Optimal nonlinear filtering of quantum state V.I. Man’ko ∗ P.N. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospect 53, Moscow 119991, Russia Moscow Institute of Physics and Technology Institutskii Per. 9, Dolgoprudny Moscow Region 141700, Russia L.A. Markovich † Moscow Institute of Physics and Technology 7 Institutskii Per. 9, Dolgoprudny Moscow Region 141700, Russia 1 V.A. Trapeznikov Institute of Control Sciences, Moscow, 0 Profsoyuznaya 65, 117997 Moscow, Russia 2 Institute for information transmission problems, Moscow, Bolshoy Karetny per. 19, n build.1, Moscow 127051, Russia a J (Dated: January 24, 2017) 2 2 We extend the optimal filtering equation known from the Stratonovich filtering theory on the quantum process case. The used observation model is based on an indirect measurement method, ] h wherethemeasurement is performed on an ancilla system that isinteracted with an unknownone. p Observation model for single quditsystem is proposed. - PACS numbers: 03.65.Ud, 03.67.Mn t n a u I. INTRODUCTION the same system being in the same state. It is obvi- q ous that in practice many identical copies of the system [ are impossible to implement and hence, this method can Therehasbeenrecentinterestinquantumfilteringand 1 state estimation problems in quantum information the- not be applied to the dynamically evolving states, as in v the quantum control [26]. The measurement scheme de- ory and quantum control [1–3]. Last experimental and 0 scribedaboveis basedontwo-qubitsystems. This isdue theoretical advances in quantum technology provide a 3 to fast developmentof quantum technologiessuch as the strong motivation for the quantum control including the 1 quantumcomputers,quantumcryptographyandtelepor- 6 engineering of quantum states, the stability theory, the tation. These technologies promise in the future to lead 0 quantum error correction, the robust control and quan- toarevolutioninthetechnologyandthecommunication. . tumnetworks[4,13–16]. Also,thequantumcontrolplays 1 a fundamental role in the development of new quantum 0 Thequantumcomputing isbuiltfromelementarypro- 7 technologies like the quantum computation. For exam- cessing elements, namely, quantum bits - qubits. It is 1 ple,thequantumfilteringincoherentstatesisintroduced knownthatclassicalcomputerselements (bits)takeonly : in [17] and the tomographical approach is used for the v two values, logic zero and logic one. On the contrary, i signal recognition and denoising in used in a number of qubitsasquantumobjectscanbelocatedalsoinacoher- X recent papers [18–20]. entsuperpositionofthesetwostates. Thustheydescribe r It is known, that for the quantum state estimation a the intermediate state between the logic zero and one. one has to give the measurement strategy that is used By measuring the qubit, we get zero or one with some to get information, and the estimator mapping the mea- probabilities. The quantum computers will be able to a surement data to the state space. In this paper, we con- finitetimetosolveproblems,tothesolutionofwhichthe sider the weak measurement model where an unobserv- classicalsupercomputers will take anunacceptable time. able quantum system is coupled with a ancilla system ”Breaking” of a cryptographic RSA algorithm based on that can be measured (a probe system). A von Neu- finding the decomposition of large numbers into primes mann measurementis used, describedby a setof projec- is the famous example. The classicalcomputer solving a tion operators P = n n . Every operator describes n similar problem using brute force, would have spent an { | ih |} what happens on one of possible outcomes of the mea- enormoustimecomparabletothelifetimeoftheuniverse, surement. The method is based on the set of copies of whilequantumcomputingsystemcansolveitwithinsev- eral minutes. An implementation of such computer has several obstacles. Quantum states of ions, electrons and Josephson junctions used as qubits are extremely unsta- ∗ [email protected] ble and can not be stored in one state for a long time. † [email protected] Forthe implementationofcomputationalalgorithmsone 2 needsasetofintereractedqubitsinaparticularstate. In The paper is organized as follows. In Sec. III we give recent years, the stability of the states of qubits was in- thebriefoverviewonthenotionoftheweakmeasurement creased from nanoseconds to milliseconds, but this task for the system of two qubits. In Sec. IV the method of is still extremely complicated, especially for the multi- the invertible mapping is used to obtain the non-linear partite systems with a large number of qubits. measurement model for the single qudit system. The Onewaytoresolvethisproblemistouseasinglequdit physical meaning of the correlations in such system is system as the quantum objects instead of qubits. Such given. In the next section the optimal filtering equa- single qudit systems have more states than two-qubit tionis described. The latter approachis extended to the ones. Thus, the dimension of the system is greatly re- case of the nonlinear quantum models. It is shown that duced. A wide number of papers exists devoted to the by means of the optimal filtering equation one can find study of various kinds of characteristicsof quantum cor- the optimal solution of the nonlinear quantum filtering relations in systems with subsystems such as the two problem without using any linearization procedures like qubit system. This quantum system can have a corre- in [42]. In Sec. VI the observation model is rewritten lationresponsibleforthe entanglementphenomenon[27] in term of the tomogram. The latter approach is used and for the violation of Bell’s inequality [28]. These cor- to write the Shannon entropies and information depend- relations may also be responsible for the quantum dis- ing on a time step of the observation model. Thus, it cord[29, 30]. However,inliterature the systems without is possible to observe the time evolution of the informa- subsystems (one qutrit, qudit) are payed less attention. tionofthequantumnonlinearprocess. TheWernerstate Recently,in[31–34]itwasshownthatthequantumprop- example of the information time evolution is provided. erties of systems without subsystems can be formulated using the method of an invertible mapping. In [34] the entanglementconceptandcorrelationsinthesinglequdit III. THE WEAK MEASUREMENT statearediscussed. Usingthelattermapping,thenotion oftheseparabilityandtheentanglementwasextendedin In spirit of [42] let us consider the discrete time case [35] to the case of the single qudit X-state with j =3/2. of the indirect measurement. We suppose that the un- observable and the observable measurements of quan- tum systems are quantum bits. Under the weak mea- II. CONTRIBUTIONS OF THIS PAPER surement we mean that the projective measurements are done on the extra ancilla system that is in state The problems related to the quantum state estima- θM = (θM1,θM2,θM3) coupled with the system θS = tion and the quantum state filtering are fundamental for (θS1,θS2,θS3) that we are interested in. Two Bloch vec- the quantum information theory and quantum control. tor representations of the latter states are In [42] the well-known procedure in the field of classi- ρ (k)=(I+θ (k)σM)/2, (1) calcontroltheory,namely the Kalmanfiltration[39]was M M applied to the quantum filtering area. The problem of ρS(k)=(I +θS(k)σS)/2, the filtering of unknown signals from the mixture with where the σS and σM are symbolic vectors constructed noise is well studied in classical probability theory. The from the Pauli operators acting on Hilbert spaces H Kalman filter is known provides the optimal solution for S and H , respectively. The indirect measurement is pro- the linear recursive model of the observation. However, M ceeded by the following way. At the time step k we fornon-linearmodelsthatappearinpracticetheKalman prepare the ancilla qubit in a known state. We cou- filter is not applicable. ple it to an unknown system. The composite system is The aim of this work is to propose the filtering method represented by the 4-dimensional square density matrix optimal for non-linear quantum processes. To this end, ρ (k). Let us take it as a direct product of two later the general filtering equation introduced in [43, 44] was S+M states,i.e. ρ (k)=ρ (k) ρ (k). Bothqubitsevolve extendedto the quantumobservationmodel forthe two- S+M S M ⊗ according to bipartite dynamics at sampling time h. At qubitsystem. Inauthor’spaper[45]itisprovedthatthe the end we do the von Neumann measurement on the optimalfilteringequationisnothingelsebuttheKalman ancillaqubit. Generally,the vonNeumannmeasurement filter in the case of the linear model. Note, that the is the measurement of Pauli operators. For example, if optimal filtering equation does not contain the explicit we are interested in the measurement of the observable probabilisticcharacteristicsoftheunknownunobservable σ ,then possibleoutcomesareits eigenvalues( 1). The sequence. This allows us to find the optimal state esti- x ± algorithm is repeated at the next time step k+1 . mate knowing only observable quantities. A further aim is to propose a quantum measurement The probabilities of two different outcomes model for the single qudit state. The optimal filtering 1 1 method proposed for the two-qubit models can be ex- A =I σ = 1 1 + 2 2, x x × 2 | ih | −2 | ih | tended to the latter case. The construction of such type (cid:18) (cid:19) (cid:18) (cid:19) oftheobservationmodelsisusefulinthelightofpossible 1 1 A 1 1 = 1 1, A 2 2 = 2 2 practical use of the single qudit systems. x| ih | 2 | ih | x| ih | −2 | ih | (cid:18) (cid:19) (cid:18) (cid:19) 3 of the von Neumann measurement are the following Let us select the unitary evolution matrix of the fol- lowing view 1 1 1 1 P(+1)= 1 1 =I , TrP =2, 2 | ih | × 2 1 1 cosϕ 0 0 sinϕ (cid:18) (cid:19) (cid:18) (cid:19) 0 1 0 0 1 1 1 1 W = (5) P( 1)= 2 2 =I − , P2 =P.  0 0 1 0  − −2 | ih | × 2 1 1 (cid:18) (cid:19) (cid:18)− (cid:19) sinϕ 0 0 cosϕ −    The evolution of the system is controlled by the unitary The numerator and the denominator elements of (3) are operator. Let the matrix of this operator has the basic given in appendix. For simplicity, the angle ϕ can be view selected equal to zero and then the elements of (3) are u u u u 11 12 13 14 ρ (1,1)=(θ +1)(θ +1)/4, u u u u S m1 s3 W = 21 22 23 24 . (2) ρ (1,2)=(θ θ i)(θ +1)/4,  u31 u32 u33 u34  S s1 − s2 m1  u41 u42 u43 u44  ρfS(2,1)=(θs1 +θs2i)(θm1 +1)/4,   ρf(2,2)= (θ +1)(θ 1)/4 The state of the composite system after the interaction S − m1 s3 − f is the following and the denominator is (θ +1)/2. f m1 For the special case of the evolution matrix that de- ρS+M(k+1)=WρS+M(k)W† pends only on one angle we get and the reduced density matrix of the system we are in- cosϕ 0 0 sinϕ terested in is 0 0 0 0 W = . (6) ρ (k+1)=Tr ρ (k+1)  0 0 0 0  S M S+M sinϕ 0 0 cosϕ =TrMWρS(k)⊗ρM(k)W†.  −  Elementsof (3)aregiveninappendix. Wecanselect,for Since, we areinterestedin measuringσ , the states after x example, the angle ϕ=0 and write the measurement are ρ( 1)= ρS+MP± ρS(1,1)=(θm3 +1)(θs3 +1)/4, (7) ± Tr(ρ P ) ρ (1,2)=(θ θ i)(θ θ i)/4, S+M ± S m1 − m2 s1 − s2 ρf(2,1)=(θ +θ i)(θ +θ i)/4, and the eigenstates of the measurement are S m1 m2 s1 s2 ρf(2,2)=(θ 1)(θ 1)/4. Tr (ρ P ) S m3 − s3 − θS(±1)= Tr(TMrM(Sρ+SM+M±P±)) (3) The denoffminator is then (θm3θs3 +1)/4. ρ S = ± . Tr(Tr (ρ P )) M S+M IV. SINGLE QUDIT OBSERVATION MODEL ± e A. Examples of the evolution matrices Recently, it was observedin [31–33] that the quantum propertiesofthesystemswithoutsubsystemscanbefor- mulated using the invertible map of integers 1,2,3... As an example, the evolution matrix can be taken as onto the pairs (triples, etc) of integers (i,k), i,k = W = e−ih(ayσyS⊗σyM) (see [42]), where ay is the coupling 1,2,... (or semiintegers). For example, the single qu- parameter and h is the sampling time. For the measure- dit state j =0,1/2,1,3/2,2,...can be mapped onto the ment A = I σ and a h = π/2, the probabilities of x ⊗ x y densityoperatorofthesystemcontainingthesubsystems two different outcomes are like the state of two qudits. P(+1)=(1+θ θ ), P( 1)=(1 θ θ ). LetthequantumstateinfourdimensionalHilberspace S2 M3 − − S2 M3 be described by the density matrix H The post measurement states are the following ρ ρ ρ ρ 11 12 13 14 θS(±1)= ±θθSS3311θθθ±±MSM2θθ2SS1±±22−θθθθMθSMMS13331θθMM11 . (4) ρ=ρρρ234111 ρρρ234222 ρρρ234333 ρρρ234444 . (8)  1±θS1θM2  thatρ=ρ†,Trρ=1anditseigenvaluesarenonnegative. Since the probability of the new state depends on both The latter matrix can describe the two-qubit system. measurementsθ andθ wecanretrievetheinformation To this end, let us use the following invertible mapping S M abouttheusefulstateusingonlytheobservablemeasure- 1 1/2 1/2; 2 1/2 1/2; 3 1/2 1/2; 4 ↔ ↔ − ↔ − ↔ ment. 1/2 1/2 and rewrite the density matrix ρ as − − 4 ρ ρ ρ ρ 1/2 1/2,1/2 1/2 1/2 1/2,1/2 1/2 1/2 1/2, 1/2 1/2 1/2 1/2, 1/2 1/2 ρ ρ − ρ − ρ − − ρ1/2 = ρ1/2 −1/2,1/2 1/2 ρ1/2 −1/2,1/2 −1/2 ρ1/2 −1/2,−1/2 1/2 ρ1/2 −1/2,−1/2 −1/2 . (9) 1/2 1/2,1/2 1/2 1/2 1/2,1/2 1/2 1/2 1/2, 1/2 1/2 1/2 1/2, 1/2 1/2 ρ − ρ − − ρ − − ρ − − −  1/2 1/2,1/2 1/2 1/2 1/2,1/2 1/2 1/2 1/2, 1/2 1/2 1/2 1/2, 1/2 1/2   − − − − − − − − − − − −  Let = 1 2, where 1 and 2 are the two di- of the twoqubit system (the subsystems) can be defined H H ⊗ H H H mensional Hilbert spaces. The reduced density matrices using the partial trace of (9) as ρ +ρ ρ +ρ ρ1 = ρ 1/2 1/2,1/2 1/2+ρ1/2 −1/2,1/2 −1/2 ρ 1/2 1/2,−1/2 1/2+ρ1/2 −1/2,−1/2 −1/2 , (10) 1/2 1/2,1/2 1/2 1/2 1/2,1/2 1/2 1/2 1/2, 1/2 1/2 1/2 1/2, 1/2 1/2 (cid:18) − − − − − − − − − − (cid:19) ρ +ρ ρ +ρ ρ2 = ρ 1/2 1/2,1/2 1/2+ρ−1/2 1/2,−1/2 1/2 ρ 1/2 1/2,1/2 −1/2+ρ−1/2 1/2,−1/2 −1/2 . (11) 1/2 1/2,1/2 1/2 1/2 1/2, 1/2 1/2 1/2 1/2,1/2 1/2 1/2 1/2, 1/2 1/2 (cid:18) − − − − − − − − − − (cid:19) The density matrix ρ can also describe the single Hence, the correlation between the two observations is 1/2 quditsystemifwerewriteitusingthefollowinginvertible given by mapping 1 3/2, 2 1/2, 3 1/2, 4 3/2 as ↔ ↔ ↔− ↔− m ,m = m m ω(m ,m ). 1 2 1 2 1 2 h i ρ3/2,3/2 ρ3/2,1/2 ρ3/2, 1/2 ρ3/2, 3/2 mX1,m2 ρ ρ ρ − ρ − ρ3/2 = ρ 1/2,3/2 ρ 1/2,1/2 ρ 1/2,−1/2 ρ 1/2,−3/2 . However,wecanbeinterestednotinthewholesystem, 1/2,3/2 1/2,1/2 1/2, 1/2 1/2, 3/2 but only incases whenthe firstcoinfalls on the one side ρ− ρ− ρ− − ρ− −  3/2,3/2 3/2,1/2 3/2, 1/2 3/2, 3/2  andthesecondcoinisnotinterestingforus. Wehavetwo  − − − − − −  new outcomes ω with probabilities p =p +p and 1 11 12 The matrix saves the standard properties of the density { } p = p +p . Analogically, if we are interested only 2 22 21 matrix,i.e. ρ3/2 =ρ†3/2,Trρ3/2 =1holdanditseigenval- in the second coein, we have two neweoutcomes ω with ues are nonnegative. Using the partial trace two ”artifi- probabilitiesp =p +p andp =p +p . Outcomes e 1 11 21 2 22 12 cial subsystems” (”artificial qubits”) can be introduced. ω and ω arecorrelated. Thelatterexampleishelpful It means that all equalities and inequalities known for {to}show t{he}existence of correlations in systems without matrix ρ1/2 (the two-qubit system) are valid for matrix subsystemes. ρ3/2 (the single qudit system). If we have a single qudit system with the spin j = Hence, one can think that density matrix ρ (k) 3/2, we can write the sample space Ω of four out- S+M describes the single qudit state. Applying the latter comes ω Ω, the values of the spin projections m >= ∈ | method,thetwo”artificialsubsystems”ρ (k)andρ (k) 3/2 >, 1/2 >, 1/2 >, 3/2 > with probabili- S M {| | | − | − } canbe constructedsuchthatρ (k)=ρ (k) ρ (k). ties p ,p ,p ,p . Then we have one four-level Then one canconstructthe obsSe+rMvationmoSdela⊗s foMr the atom,3/e2.g.1/t2he−m1/2>=−3/32/2 > corresponds to the case | | two-qubit system. However, it does not contain the real when the highest (fourth) level is filled. If we are inter- observable ancilla qubit and the unobservable one. ested only in outcomes when the fourth or the second levels of the four-level atom are filled, then we can as- sume that we have a new set of two outcomes ω with 1 { } probabilities p =p +p , p =p +p . If we 1 3/2 1/2 2 3/2 1/2 A. Physical meaning of the ”artificial qubits” are interested only in outco−mes when t−he fourth and the third levels are filled, then we have another set of out- Letusstartfromtheshortexampleoftwocoinswhich comes ω and their probabilities are p = p +p , 2 1 3/2 1/2 { } candropontheone(1)oronthesecond(2)side. Hence, p = p +p . The outcomes ω and ω are 2 3/2 1/2 1 2 there are two random variables m1, m2 and four oppor- correlat−ed. Hen−ce, correlations in sin{gele}qudit{sys}tems tunities(m1,m2)={(11),(12),(21),(22)}withprobabil- aere between different combinations of outcomes. ities pij, i,j = 1,2. Let ω(m1,m2) be the joint prob- With respectto our problem,the experimentcould be ability function of these two random variables. Their designed so that we can measure the population only on marginal probability functions can be defined as certain levels, while others are not available to measure. This may be due to their short lifetime or a diversity in ω (m )= ω(m ,m ), ω (m )= ω(m ,m ). thefrequencybandreception. Forexample,wecanmea- 1 1 1 2 2 2 1 2 sure only the first and the second levels of the four-level Xm2 Xm1 5 atom and the third and the fourth levels are unobserv- where(η R) isani.i.drandomsequence,(s ) is k k 1 k k 1 able. Then we can think about the observable levels as a Markov s∈equen≥ce and ϕ is some function. Realizat≥ions about ”artificial ancilla qubit” and about other two lev- of random variables s R and x R are k k k n els as about ”artificial unobservable qubit”. Thus, we denoted by sk = (s ,..∈.,Ss )T⊆and xk =∈(xX,..⊆.,x )T, 1 1 k 1 1 k canconstructthe observationmodeljustlikeforthe real respectively. two-qubit system. In case when (14) has the recursive linear form s =as +bξ , (15) k k 1 k − V. FILTERING OF UNKNOWN SIGNALS x =As +Bη , k k k where s ,x R for all k, ξ and η are mutually in- The problem of filtering of unknown signals from the k k ∈ k k dependentrandomvariableswith the standardGaussian mixture with anoise hasa wide rangeofapplicationsin- distribution, cluding control of linear and nonlinear systems. In the following,weconsiderthattheBlochvectorofunobserv- b2 s (0,σ2), σ2 = able qubit θS2 is sk, where k is the time step. The ob- 0 ∈N 1 a2 servableancilla qubit willbe characterizedby parameter − s (0,1), n=1,2,3..., n c=θ . Hence,wecanrewrite,forexample,thesecond ∈N e e M3 row in (4) as coefficientsA,B,a,baregivenbyrealnumbersand a < | | 1, the Kalman filter is applied as optimal method [39]. s c k 1 sk = 1 −cs± . Remark V.1 In [42] the distribution of x is approx- k 1 k ± − imated by the Gaussian distribution x (Ncs ;N) Let us rewrite the latter process under the assumption k ∼ N k and the Kalman-like filter is applied. However, we would that c is small enough. Then the process will be the like to observe general case of nonlinear models that are following more important for practice. In that case the Kalman s =s c(1 s2 )+O(c) filter does not provide the optimal filtration solution. k k−1± − k−1 or if we are interestedin the system changeonly after N time steps we get A. Nonlinear process s =s +x c(1 s2 ), (12) k k−1 k−1 − k−1 The optimal approach for nonlinear processes is pro- nwohteerethxekp=lusxak+nd−thxek−m,iNnus=ouxtkc+om+exska−s xhko+ld.anWdexkde-, mpoasteodriinn[f4o3r]m. Toof tehsetimcoantdeitsinontahlemoepatnimal Bayesian esti- respectively. − Analogicallytothepreviousexample,forrotationma- s =E(s xn)= s w (s xn)ds , (16) trix (6) and probabilities (7) we can write n n| 1 n n n| 1 n Z Sn s (c+1)+c+1 b sk = k−11+cs has been used. The wn(sn|xn1) is the posterior proba- k 1 bility density function that satisfies the Stratonovich’s − (sk 1(c+1)+c+1)(1 csk 1) recurrence equation [46] given by = − − − (1+cs )2 k 1 f(x s )p(s ) − 1 1 1 =sk−1+(c+1)(1−cs2k−1)+O(c). w1(s1|x1)= f(x1||s1)p(s1)ds1, and the process is SR1 f(x s ) sk =sk−1+(c+1)(1−cs2k−1). wn(sn|xn1)= f(xn|nx|1nn−1) (17) If we are interested in the system change only after N time steps, then the latter process can be rewritten as · p(sn|sn−1)wn−1(sn−1|x1n−1)dsn−1, n≥2. Z s =s +x (c+1)(1 cs2 ). (13) Sn−1 k k−1 k−1 − k−1 Here we denote as p(s s ) the transition probability n n 1 We can get the whole class of processes like (12) and density function of the|M−arkov sequence (S ) and n n 1 (T1h3u)s,dwepeehnadvinegaopnartthiaellychoobisceervoafbtleheMeavroklouvtiroanndmoamtrsixe-. f(xSnin|xce1n−th1)e,pfo(sxtne|rsionr) ddeennsoitteycwon(dsitioxnna)lddeepnesnitdiess≥.on the quence (s ,x ) , where the sequence s = (s ) is n n| 1 k k n 1 k k 1 unknown prior distribution function p(s ) and the tran- unobservable and≥ the sequence x = (x ) is ob≥serv- 1 k k 1 sition probability p(s s ) of the Markov sequence able. The connectionbetweenthese variabl≥esis givenby n| n−1 (s ) , we cannot use formula (18) to estimate ϑ . To the following nonlinear or linear expression n n>1 n overcome this problem the general filtering equation is xk =ϕ(sk,ηk), (14) proposed in [44]. b 6 Theexactcoincidencesofthegeneralfilteringequation which is the Kalman filter. (for the unobservableMarkovsequence(s )definedby a Since the observation model given by equations (13) n linear equation with a Gaussian noise) with Kalman fil- and (14) is nonlinear we propose to use general filtra- ter and the conditional expectation E(Q(s )xn) defined tion equation (19) for the state estimation. It provides n | 1 by Theorem of normal correlation [47] is proved in [45]. the optimal solution knowing only the observed random Thus, the general filtering equation is nothing else but variables. We do not need any simplifications, lineariza- the Kalman filter in case of linear model (15). However, tions or assumptions on the distribution of the observed fornonlinearprocessesthegeneralfilteringequationpro- random variables as it is done for example in [42]. videstheoptimalsolutionincontrasttotheKalmanfilter VI. INEQUALITIES FOR QUANTUM that cannot be applied to nonlinear models. TOMOGRAPHIC MUTUAL INFORMATION The tomographic probability representation of spin B. Equations of optimal filtering (qudit) states is introduced in [48, 49] . In this repre- sentation qudit states with density matrices ρ (k) and S Let us assume that the conditional density f(xn|sn) ρM(k) are identified with spin-tomograms which are the belongs to the exponential family of distributions, i.e. probabilitydistributionfunctionsdeterminedbytheden- sity operators of the states f(x s )=C(s )h(x )exp(T(x )Q(s )), (18) n n n n n n | where C(s ) is a normalization constant and ρS(k)⇔ωS(m′,US,k)=hm′|US ·ρS(k)·US†|m′i, n e h(xn),T(xn),Q(sn) are known functions. The gen- ρM(k)⇔ωM(m,UM,k)=hm|UM ·ρM(k)·UM† |mi, eral filtraetion equation is where m,m = j, j + 1,...,j, j = 0,1/2,1... are ′ − − E(Q(sn)|xn1)·Tx′n(xn)= ln f(xhn(|xx1n)−1) ′ .(19) sUp(iθnMp,rϕoMje,cψtiMon)saarnedthUeSro≡tatiUon(θSm,aϕtSri,cψesS)ofainrdredUuMcib≡le (cid:18) (cid:18) n (cid:19)(cid:19)xn representations of SU(2) - group Note that equation (19) does not contain explicit prob- abilistic characteristics p(s1) and p(sn sn 1) of the un- U(θ ,ϕ ,ψ )= cosθ2Sei(ϕS2+ψS) sinθ2Sei(ϕS2−ψS) . kesntoimwnatsoerq(u1e6n)cekn(oswn)i.ngTohnislyalolbowsesrvuasbtloe|qfiun−adntthiteieospotfimxna.l S S S −sinθ2Sei(ψS2−ϕS) cosθ2Se−i(ϕS2+ψS) ! 1 As an example of the exponential family (18) we can For density matrix ρS+M we can determine the tomo- take the Gaussian density gram f(x s )= 1 exp (xn−Asn)2 . (20) ρS+M(k+1)⇔ωS+M(m′,m,US+M,k+1) n| n √2πB (cid:18)− 2B2 (cid:19) =hm′m|US+M ·ρS+M(k+1)·US†+M|m′mi. Thentheobservationmodelisdefinedbythelinearequa- RotationmatrixU canbedefinedasthedirectprod- S+M tion uct of two matrices U and U S M x =As +Bη , (21) n n n U =U(θ ,ϕ ,ψ ) U(θ ,ϕ ,ψ ). S+M S S S M M M ⊗ where η are independent identically distributed ran- { n} ThematricesUS andUM dependonlyontheEulerangles dom variables with the Gaussian distribution and coeffi- θ ,ϕ ,ψ ,i S,M whichdetermine directionsofthe i i i cients A and B are real numbers. Hence, we can rewrite { } ∈{ } quantization, e.g., points on the Bloch sphere. Hence, equation (19) in a special form we use following notations U = U(~n ), U = U(~n ), S S M M E(s xn)= B2fx′n(xn|x1n−1) + xn (22) awxheesr.eH~neSnacen,dt~nhMe ladtetteerrmtoimneogdriraemcticoannsboefsrpeiwnrpitrtoejnecatsion n| 1 A f(xn|x1n−1) A ω (m,m,~n ,~n ,k+1)=<mmU(~n ) U(~n )ρ (k+1)U (~n ) U (~n )mm> (23) S+M ′ S M ′ S M S+M † S † M ′ | ⊗ ⊗ | =<m′mU(~nS) U(~nM)WρS(k) ρM(k)W†U†(~nS) U†(~nM)m′m>. | ⊗ ⊗ ⊗ | Itistheconditionalprobabilityofprojectionsofspinsm, function (23) has the property of a no-signaling. Hence, ′ m on vectors ~n , ~n on the Bloch sphere. Probability marginal probability distributions of the first system is S M given by 7 ω (m,U ,k+1)=<m U(~n )ρ (k+1)U (~n )m >=<m U(~n )Tr Wρ (k) ρ (k)W U (~n )m >.(24) S ′ S ′ S S † S ′ ′ S M S M † † S ′ | | | ⊗ | Since the ancilla qubit does not change in time we can A. Example write ωM(m,UM)=<mU(~nM)ρMU†(~nM)m>. As an example let us take the Werner state [50]. The | | two-qubitsystemorthesinglequditsystemwiththespin Diagonalelements ofthe latter tomogramsare ω (1,k+ S j =3/2canbedescribedbythefollowingWernerdensity 1),ω (2,k+1),ω (1),ω (2). BydefinitionoftheShan- S M M matrix nonentropy,wecanconstructthetomographicentropies of the subsystems as H (k+1)= ω (1,k+1)lnω (1,k+1) S − S S 1+p 0 0 p 4 2 HM(k+1)−=ω−Sω(M2,(k1,+k1+)l1n)ωlnSω(2M,k(1+,k1+),1) ρS+M(0)= 00 1−04p 1−04p 00 , (26) ω (2,k+1)lnω (2,k+1)  p 0 0 1+p  − M M  2 4    Analogically, we can find the diagonal elements of US+MρS+MUS†+M which are the tomograms ω (1,1,k), ω (2,2,k), ω (3,3,k) and where the parameter p satisfies the inequality 1/3 S+M S+M S+M − ≤ ω (4,4,k). The Shannon entropy of the com- p 1. The parameter domain 1/3 < p 1 corresponds S+M ≤ ≤ bined system is to the entangled state. The reduced density matrices of thefirstandthesecondqubits(orthe”artificialqubits”) H (k)= ω (1,1,k)lnω (1,1,k) S+M S+M S+M are − ω (2,2,k)lnω (2,2,k) S+M S+M − ω (3,3,k)lnω (3,3,k) S+M S+M − ωS+M(4,4,k)lnωS+M(4,4,k). 1 0 1 0 − ρ (0)= 2 , ρ = 2 . S 0 1 M 0 1 The Shannon information depending on the time step is (cid:18) 2 (cid:19) (cid:18) 2 (cid:19) I(k)=H (k)+H (k) H (k) 0. S M S+M − ≥ In case when the state of the ancilla qubit does not Letustakeevolutionmatrix(5)withϕ=π/8andthean- change in time the latter inequality can be rewritten as gles of the rotation matrices θ,ϕ,ψ = 2π,π,π both { } { } for U and U . On the time step k = 0 and k = 1 the S M I(k)=HS(k)+HM HS+M(k) 0. (25) information (25) is − ≥ 1 p (p 1) 1+p (p+1) I(0)=2ln2+ ln − − +ln , − 4 2 4 2 (cid:18) (cid:19) (cid:18) (cid:19) (p+√2p+1) (p+√2p+1) (p √2p+1) (p √2p+1) I(1)=ln2+ln +ln − − 4 4 4 4 ! ! 1 p (p 1) 1 √2p (√2p 2) 1+ √2p (√2p+2) ln − − +ln − 2 − 2 . − (cid:18) 4 (cid:19) 2 2 ! 4 − 2 ! 4 The information (25) against parameter p is shown in VII. CONCLUSION Fig. 1 for time steps k = 0,1,2 . { } To conclude let us point out the main results of our work. Using the invertible method of indices we extende the observation model based on the indirect measure- ment knownfor the two-qubitsystemto the single qudit 8 0.7 k = 0 0.6 k = 1 0.5 k = 2 0.4 0.3 0.2 0.1 0 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 FIG. 1. Information (25) against parameter p dependingon thetime step k={0,1,2} 1 system. Since the models can be nonlinear the known ρ (1,2)= cosϕ (sinϕ(θ 1)(θ 1) Kalman filter approach does not give the optimal solu- S 8 ∗ m3 − s3 − tion. Hence, we propose to use for these nonlinear quan- +cfosϕ(θm1 −(cid:16)θm2i)(θs1 −θs2i))−sinϕ∗ tum models the general filtering equation that gives us (cosϕ(θ +1)(θ +1)+sinϕ(θ +θ i) the optimalsolutionfor the nonlinear filtering problems. · m3 s3 m1 m2 It provides the optimal solution to the state estimation · (θs1 +θs2i))−(θm1 +θm2i)(θs1 −θs2i)−cosϕ∗ (θ 1)(θ θ i) sinϕ (θ +1)(θ +θ i) problemknowingonlytheobservedrandomvariables. In · m3 − s1 − s2 − ∗ s3 m1 m2 contrast to the known in the literature state estimation +cosϕ(θm3 +1)(θs1 −θs2i) methods we do not need any simplifications, lineariza- sinϕ(θ 1)(θ +θ i) , tions or assumptions on the distribution of the observed − s3 − m1 m2 random variables. Therefore, having the nonlinear ob- (cid:17) servation model one can forget about its physical nature 1 and apply the filtering method described above. ρS(2,1)= 8 −cosϕ∗(sinϕ(θm3 +1)(θs3 +1) Moreover,thetomographicapproachisusedtowritethe cosϕ(θ +(cid:16)θ i)(θ +θ i))+sinϕ Shannonentropiesfor the examinedstates depending on −f m1 m2 s1 s2 ∗ the time step of the quantum observation model. Using · (cosϕ(θm3 −1)(θs3 −1)−sinϕ(θm1 −θm2i) the latter entropies, the time step dependent informa- (θ θ i))+(θ θ i)(θ +θ i) tioninequalityisobtained. Asanexampleweselectethe · s1 − s2 m1 − m2 s1 s2 +cosϕ (θ +1)(θ +θ i) sinϕ (θ 1) Werner state to show the time evolution of the informa- ∗ m3 s1 s2 − ∗ s3 − (θ θ i) cosϕ(θ 1)(θ +θ i) tion. · m1 − m2 − m3 − s1 s2 sinϕ(θ +1)(θ θ i) , − s3 m1 − m2 (cid:17) ACKNOWLEDGEMENTS 1 The study in Sections 2 and 3 by MarkovichL.A. was ρS(2,2)= 8 −(θm3 +1)(θs3 −1)+cosϕ∗ supported by the RussianScience Foundation grant(14- · (fcosϕ(θm3 −(cid:16)1)(θs3 −1)−sinϕ(θm1 −θm2i) 50-00150). · (θs1 −θs2i))+sinϕ∗(sinϕ(θm3 +1)(θs3 +1) cosϕ(θ +θ i)(θ +θ i)) cosϕ (θ 1) − m1 m2 s1 s2 − ∗ s3 − (θ θ i) sinϕ (θ +1)(θ +θ i) VIII. APPENDIX · m1 − m2 − ∗ m3 s1 s2 cosϕ(θ 1)(θ +θ i) − s3 − m1 m2 The numerator elements (3) for evolution matrix (5) −sinϕ(θm3 +1)(θs1 −θs2i) are (cid:17) and the denominator is the following 1 ρ (1,1)= (θ 1)(θ +1)+cosϕ S 8 − m3 − s3 ∗ Tr(Tr (ρ (k+1)A ))= M S+M x (cosϕ(θ +(cid:16)1)(θ +1)+sinϕ(θ +θ i)(θ +θ i)) · f m3 s3 m1 m2 s1 s2 = 1 cos(ϕ ϕ)+θ (cosϕ+cosϕ ) +sinϕ∗(sinϕ(θm3 −1)(θs3 −1)+cosϕ(θm1 −θm2i) 4 ∗− m1 ∗ (θ θ i))+cosϕ (θ +1)(θ +θ i) +θ(cid:16)(sinϕ sinϕ )i+θ θ (cosϕ cosϕ)i · s1 − s2 ∗ s3 m1 m2 s2 − ∗ m2 s3 ∗− sinϕ (θ 1)(θ θ i)+cosϕ(θ +1) θ θ (cos(ϕ ϕ) 1) θ θ (sinϕ +sinϕ) − ∗ m3 − s1 − s2 s3 · m3 s3 ∗− − − m3 s1 ∗ (θ θ i) sinϕ(θ 1)(θ +θ i) , (θ θ +θ θ )sin(ϕ ϕ)i+1 . · m1 − m2 − m3 − s1 s2 − m1 s2 m2 s1 ∗− (cid:17) (cid:17) 9 The numerator elements (3) for evolutionmatrix (6) are For the real ϕ the latter formulas may be reduced to 1 ρ (1,1)= cosϕ (cosϕ(θ +1)(θ +1)+sinϕ 1 ·(fθSm1 +θm28i)(cid:16)(θs1 +∗θs2i))+smin3ϕ∗(sinsϕ3(θm3 −1) ·ρ(fθSm(11,+1)θ=m28i)(cid:16)(θcso1s+ϕ(θcso2si)ϕ)(+θms3in+ϕ1(s)i(nθsϕ3(+θm13)−+1s)inϕ (θ 1)+cosϕ(θ θ i)(θ θ i)) , · s3 − m1 − m2 s1 − s2 (cid:17) ·(θs3 −1)+cosϕ(θm1 −θm2i)(θs1 −θs2i)) , (cid:17) 1 ρS(1,2)= 8 cosϕ∗(sinϕ(θm3 −1)(θs3 −1)+cosϕ 1 ·(fθm1 −θm2i)(cid:16)(θs1 −θs2i))−sinϕ∗(cosϕ(θm3 +1) ρS(1,2)= 8 cosϕsinϕ(θm3 −1)(θs3 −1)+cosϕ ·(θs3 +1)+sinϕ(θm1 +θm2i)(θs1 +θs2i)) , ·(fθm1 −θm2i)(cid:16)(θs1 −θs2i))sinϕ(cosϕ(θm3 +1) (cid:17) (θ +1)+sinϕ(θ +θ i)(θ +θ i)) , · s3 m1 m2 s1 s2 1 ρ (2,1)= cosϕ (sinϕ(θ +1)(θ +1) cosϕ (cid:17) S 8 − ∗ m3 s3 − (θ +θ i)(cid:16)(θ +θ i))+sinϕ (cosϕ(θ 1) ·fm1 m2 s1 s2 ∗ m3 − 1 ρ (2,1)= cosϕ(sinϕ(θ +1)(θ +1) cosϕ ·(θs3 −1)−sinϕ(θm1 −θm2i)(θs1 −θs2i)) , S 8 − m3 s3 − (cid:17) ·(fθm1 +θm2i)(cid:16)(θs1 +θs2i))+sinϕ(cosϕ(θm3 −1) 1 (θ 1) sinϕ(θ θ i)(θ θ i)) , ρS(2,2)= 8 cosϕ∗(cosϕ(θm3 −1)(θs3 −1)−sinϕ · s3 − − m1 − m2 s1 − s2 (cid:17) (θ θ i)(cid:16)(θ θ i))+sinϕ (sinϕ(θ +1) ·fm1 − m2 s1 − s2 ∗ m3 ·(θs3 +1)−cosϕ(θm1 +θm2i)(θs1 +θs2i)) , ρ (2,2)= 1 cosϕ(cosϕ(θ 1)(θ 1) sinϕ (cid:17) S 8 m3 − s3 − − and the denominator is (θ θ i)(cid:16)(θ θ i))+sinϕ(sinϕ(θ +1) 1 ·fm1 − m2 s1 − s2 m3 4 cos(ϕ∗−ϕ)(1+θm3θs3) ·(θs3 +1)−cosϕ(θm1 +θm2i)(θs1 +θs2i)) , (cid:16) (cid:17) (θ θ +θ θ )sin(ϕ ϕ)i . − m1 s2 m2 s1 ∗− (cid:17) [1] M. 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