) Proo f. Supp ose first that α 0 6 = 0. Setting j = k = 1 in equation ( 64) ab ov e, and noting that K ∈ Y s w a s o therwise arbitrary , w e deduce that π 11 t ( X ) is the desired conditional exp ectation for the single photon field state, a s c haracterized b y equation (8 ). The differential equations (66)-(69) follo w from the definition (63), the filter (55) for the extended system, and the Ito rule. Next, we note that the co efficien t s of the QSDEs (66)- (69), the initial conditions, and Y t do not dep end o n α 0 and α 1 . Henc e, the solutions π j k t ( X ) of this system of equations are indep enden t of α 0 and α 1 . Therefore, π j k t ( X ) can b e defined fo r α j ∈ { 0 , 1 } , j = 0 , 1, and is in fa ct iden tical for all 0 ≤ | α 0 | , | α 1 | ≤ 1. Single Photon Quantum Filter 15 W e now prov e that W ( t ) is a Y t -martingale, that is, E 11 [ W ( t ) − W ( s ) | Y s ] = 0. T o this end, let K ∈ Y s . Then E 11 [ { W ( t ) − W ( s ) } K ] = E 11 [ { Y ( t ) − Y ( s ) − Z t s ( π 11 r (( L + L ∗ ) + π 10 r ( S ) ξ ( t ) + π 01 r ( S ∗ ) ξ ∗ ( t )) dr } K ] = E 11 [ { Z t s ( L ( r ) + L ∗ ( r )) dr + V ( t ) − V ( s ) − Z t s ( π 11 r (( L + L ∗ ) + π 10 r (1) ξ ( t ) + π 01 r (1) ξ ∗ ( t )) dr } K ] = E 11 [ { Z t s ( L ( r ) + L ∗ ( r ) − π 11 r (( L + L ∗ )) dr } K ] + E 11 [ { V ( t ) − V ( s ) − Z t s ( π 10 r ( S ) ξ ( t ) + π 01 r ( S ∗ ) ξ ∗ ( t )) dr } K ] ho w ev er, this v anishes from (64), Lemma 3.5, and E 11 [ { Z t s ( π 10 r ( S ) ξ ( t ) + π 01 r ( S ∗ ) ξ ∗ ( t )) dr } K ] = E 10 [ Z t s S ( r ) ξ ( r ) dr K ] + E 01 [ Z t s S ∗ ( r ) ξ ∗ ( r ) dr K ] . Finally , since dW ( t ) dW ( t ) = dt , Levy’s Theorem implies that W ( t ) is a Y t Wiener pro cess.  3.6 Reference Metho d for Filtering in the Extended System The reference metho d is one of the standard appro a c hes to filtering t heory , with its o r ig ins in the work of Duncan, Mortensen, Zak ai, Holev o and Belav kin, see [1 2], [2 1 ], [5], [8 ], [9]. In this sec tion w e apply this approach, as describ ed in [9, sec. 6], to the filtering problem in the extended system, giving a n indep enden t deriv ation of the fundamen tal filtering equation. Our first step is the f ollo wing. Lemma 3.9 Assume α 0 6 = 0 . T hen we have E Σ [( A ⊗ X ( t ))] = E Σ [ F ∗ ( t )( A ⊗ X ) F ( t )] (72) wher e F ( t ) ∈ ( I ⊗ Z t ) ′ is g i v en by dF ( t ) = ( G 0 ( t ) dt + G 1 ( t ) d Z ( t )) F ( t ) , (73) F (0) = I , and G 0 ( t ) = − I ⊗ ( 1 2 L ∗ L + iH ) − σ + ⊗ ( L + L ∗ S ) ν ξ ( t ) , (74) G 1 ( t ) = I ⊗ L + σ + ⊗ ( S − I ) ν ξ ( t ) . (75) Single Photon Quantum Filter 16 Proo f. Let us supp o se that F ( t ) satisfies (73) with co efficien ts G 0 ( t ), G 1 ( t ) whic h b oth comm ute with σ + ⊗ I , that is, G i ( t ) = I ⊗ g i 0 ( t ) + σ + ⊗ g i 1 ( t ) . Then E Σ [ d { F ( t ) ∗ A ⊗ X F ( t ) } ] = E Σ [ F ( t ) ∗ { T t ( A ⊗ X ) d t + Q t ( A ⊗ X ) d Z ( t ) } F ( t )] , where T t ( A ⊗ X ) = G 1 ( t ) ∗ A ⊗ X G 1 ( t ) + A ⊗ X G 0 ( t ) + G 0 ( t ) ∗ A ⊗ X , (76) Q t ( A ⊗ X ) = A ⊗ X G 1 ( t ) + G 1 ( t ) ∗ A ⊗ X . (77) Using lemma 3.3 we see that this equals E Σ [ F ( t ) ∗ { T t ( A ⊗ X ) + Q t ( A ⊗ X ) ν ξ ( t ) ( σ + ⊗ I ) + ν ∗ ξ ( t ) ∗ ( σ − ⊗ I ) Q t ( A ⊗ X ) } F ( t )] d t. W e no w require that this equals E Σ [ G t ( A ⊗ X )] dt . This implies the f o ur iden tities g 11 ∗ X g 11 + ν ξ g ∗ 11 X + ν ∗ ξ ∗ X g 11 = | ν ξ | 2 ( S ∗ X S − X ) (78) g ∗ 10 X g 11 + X g 01 + ν ξ g ∗ 10 X + ν ξ X g 10 = ν ξ [ L ∗ , X ] S (79) g ∗ 11 X g 10 + g 01 ∗ X + ν ∗ ξ ∗ g 10 ∗ X + ν ∗ ξ ∗ X g 10 = ν ∗ ξ ∗ S ∗ [ X , L ] (80) g ∗ 10 X g 10 + g ∗ 00 X + X g 00 = L ( X ) . (81) The first iden tity (7 8) is satisfied if g 11 = ν ξ ( S − 1). Substituting X = I into (7 9) w e deduce that g 01 = − ν ξ g ∗ 10 S − ν ξ g 10 and th us g ∗ 10 X g 11 + X g 01 + ν ξ g ∗ 10 X + ν ξ X g 10 = ν ξ [ g ∗ 10 , X ] S. Therefore (79 ) is satisfied if g 10 = L , and consequen tly g 01 = − ν ξ ( L + L ∗ S ). It then follo ws that (80) will b e automatically satisfied, while (81) then only requires that g 00 = − ( 1 2 L ∗ L + iH ) in order to obtain the Lindblad generator L ( X ). This leads us precisely to the co efficien ts G i ( t ) stat ed in the lemma, and the iden tit y T t ( A ⊗ X ) + Q t ( A ⊗ X ) ν ξ ( t ) ( σ + ⊗ I ) + ν ∗ ξ ( t ) ∗ ( σ − ⊗ I ) Q t ( A ⊗ X ) = G t ( A ⊗ X ) (82 ) The following Bay es’ relation is prov en along similar lines to [9, Theorem 6.2]. Lemma 3.10 F or α 0 6 = 0 , define ς t ( A ⊗ X ) = ( I ⊗ U ∗ ( t )) E Σ [ F ∗ ( t )( A ⊗ X ) F ( t ) | I ⊗ Z t ]( I ⊗ U ( t )) . (83) Then π t ( A ⊗ X ) = ς t ( A ⊗ X ) ς t ( I ⊗ I ) . (84) Single Photon Quantum Filter 17 In order to determine the differential equation for ς t ( A ⊗ X ), w e first define, for A ⊗ X ∈ B ( C 2 ⊗ h S ) the pro cess γ t ( A ⊗ X ) = E Σ [ F ( t ) ∗ A ⊗ X F ( t ) | I ⊗ Z t ] , (85) so that ς t ( A ⊗ X ) ≡ ( I ⊗ U ( t )) ∗ γ t ( A ⊗ X )( I ⊗ U ( t )). W e then ha v e Lemma 3.11 L et α 0 6 = 0 . T he p r o c ess γ t ( A ⊗ X ) satisfies the Q SDE dγ t ( A ⊗ X ) = τ t ( A ⊗ X ) d t + β t ( A ⊗ X ) d Z ( t ) (86) wher e τ t ( A ⊗ X ) , β t ( A ⊗ X ) ∈ Z t , ar e given by β t ( A ⊗ X ) = γ t ( Q t ( A ⊗ X )) + ν ξ ( t ) γ t ( A ⊗ X ( σ + ⊗ I )) + ν ∗ ξ ( t ) ∗ γ t (( σ − ⊗ I ) A ⊗ X ) − γ t ( A ⊗ X ) θ t , τ t ( A ⊗ X ) = γ t ( T t ( A ⊗ X )) + ν ξ ( t ) γ t ( Q t ( A ⊗ X ) σ + ) + ν ∗ ξ ( t ) ∗ γ t ( σ − Q t ( A ⊗ X )) − β t ( A ⊗ X ) θ t , with θ t = E ψ [( ν ξ ( t ) σ + + ν ∗ ξ ∗ ( t ) σ − ) ⊗ I | I ⊗ Z t ] . Proo f. Setting R t = F ( t ) ∗ ( A ⊗ X ) F ( t ), we hav e tha t dR t = F ( t ) ∗ T t ( A ⊗ X ) F ( t ) dt + F ( t ) ∗ Q t ( A ⊗ X ) F ( t ) d Z ( t ) and our aim is compute γ t ( A ⊗ X ) = E Σ [ R t | I ⊗ Z t ]. In part icular, E Σ [( R t − γ t ( A ⊗ X )) D t ] = 0 (87) for ev ery D t ∈ I ⊗ Z t and we now apply a tec hnique similar to the c haracteristic f unction metho d, this time using the input pro cess Z and ta king the pro cess D t to satisfy the QSDE dD t = f ( t ) D t d Z ( t ) with D 0 = I , for g iven integrable f . F rom the Ito pro duct rule we then ha v e 0 = E Σ [( dR t − dγ t ( A ⊗ X )) D t + ( R t − γ t ( A ⊗ X )) d D t + ( dR t − dγ t ( A ⊗ X )) d D t ] (88) and making the ansatz that d γ t ( A ⊗ X ) = τ t ( A ⊗ X ) d t + β t ( A ⊗ X ) d Z ( t ) for unknow n co efficien ts τ t ( A ⊗ X ) and β t ( A ⊗ X ) we see that 0 = E Σ [( F ( t ) ∗ T t ( A ⊗ X ) F ( t ) − τ t ( A ⊗ X )) D t dt +( F ( t ) ∗ Q t ( A ⊗ X ) F ( t ) − β t ( A ⊗ X )) D t d Z ( t )] + E ψ [( R t − γ t ( A ⊗ X )) D t f ( t ) d Z ( t )] + E ψ [[ F ( t ) ∗ Q t ( A ⊗ X ) F ( t ) − β t ( A ⊗ X )] D t f ( t ) dt ( t )] . W e now mak e use of Lemma 3.3 again and a pply the commutation relations F ( t ) σ + = σ + F ( t ), σ − F ( t ) ∗ = F ∗ ( t ) σ − . (Note that σ + will not comm ute with F ∗ ( t ).) Inserting Single Photon Quantum Filter 18 E Σ [ ·| I ⊗ Z t ] under the exp ectation sign, then separating co efficien ts of D t and D t f ( t ), we obtain the equations 0 = γ t ( T t ( A ⊗ X )) − τ t ( A ⊗ X ) + ν t γ t ( Q t ( A ⊗ X ) ( σ + ⊗ I )) + ν t γ t (( σ − ⊗ I ) Q t ( A ⊗ X )) − β t ( A ⊗ X ) E Σ [ ν ξ ( t ) σ + ⊗ I + ν ∗ ξ ( t ) ∗ σ − ⊗ I | I ⊗ Z t ] , 0 = γ t ( Q t ( A ⊗ X )) − β t ( A ⊗ X ) + ν ξ ( t ) γ t ( A ⊗ X ( σ + ⊗ I )) + ν ∗ ξ ( t ) ∗ γ t (( σ − ⊗ I ) A ⊗ X ) − γ t ( A ⊗ X ) E Σ [ ν ξ ( t ) σ + ⊗ I + ν ∗ ξ ( t ) ∗ σ − ⊗ I | I ⊗ Z t ] . Rearranging these expressions yields the relations in the statemen t o f the lemma. Theorem 3.12 L et α 0 6 = 0 . Th e unnorma lize d c onditional exp e ctation ς t ( A ⊗ X ) define d by (83) satisfies the e quation dς t ( A ⊗ X ) = ς t ( G t ( A ⊗ X )) d t + λ t ( A ⊗ X ) d ˜ Y ( t ) , (89) wher e λ t ( A ⊗ X ) = ς t ( A ⊗ X ˜ L t + ˜ L ∗ t A ⊗ X ) − ς t ( A ⊗ X ) κ t , (90) ˜ L t = I ⊗ L + ν t ξ ( t ) σ + ⊗ S, (91) d ˜ Y ( t ) = d Y ( t ) − κ t dt, ˜ Y (0) = 0 , (92) κ t = ς t (( ν ξ ( t ) σ + + ν ∗ ξ ( t ) ∗ σ − ) ⊗ I ) . (93) Proo f. W e remark that b y insp ection the co efficien ts in t he QSDE for γ t ( A ⊗ X ) simplify to β t ( A ⊗ X ) = γ t ( Q t ( A ⊗ X )) + ν ξ ( t ) γ t ( A ⊗ X ( σ + ⊗ I )) + ν ∗ ξ ( t ) ∗ γ t (( σ − ⊗ I ) A ⊗ X ) − γ t ( A ⊗ X ) θ t = γ t ( A ⊗ X ( I ⊗ L + ν t σ + ⊗ S ) + ( I ⊗ L ∗ + ν ∗ ξ ( t ) ∗ σ − ⊗ S ∗ ) A ⊗ X ) − γ t ( A ⊗ X ) θ t , τ t ( A ⊗ X ) = γ t ( T t ( A ⊗ X )) + ν ξ ( t ) γ t ( Q t ( A ⊗ X ) σ + ) + ν ∗ ξ ( t ) ∗ γ t ( σ − Q t ( A ⊗ X )) − β t ( A ⊗ X ) θ t ≡ γ t ( G ( A ⊗ X )) − β t ( A ⊗ X ) θ t , and therefore dγ t ( A ⊗ X ) = γ t ( G ( A ⊗ X )) dt + β t ( A ⊗ X ) [ d Z ( t ) − θ t dt ] . The QSDE for ς t ( A ⊗ X ) ≡ ( I ⊗ U ( t )) ∗ γ t ( A ⊗ X )( I ⊗ U ( t )) is then readily deduced from the unitary ro tation noting that κ t ≡ ( I ⊗ U ( t )) ∗ θ t ( I ⊗ U ( t )) a nd λ t ( A ⊗ X ) ≡ ( I ⊗ U ( t )) ∗ β t ( A ⊗ X ) ( I ⊗ U ( t )). Single Photon Quantum Filter 19 Corollary 3.13 F or α 0 6 = 0 , the c onditional exp e ctation π t ( A ⊗ X ) de fine d by (5 0) and given by (84) satisfies e quation ( 55) derive d in The or em 3.6. Proo f. W e see that dς t ( I ⊗ I ) = λ t ( I ⊗ I ) d ˜ Y ( t ) and so d 1 ς t ( I ⊗ I ) = − λ t ( I ⊗ I ) ς t ( I ⊗ I ) 2 d ˜ Y ( t ) + 1 ς t ( I ⊗ I ) 3 λ t ( I ⊗ I ) 2 dt. Ho w ev er, w e note from (90) that λ t ( I ⊗ I ) ς t ( I ⊗ I ) ≡ π t  ˜ L t + ˜ L ∗ t  − κ t . By an application of the Ito pro duct r ule, the normalized filter therefore satisfies dπ t ( A ⊗ X ) = π t ( G t ( A ⊗ X )) d t +  λ t ( A ⊗ X ) − π t ( A ⊗ X ) λ t ( I ⊗ I ) ς t ( I ⊗ I )   d Y ( t ) − κ t dt − λ t ( I ⊗ I ) ς t ( I ⊗ I ) dt  = π t ( G t ( A ⊗ X )) d t + { π t  A ⊗ X ˜ L t + ˜ L ∗ t A ⊗ X  − π t ( A ⊗ X ) π t  ˜ L t + ˜ L ∗ t  } h d Y ( t ) − π t  ˜ L t + ˜ L ∗ t  dt i ≡ π t ( G t ( A ⊗ X )) d t + H t ( A ⊗ X ) d W ( t ) , since w e hav e from (91), (56) and(57) H t ( A ⊗ X ) ≡ π t  A ⊗ X ˜ L t + ˜ L ∗ t A ⊗ X  − π t ( A ⊗ X ) π t  ˜ L t + ˜ L ∗ t  , dW ( t ) ≡ d Y − π t  ˜ L t + ˜ L ∗ t  dt. Therefore w e reco v er equation (5 5). 4 Fields i n a Sup erp osi tion of Cohe ren t St ates 4.1 Sup erp osition of Coheren t States In this section we tak e the field to b e in a sup erp osition state | Ψ i = X j α j | f j i , (94) where | f j i ar e coheren t states and the complex n umbers α j ( j = 1 , . . . , n ) are non-zero normalized w eights (describ ed further b elow). Single Photon Quantum Filter 20 Coheren t vec tors | f i may b e expressed in terms o f the v acuum vec tor using t he W eyl (or displacemen t) op erator [28] W ( f ) whic h serv es as a “density ”: | f i = W ( f ) | 0 i . (95) While the collection o f all coherent v ectors is dense in the F o c k space, they are not orthog- onal, and indeed the inner pro duct (in the F o ck space) is give n b y h f | g i = exp( − 1 2 k f k 2 2 − 1 2 k g k 2 2 + h f , g i 2 ) . (96) Here, k f k 2 2 = h f , f i 2 and h f , g i 2 are the L 2 ([0 , ∞ ) , C ) norm and inner pro duct resp ectiv ely . The sup erp osition stat e | ψ i giv en b y (94) is sp ecified b y a c ho ice of coherent v ectors | f j i , with w eigh ts α j ensuring normalization: h ψ | ψ i = P j k α ∗ j α k g j k = 1, where g j k = h f j | f k i . F or a system op erator X acting on H S , and F is a field op erat o r acting on the F o ck space F , the exp ectation with resp ect t o t he state | η i ⊗ | Ψ i is defined by E η Ψ [ X ⊗ F ] = h η Ψ | ( X ⊗ F ) | η Ψ i = h η | X | η ih Ψ | F | Ψ i = h η | X | η i X j k α ∗ j α k h f j | F | f k i = X j k α ∗ j α k E j k [ X ⊗ F ] , (97 ) where E j k [ X ⊗ F ] = h η | X | η i h f j | F | f k i (98) for j, k = 1 , . . . , n . W e write E 00 [ X ⊗ F ] = h η | X | η ih 0 | F | 0 i for the v acuum case. Consider no w the exp ectation of an a dapted o p erator K ( t ) on the comp osite system H = ( H S ⊗ F t ] ) ⊗ F ( t ; this means that K ( t ) acts trivially on the future comp onen t F ( t . Let χ [0 ,t ] is the indicator function f or the t ime in t erv al [0 , t ]. No w coherent v ectors and W eyl op erat ors factorize as | f i = | f χ [0 ,t ] i ⊗ | f χ ( t, ∞ ) i and W ( f ) = W ( f χ [0 ,t ] ) ⊗ W ( f χ ( t, ∞ ) ), resp ectiv ely . W rite W − t ( f ) = W ( f χ [0 ,t ] ) , W + t ( f ) = W ( f χ ( t, ∞ ) ) . (99) Then we can express the coheren t exp ectations of adapted pro cesses K ( t ) in terms of the v acuum: E j k [ K ( t )] = E 00 [ W −∗ t ( f j ) K ( t ) W − t ( f k )] r j k ( t ) (100) where r j k ( t ) = h 0 | W + ∗ t ( f j ) W + t ( f k ) | 0 i satisfies ˙ r j k ( t ) = − ( f ∗ j ( t ) f k ( t ) − 1 2 | f j ( t ) | 2 − 1 2 | f k ( t ) | 2 ) r j k ( t ) , r j k (0) = 1 . (101) Note that j = k is the standard coheren t exp ectation, in whic h case r j j ( t ) = 1. The follow ing lemma sho ws how exp ectations of sto chastic in tegrals with resp ect to the sup erp osition state can b e ev aluated. Single Photon Quantum Filter 21 Lemma 4.1 L et K ( t ) b e a b ounde d quantum sto chastic pr o c es s define d by (18), wher e M 0 , M ± and M 1 ar e b ounde d and adapte d. Then we have E j k [ K ( t )] = E j k [ Z t 0 M 0 ( s ) ds ] + Z t 0 M − ( s ) f k ( s ) ds + Z t 0 M + ( s ) f ∗ j ( s ) ds + Z t 0 M 1 ( s ) f ∗ j ( s ) f k ( s ) ds ] . (102) Proo f. Equation (102) follow s from the follo wing eigenstate prop ert y of coheren t v ectors: dB ( t ) | f i = f ( t ) | f i dt, d Λ( t ) | f i = d B ∗ ( t ) f ( t ) | f i . (103)  4.2 Em b edding F or the sup erp o sition of n coherent states, we use an n -lev el ancilla system, leading to the extended space ˜ H = C n ⊗ H = H ⊕ H ⊕ · · · ⊕ H ( n times) . (104) As in the single photon case, w e allo w the extended system to ev olv e unitarily according to I ⊗ U ( t ), where U ( t ) is the unitary op erato r for the syste m and field, giv en b y the Sc hr¨ odinger equation (2). Let | e j i , j = 1 , . . . , n , b e an or thonormal basis for C n . W e initialize the extended system in the state | Σ i = 1 | α | X j α j | e j i ⊗ | η i ⊗ | f j i , (105) where α j 6 = 0 for all j and | α | 2 = P j α ∗ j α j (so that h Σ | Σ i = 1). This state ev olv es according to | Σ( t ) i = ( I ⊗ U ( t )) | Σ i . Let A b e an op erator acting on C n , i.e. a complex n × n matrix, A = ( a j k ), j, k = 1 , . . . , n . Then exp ectation in the extended system is defined by E Σ [ A ⊗ X ⊗ F ] = h Σ | ( A ⊗ X ⊗ F ) | Σ i = 1 | α | 2 X j k a j k α ∗ j α k E j k [ X ⊗ F ] . (106) Exp ectatio ns of quan tum sto c hastic in tegrals can b e compactly expressed in the ex- tended system, as the fo llo wing lemma show s. Single Photon Quantum Filter 22 Lemma 4.2 L et M ( t ) b e adapte d. Then E Σ [ Z t 0 A ⊗ M ( s ) dB ( s )] = E Σ [ Z t 0 ( AC ( s )) ⊗ M ( s ) ds ] , (107) E Σ [ Z t 0 A ⊗ M ( s ) dB ∗ ( s )] = E Σ [ Z t 0 ( C † ( s ) A ) ⊗ M ( s ) ds ] , (108) E Σ [ Z t 0 A ⊗ M ( s ) d Λ( s ) ] = E Σ [ Z t 0 ( C † ( s ) AC ( s )) ⊗ M ( s ) d s ] , (109) wher e C ( t ) = diag[ f 1 ( t ) , . . . , f n ( t )] . (110) Notice tha t the exp ectations of the sto c hastic integrals are expres sed in terms of the action of the matrix C ( t ) on the ancilla factor A . 4.3 Master Equation In this section we sho w ho w the the unconditional exp ectation µ t ( X ) = E η Ψ [ X ( t )] (1 11) ma y b e computed from a collection o f differen tial equations. W e do this thro ug h a differ- en t ia l equation for the unconditional exp ectation ˜ µ t ( A ⊗ X ) = E Σ [ A ⊗ X ( t )] (112) for the extended system. Let R b e an n × n ma t r ix defined b y R j k = 1 for all j, k = 1 , . . . , n , and define G t ( A ⊗ X ) = A ⊗ L ( X ) + ( AC ( t )) ⊗ [ L ∗ , X ] S + ( C † ( t ) A ) ⊗ S ∗ [ X , L ] +( C † ( t ) AC ( t )) ⊗ ( S ∗ X S − X )) . (113) Lemma 4.3 The unc onditional exp e ctation (111) with r esp e ct to the sup erp osition state | Ψ i (define d by (9 4)) is giv e n by µ t ( X ) = ˜ µ t ( R ⊗ X ) ˜ µ t ( R ⊗ I ) , (114) and the master e quation for the exp e c tation (112) in the e x tende d system is d dt ˜ µ t ( A ⊗ X ) = ˜ µ t ( G t ( A ⊗ X )) , (115) with initial c ondition ˜ µ 0 ( A ⊗ X ) = 1 | α | 2 h η | X | η i P j k a j k α ∗ j α k . Single Photon Quantum Filter 23 Proo f. By definitions (106 ) and (97) w e hav e E Σ [ R ⊗ X ( t )] = 1 | α | 2 X j k α ∗ j α k E j k [ X ( t )] (116) = 1 | α | 2 E η Ψ [ X ( t )] , (117) and in particular E Σ [ R ⊗ I ] = 1 | α | 2 . (118) F rom these expressions, w e see that E η Ψ [ X ( t )] = | α | 2 E Σ [ R ⊗ X ( t )] = E Σ [ R ⊗ X ( t )] E Σ [ R ⊗ I ] , (119) whic h prov es (114 ) . The differen tial equation (115) follows from the QSDE (3) for X ( t ) = j t ( X ) a nd rela- tions (107)-(109) up on ev aluating the differential d E Σ [ A ⊗ X ( t )].  Theorem 4.4 The unc onditional exp e ctation µ t ( X ) when the field is in the sup erp osition state | Ψ i (define d by ( 9 4)) is given by µ t ( X ) = P j k α ∗ j α k µ j k t ( X ) P j k α ∗ j α k µ j k t ( I ) , (120) wher e µ j k t ( X ) i s given by the system o f e quations d dt µ j k t ( X ) = µ j k t ( G j k t ( X )) , (121) and wher e G j k t ( X ) = L ( X ) + S ∗ [ X , L ] f ∗ j ( t ) + [ L ∗ , X ] S f k ( t ) + ( S ∗ X S − X ) f ∗ j ( t ) f k ( t ) . (122) The initial c ond i tion s ar e µ j k 0 ( X ) = h η | X | η i g j k . (123) Proo f. Define µ j k t ( X ) = E j k [ X ( t )] . (124) Then as in the pro of o f Lemma 4.3 w e may show tha t µ j k t ( X ) = | α | 2 α ∗ j α k ˜ µ t ( | e j ih e k | ⊗ X ) . (125) The the relation (120) follows from ( 114). The differen tial equation (121) follo ws from equation (115) with A = | e j ih e k | .  Single Photon Quantum Filter 24 4.4 Sup erp osition State Filter In this section we sho w ho w t he conditional exp ectation ˆ X ( t ) = π t ( X ) defined b y (7 ) can b e ev aluat ed using a sys tem o f conditional equations. This will mak e use of the conditional exp ectatio n ˜ π t ( A ⊗ X ) = E Σ [ A ⊗ X ( t ) | I ⊗ Y t ] . (126) for the extended system. Lemma 4.5 The c onditional exp e ctation ˆ X ( t ) = π t ( X ) defi ne d by (7) with r esp e ct to the sup erp osition state | Ψ i is given by π t ( X ) = ˜ π t ( R ⊗ X ) ˜ π t ( R ⊗ I ) . (127) The quantum filter for the c onditional exp e ctation ˜ π t ( A ⊗ X ) is d ˜ π t ( A ⊗ X ) = ˜ π t ( G t ( A ⊗ X )) d t + H t ( A ⊗ X ) dW ( t ) (128) with initial c ondition ˜ π 0 ( A ⊗ X ) = 1 | α | 2 h η , X η i P j k a j k α ∗ j α k , wher e H t ( A ⊗ X ) = ˜ π t ( A ⊗ X ( I ⊗ L + C ( t ) ⊗ S ) + ( I ⊗ L ∗ + C † ( t ) ⊗ S ∗ ) A ⊗ X ) − ˜ π t ( A ⊗ X ) ˜ π t ( I ⊗ L + C ( t ) ⊗ S + I ⊗ L ∗ + C † ( t ) ⊗ S ∗ ) (129) and W ( t ) is a Y t -Wiener pr o c ess give n b y dW ( t ) = d Y ( t ) − ˜ π t ( I ⊗ L + C ( t ) ⊗ S + I ⊗ L ∗ + C † ( t ) ⊗ S ∗ ) dt, W (0 ) = 0 . (130) The initial c ond i tion is ˜ π 0 ( A ⊗ X ) = 1 | α | 2 h η | X | η i P j k a j k α ∗ j α k . Proo f. Let K ∈ Y t . Then w e ha v e E Σ [ ˜ π t ( R ⊗ X )( I ⊗ K )] = E Σ [( R ⊗ X ( t ))( I ⊗ K )] = 1 | α | 2 X j k α ∗ j α k E j k [ X ( t ) K ] = 1 | α | 2 E η Ψ [ X ( t ) K ] = 1 | α | 2 E η Ψ [ π t ( X ) K ] = E Σ [ R ⊗ π t ( X ) K ] = E Σ [ E Σ [ R ⊗ π t ( X ) K | I ⊗ Y t ]] = E Σ [ ˜ π t ( R ⊗ I )( I ⊗ π t ( X ))( I ⊗ K )] . (131) This prov es (127). Single Photon Quantum Filter 25 The filtering equation (128) is deriv ed using the characteristic function metho d [29], [7], [5]. W e p o stulate that the filter has the form d ˜ π t ( A ⊗ X ) = F t ( A ⊗ X ) d t + H t ( A ⊗ X ) I ⊗ d Y ( t ) , (132) where F t and H t are t o b e determined. Let f b e square in tegrable, and define a pro cess c f ( t ) b y dc f ( t ) = f ( t ) c f ( t ) d Y ( t ) , c f (0) = 1 . (133) Then I ⊗ c f ( t ) is adapted to I ⊗ Y t , and the definition of quantum conditional exp ectation [9, sec. 3.3] implies that E Σ [ A ⊗ ( X ( t ) c f ( t ))] = E Σ [ ˜ π t ( A ⊗ X ( t ))( I ⊗ c f ( t ))] (134) holds for all f . By calculating the differen tials of b oth sides, taking exp ectatio ns and conditioning w e obtain E Σ [ A ⊗ ( d X ( t ) c f ( t ))] = E Σ [( I ⊗ c f ( t )) ˜ π t ( G ( A ⊗ X )) (135) +( I ⊗ f ( t ) c f ( t )) ˜ π t (( A ⊗ X )( I ⊗ L + C ( t ) ⊗ S ) +( I ⊗ L + C † ( t ) ⊗ S ∗ )( A ⊗ X )) ] d t and E Σ [ A ⊗ ( d ˜ π t ( A ⊗ X ) c f ( t ))] (136) = E Σ [( I ⊗ c f ( t ) {F t ( A ⊗ X ) + H t ( A ⊗ X ) ˜ π t ( I ⊗ L + C ( t ) ⊗ S + I ⊗ L ∗ + C † ( t ) ⊗ S ∗ ) } +( I ⊗ f ( t ) c f ( t )) { ˜ π t ( A ⊗ X ) ˜ π t ( I ⊗ L + C ( t ) ⊗ S + I ⊗ L ∗ + C † ( t ) ⊗ S ∗ ) + H t ( A ⊗ X ) } ] d t. No w equating co efficien ts of c f ( t ) and f ( t ) c f ( t ) we solv e for F t ( A ⊗ X ) and H t ( A ⊗ X ) to obtain the filtering equation (128). W e no w sho w that W ( t ) is a Y t -martingale, and since d W ( t ) dW ( t ) = dt , then b y Levy’s theorem [12] we hav e that W ( t ) is a Y t -Wiener pro cess. Indeed, for any K ∈ Y t w e ha v e E Σ [( I ⊗ dW ( t )) ( I ⊗ K )] = E Σ [( I ⊗ d Y ( t ) − ˜ π t ( I ⊗ L + C ( t ) ⊗ S + I ⊗ L ∗ + C † ( t ) ⊗ S ∗ ) dt )( I ⊗ K )] = E Σ [ I ⊗ ( L ( t ) + L ∗ ( t )) + C ( t ) ⊗ S + C ∗ ( t ) ⊗ S ∗ − ˜ π t ( I ⊗ L + C ( t ) ⊗ S + I ⊗ L ∗ + C † ( t ) ⊗ S ∗ ) dt )( I ⊗ K )] dt = 0 . (137) This completes the pro of.  Theorem 4.6 The unc onditional exp e ctation µ t ( X ) when the field is in the sup erp osition state | Ψ i (define d by ( 9 4)) is given by π t ( X ) = P j k α ∗ j α k π j k t ( X ) P j k α ∗ j α k π j k t ( I ) , ( 1 38) Single Photon Quantum Filter 26 wher e the c ond i tion al q uan tities π j k t ( X ) a r e given by dπ j k t ( X ) = π j k t ( G j k ( X )) dt + ( π j k t ( X ( L + S f k ( t )) + ( L ∗ + S ∗ f ∗ j ( t )) X ) − π j k t ( X ) X j | α j | 2 | α | 2 π j j t ( L + S f j ( t ) + L ∗ + S ∗ f ∗ j ( t ))) dW ( t ) The in n ovations pr o c ess W ( t ) is a Y t Wiener pr o c ess w i th r esp e ct to the sup erp osition state | Ψ i and is given by dW ( t ) = d Y ( t ) − X j | α j | 2 | α | 2 π j j t ( L + S f j ( t ) + L ∗ + S ∗ f ∗ j ( t )) dt. ( 1 39) The initial c ond i tion s ar e π j k 0 ( X ) = h η | X | η i g j k . (140) Proo f. These assertions fo llo w up on substitution of π j k t ( X ) = | α | 2 α ∗ j α k ˜ π t ( e j e ∗ k ⊗ X ) . (141) in to the relev ant expressions from Lemma 4.5.  5 Discuss ion and C onclus ion In this pap er w e hav e deriv ed t he master equation and quan tum filter for a class of op en quan tum systems that are coupled to con tinuous-mo de fields in non-classical states: (i) single photon states, and (ii) sup erp ositions of coherent states. The quantum filter in b o th of the cases w e consider consists of coupled equations that determine the ev olutio n of the conditional state o f the system under contin uous (w eak) measuremen t p erformed on the output field, in contrast to the familiar single filtering equation for op en Mark o v quan tum systems that are coupled to coherent b oson fields. This coupled equations structure of the master a nd filter equations is a reflection o f the non-Marko v nature o f systems coupled to the non-classical fields. Ind eed, a k ey feature of our approach is the em b edding of the system in to a larger extended system, a tec hnique often employ ed in the a na lysis of non-Mark o v systems, providing an elegan t framew ork within whic h to study the the dynamics, bot h unconditional and conditional, of the system. In con trast to Mark ov ian em b eddings [10], [1 9], [20], the extende d system (including the field) is initialized in a sup erp osition state. This embedding pro vides a fra mew ork within whic h the to ols of the quan tum sto chastic calculus ma y b e efficien tly applied to determine quantum filtering equations. W e exp ect that the use of suitable em b eddings, b oth Mark ovian and non- Mark ovian, could b e adapted to study quan tum systems t ha t are coupled to other types of highly non-classical fields. Single Photon Quantum Filter 27 Ac kn o w ledgement The author s wish to thank J. Hop e for helpful discussions and for p ointing o ut reference [10] to us. W e also wish to thank A. Doherty , H. Wiseman, E. Huntington and J. Com b es for help discussions a nd suggestions. References [1] G. Aulett a, M . F or tuna to, and G. P arisi , Q uantum Me cha nics , Cambridge Univ ersity Press, Cam bridge, UK, 2009. [2] H. Bachor and T. Ralph , A Guide to Exp eriments in Quantum O ptics , Wiley- V CH, W einheim, German y , second ed., 2004. [3] A. Barchielli , Continual me asur ements in quan tum me chanics , Summer School on Quan tum Op en Systems 2003. [4] A. Barchielli and V. Bela vkin , Me asur em ents c ontinuous in time and a p o ste- riori states in quantum me chanics , J. Ph ys. A: Math. Gen., 2 4 (1991), pp. 149 5–1514. [5] V. Bela vkin , Quantum c ontinual me asur em e nts and a p osteriori c ol lapse on CCR , Comm un. Math. Phys ., 146 (1992) , pp. 611 – 635. [6] , Quantum sto chastic c alculus and quantum nonlin e ar filtering , J. Multiv a r iate Analysis, 42 (1992) , pp. 171–201. [7] , Quantum diffusion , me asur ement, and filtering , Theory Probab. Appl., 38 (1994), pp. 573–58 5. [8] L. Bouten and R . v an Handel , On the sep ar a tion principle of quantum c o n- tr ol , in Pro ceedings of the 2006 QPIC Symp osium, M. G uta, ed., W or ld Scien tific, math-ph/05110 2 1 2006. [9] L. Bouten, R. v an Handel, and M. James , A n intr o duction to quantum filtering , SIAM J. Con trol a nd Optimization, 46 (20 07), pp. 2199–224 1. [10] H. Breuer , Genuine quantum tr aje ctories for non-Markov ian pr o c ess e s , Ph ys. Rev. A, 7 0 ( 2 004), p. 012 106. [11] H. Carmichael , An Op en Systems Appr o ach to Quantum Optics , Springer, Berlin, 1993. [12] R. Elliott , Sto cha stic C alculus and Applic a tion s , Springer V erlag, New Y or k, 198 2 . [13] F. F agno la , On quantum sto chastic differ ential e quations with unb ounde d c o effi- cients , Probab. Th. Rel. F ields, 8 6 ( 1 990), pp. 501 –516. Single Photon Quantum Filter 28 [14] F. F a gnola and S. J. Wills , Solving quantum sto chastic differ en tial e quations with unb ounde d c o efficients , Journa l of F unctional Analysis, 1 98 (2003), pp. 2 79–310. [15] C. Gardiner and M. Collett , In put and output in damp e d quantum systems: Quantum sto ch a stic diffe r ential e quations and the master e quation , Ph ys. Rev. A, 31 (1985), pp. 3761–3 774. [16] C. G ardiner and P. Zoller , Quantum Noise , Springer, Berlin, 200 0. [17] K. Gher i, K. Ellinger, T. Pellizzari, and P. Zoller , Pho ton-wavep ack e ts as flying quantum bits , F ortsc hr. Ph ys., 46 (1998 ), pp. 40 1–415. [18] N. Gisin, G. Ribord y, W. Tittel, and H. Zbinden , Quantum crypto g r aphy , Reviews of Mo dern Ph ysics, 74 (200 2 ), p. 14 5. [19] J. Gough, M. James, and H. Nurdin , Quantum master e q uation and filter for systems driven by fi elds in a single pho ton state . submitted to IEEE Conference on Decision and Con trol, 2011. [20] J. Gough, M . James , H. Nurdin, and J. Combes , In put-output the ory and quantum tr aje ctories for non -gaussian inp ut fi elds . 2011. [21] A. Holevo , Quantum sto chas tic c alculus , J. So viet Math., 56 (1 9 91), pp. 26 09–2624 . [22] R. Hudson and K. P ar thas ara thy , Quantum Ito’s formula and sto chastic evo - lutions , Commun. Math. Ph ys., 93 (1984), pp. 301– 3 23. [23] E. Knill, R. Laflamme, and G. Milburn , A sch e me for efficient quantum c om- putation with line ar optics , Nature, 409 (2 001), pp. 46– 5 2. [24] R. Loudon , The Quantum The ory of Light , Oxford Univ ersity Press, Oxford, 3 rd ed., 2000. [25] X. Maitre , E. Hagley, G . Nogues, C. Wunderlich, P. Goy, M. Br une , J. Raimond, and S. Haroche , Quantum memory with a s i n gle photon in a c avity , Ph ys. Rev. Lett., 79 (1 997), pp. 769 – 772. [26] E. Merzbacher , Quantum Me chanic s , Wiley , New Y ork, third ed., 199 8 . [27] G. J. Milburn , Coher ent c ontr ol of sin gle p h oton states , Eur. Ph ys. J. Sp ecial T opics, 159 (2 008), pp. 113– 117. [28] K. P ar thasara thy , An Intr o duction to Quantum S to chastic Calculus , Birkhauser, Berlin, 1992. [29] R. v an Handel, J. Stockton, and H. Mabuchi , F e e db a ck c ontr ol of quantum state r e duction , IEEE T rans. Automatic Con trol, 50 (20 05), pp. 768– 7 80. Single Photon Quantum Filter 29 [30] J. V olz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, K. Sa ucke, C. Kur tsiefer , and H. Weinfur ter , Observation of en tanglement of a single photon with a tr app e d atom , Ph ys. Rev. Lett., 96 (2006 ), p. 03 0404. [31] H. Wiseman and G. Milburn , Quantum the ory of field-quadr atur e m e asur ements , Ph ys. Rev. A, 47 (19 93), pp. 642– 6 63. [32] H. Wiseman and G. Milburn , Q uantum Me asur ement and Contr ol , Cam bridge Univ ersity Press, Cam bridge, UK, 2010.