Stability of continuous-time quantum filters N. H. AMINI [email protected] CNRS, Laboratory of Signals and Systems CentraleSup´ elec S´ eminaire Scube June 19th 2015

Background and motivation Introduction to continuous-time filtering in classical and quantum cases Stability of continuous-time quantum filters driven by a Wiener process Design and stability of filters driven by both Poisson and Wiener processes with imperfections

Outline Background and motivation Introduction to continuous-time filtering in classical and quantum cases Stability of continuous-time quantum filters driven by a Wiener process Design and stability of filters driven by both Poisson and Wiener processes with imperfections

Some definitions I Notations : Ket |.i : a vector in the Hilbert space H and Bra h.| : a co-vector in the dual of the Hilbert space H . I Observables : Physical quantities like position, momentum, spin, etc : noncommutative counterparts of random variables. I Hermitian operators on the Hilbert space H ; I Spectral decomposition : O = µ lµ Pµ, lµ are the eigenvalues and reals, and Pµ are the orthogonal projectors on the corresponding eigenspaces : ÂPµ = I. I States : a summary of the status of a physical system that enables the calculation of statistical quantities associated with observables. I can be represented by a vector |yi (wave function) in the Hilbert space H , with hy|yi = ||y||2 = 1. I or by a density matrix r : self-adjoint operator on H ; r 0 and Tr(r) = 1. (Noncommutative counterpart of a probability density.)

I Pure state : The unit vector |yi corresponds to a pure state. It is associated to the rank one projector r = |yihy|, which satisfies Tr(r) = Tr r2 = 1. I Mixed state : The density operator takes the following form r = Ân pn |ynihyn|, where pn is the probability that system is in the pure state |yni. We have Tr(r) = 1, however, Tr r2 < 1. I Composite system : Consider a system composed of two sub-systems A and B with states yA ↵ and yB ↵ which are represented in the Hilbert space H A and H B. The joint state is yA ↵ ⌦ yB ↵ living in H A ⌦H B.

Measurements I A measurement is a physical procedure that produces numerical results related to observables. I The allowable results take values in the spectrum spec(A) of a chosen observable A. I Given the state r, the value µ 2 spec(A) is observed with probability Tr(rPµ) =) the expectation of an observable A : hAi = Tr(rA). I Projective measurements : r + = Pµ rPµ pµ , with pµ = Tr(rPµ), P† µ = Pµ and P2 µ = Pµ I Positive operator values measure (POVM) : r + = Mµ rM† µ Tr(Mµ rM† µ ) , where µ M† µ Mµ = IS and the operators M† µ Mµ are positive. I Quantum non-demolition (QND) measurement :[H,Os] = 0.

Quantum commutative conditional expectation Spectral theorem. Any commutative collection of operators can be simultaneously diagonalized. Denote the commutative algebra by Y = span(Py ), where Py are the orthogonal projections : ÂPy = I. Denote Y 0 the collection of all operators that commute with Y . Conditional expectation : an orthogonal projection from Y 0 to Y : b X = E(X|Y ) = Â y Py ||Py || h Py ||Py || ,Xi = Â y E(Py X) E(Py ) Py . Here hA,Bi = Tr(rA⇤B) and ||A|| = p hA,Ai.

Quantum noncommutative conditional expectation Spectral theorem is no more valid for non-commutative algebra Y . FIGURE: Example : non-commutative algebra Y = M 2 (the space of complex 2 ⇥2 matrices) contains distinct quantum probability space determined by non-commutative observable sx , sy and sz. Remark : The existence of a non-commutative conditional expectation is not always guaranteed ! (see e.g., M. Takesaki, Conditional expectations in von Neumann algebras. Journal of Functional Analysis 9(3), 306-321 (1972)).

Schr¨ odinger equation Consider a time-dependent system whose state at time t is described by the state |yt i which evolves as follows i d dt |yt i = H(t)|yt i, where H is a time-varying Hermitian operator (H† = H), called the Hamiltonian. Take the initial state |y 0 i, then |yt i = Ut |y 0 i, with the linear operator Ut satisfies i d dt Ut = H(t)Ut , U 0 = I. =) U† t Ut = Ut U† t = I.

The perfect discrete-time non-linear Markov model Consider a finite-dimensional quantum system (the underlying Hilbert space H = Cd is of dimension d > 0) being measured through a generalized measurement procedure at discrete-time intervals. 1 D := {r 2 Cd⇥d | r = r†, Tr(r) = 1, r 0}. The random evolution of the state rk 2 D at time-step k is modeled through the following Markov process : rk+1 = Mµk (rk ) := Mµk rk Mµk † Tr Mµk rk Mµk † , where, I To each measurement outcome µ is attached the Kraus operator Mµ 2 Cd⇥d depending on µ. We have Âm µ=1 Mµ †Mµ = I. I µk is a random variable taking values µ in {1,··· ,m} with probability pµ,rk = Tr Mµ rk Mµ † . 1. S. Haroche and J.-M. Raimond. Exploring the quantum : atoms, cavities and photons. Oxford University Press, New York, 2006.

Example : LKB photon box 2 Experiment : C. Sayrin et. al., Nature 477, 73-77, September 2011. Theory : I. Dotsenko et al., Physical Review A, 80 : 013805-013813, 2009. R. Somaraju et al., Rev. Math. Phys., 25, 1350001, 2013. H. Amini et. al., Automatica, 49 (9) : 2683-2692, 2013. 2. Courtesy of Igor Dotsenko

I µk 2 {g,e}; I d = nmax +1; I All operators are expressed in the truncated Fock-basis (|ni)n=0,···,nmax ; I Mg = cos( fR+f 0(N+ 1 2 ) 2 ), Me = sin( fR+f 0(N+ 1 2 ) 2 ); I N = a†a is the photon number operator (N|ni = n|ni).

Outline Background and motivation Introduction to continuous-time filtering in classical and quantum cases Stability of continuous-time quantum filters driven by a Wiener process Design and stability of filters driven by both Poisson and Wiener processes with imperfections

Classical filtering Consider the following stochastic dynamics dXt = v(Xt )dt +sX (Xt )dW 1, with the following noisy observation dYt = h(Xt )dt +sY (Xt )dW 2. Filtering problem : Obtain the dynamics for the least mean squares estimate for the state dynamics, i.e., pt (f) := E ⇣ f(Xt )|F Y t] ⌘ , where FY t] is the s algebra generated by the observation processes up to time t.

Filter’s dynamics 4 Kallianpur-Striebel Formula 3 : Take the Kallianpur-Striebel likelihood function Zt (X,Y) = exp ✓Z t 0 h(Xs)T dYs 1 2 h(Xs)T h(Xs) ◆ . The conditional expectation is given by pt (f) = R f(xt )Zt P(dx) R Zt P(dx) =: st (f) st (1) . 3. Kallianpur, Striebel, 1968. 4. Davis and Marcus, 1981.

Filter’s dynamics Duncan-Motensen-Zakai equation : Unnormalized filter satisfies dst (f) = st (Lf)dt +st (fhT )dYt . Kushner-Stratonovich equation : Normalized filter satisfies dpt (f) = pt (Lf)dt +[pt (fhT ) pt (f)pt (hT )]dIt , where It is the innovation process : dIt = dYt pt (h)dt, I(0) = 0.

Model of open quantum systems Consider G = (S,L,H), with unitary S describing photon scattering phase, L describing the coupling to the creation mode of the field and H describing the system Hamiltonian. [Hudson and Parthasarathy, 1984.] The evolution is described by evolution U with the following QSDE dU(t) = ⇣ (S I)d⇤(t)+dB†(t)L L†SdB(t) (1 2 L†L +iH)dt ⌘ U(t), U(0) = I, where, I dB(t) = B(t +dt) B(t), B(t)† = R t 0 b†(s)ds and B(t) = R t 0 b(s)ds; I [b(t),b†(s)] = d(t s) and ⇤(t) = R t 0 b†(s)b(s)ds.

Model of open quantum systems Quantum Ito rule : d(X Y)=dX Y+X dY+dX dY Heisenberg-Langevin Equation : The Heisenberg dynamics of an operator X is given by jt (X) = U†(t)(X ⌦I field)U(t) Using It¯ o rules : djt (X) = jt (LX)dt +dB†(t)jt (S†[X,L]) +jt ([L†,X]S)dB(t)+jt (S†XS X)d⇤(t), where the (Gorini-Kossakowski-Sudarshan-Lindblad) generator is L(X) = 1 2 [L†,X]L + 1 2 L†[X,L] i[X,H] Input-Output relation : The Output B out is obtained from the input by B out(t) = U(t)†(I system ⌦B(t))U(t) Using It¯ o rules : dB out(t) = jt (S)dB(t)+jt (L)dt.

Derivation of quantum filters for Homodyne detection Consider dX(t) = i[X(t),H(t)]dt +LL(X(t))dt +dB†(t)[X(t),L(t)]+[L†(t),X(t)]dB(t), with LL(X) = 1 2 L†[X,L]+ 1 2 [L†,X]L. Take the quadrature measurement Y(t) = B out(t)+B⇤ out (t). I {Y(t) : t 0} is self-commuting : [Y(t),Y(s)T ] = 0 for all t s, then, we can simultaneously diagonalize all observables ; I We can estimate an observable that commutes with the observable Y(t) up to time t : the non-demolition property, i.e., [X(t),Y(t)T ] = 0, for t t.

Derivation of quantum filters in Homodyne detection Recall : Y(t) = B out(t)+B⇤ out (t), then dY(t) = (L(t)+L⇤(t))dt +d(B(t)+B⇤(t)). Let Y (t) denotes the commutative subspace of operators generated by Y(s),0  s  t. ˆ X(t) = pt (X) = E(X(t)|Y (t)). Conditional characteristic approach : cf (t) = exp( Z t 0 f(s)dY(s) 1 2 Z t 0 |f(s)|2ds), we have dcf (t) = f(t)cf (t)dY(t), with cf (0) = I.

Derivation of quantum filters for Homodyne detection Suppose that d ˆ X(t) = a(t)dt +b(t)dY(t), where a and b are to be determined from E(X(t)cf (t)) = E ˆ X(t)cf (t) . Filter’s dynamics : The best estimate satisfies : dpt (X) = pt ( i[X,H]+LL(X))dt + pt (XL +L†X) pt (L +L†)pt (X) dY(t) pt (L +L†)dt .

Derivation of quantum filters for Homodyne detection (4) A conditional density r may be defined by pt (X) = Tr(rX). Quantum filter in terms of the density operator : The density operator-value stochastic process r satisfies the following dr = i[H,r]+LL(r) dt +(Lr+rL† Tr (L +L†)r r)(dY Tr (L +L†)r dt)

Photon counting case We measure the number observable Y(t) given by Y(t) = U(t)†⇤(t)U(t) = ⇤out(t) = Z t 0 b† out (s)b out(s)ds. Then, Quantum filter for photon counting case : dr(t) = i[H,r]dt +LL(r)dt + L(r)dN(t), where (r) = LrL† Tr(rL†L) r, and dN(t) = dY Tr r(t)L†L dt.

Quantum filtering Take Y (t) as the s algebra generated by the observation process (Y(s))0st .Then, I If Y is commutative, then the filtering problem is very similar to the classical one, e.g., : quadrature case using Homodyne detection, photon counting case ; [Davies, 1960s ; Belavkin, 1980s] I If Y is non-commutative, we cannot apply the classical method to obtain the filter’s equation. [No results]

Application of commutative filtering : measurement-based feedback Estimation or filtering step consists of extracting information on the system state from the past control input and measured output values. Computation of the next control input is based on the current filter state obtained in the first step. Measurement Classical control actions Classical controller Quantum system Classical information Difficulty : any measurement necessarily modifies the system state. [Belavkin, 1983 ; Wiseman and Milburn, 2010]

Application of non-commutative filtering : coherent quantum feedback Quantum information Quantum system Quantum controller Quantum control actions The controller is also a quantum system and information flowing in the feedback loop is also quantum (e.g. via a quantum field). [Lloyd, 2000 ; James, Nurdin, Petersen, 2008 ; Mabuchi, 2008]

Outline Background and motivation Introduction to continuous-time filtering in classical and quantum cases Stability of continuous-time quantum filters driven by a Wiener process Design and stability of filters driven by both Poisson and Wiener processes with imperfections

Some definitions I Asymptotic stability=Convergence : The independence of the filter, after a long time, from the initial state estimate. I Observability means that there do not exist two different initial states which give rise to measurement outcomes with the same probability. I Stability means that a distance between the state and its associated quantum filter decreases.

The mathematical model State space : density matrix D := {r 2 CN⇥N | r = r†, Tr(r) = 1, r 0}. Quantum filter dynamics : dr = i[H,r]+LL(r) dt +(Lr+rL† Tr (L +L†)r r)(dY Tr (L +L†)r dt) dyt = Tr (L +L†)rt dt +dWt . Quantum filter estimate equation : The estimate b r 2 D of quantum system follows 5 db r = i[H,b r]+LL(b r) dt +(Lb r+b rL† Tr (L +L†)b r b r)(dY Tr (L +L†)b r dt). What can we say about the distance between r and b r ? 5. A. Barchielli. Journal of the European Optical Society Part B, 1990.

The definition of the fidelity (distance) We consider the following definition of fidelity between two density matrices r and s : 6 F(r,s) = Tr ✓q prspr ◆ 2 . Distance = 1 F I F(r,s) = 1 iff r = s ; I F(r,s) = 0 means that the support of r and s are orthogonal ; I F(r,s) coincides with their inner product Tr(rs) when at least one of the states r or s is pure (i.e., orthogonal projector of rank one). 6. M. Nielsen and I. Chuang. Quantum Computation and quantum information, 1999.

Main Result Theorem (Amini, Mirrahimi, and Rouchon, 2011) Consider the Markov processes (rt ,b rt ) satisfying dr = i[H,r]+LL(r) dt +(Lr+rL† Tr (L +L†)r r)(dY Tr (L +L†)r dt) dyt = Tr (L +L†)rt dt +dWt . db r = i[H,b r]+LL(b r) dt +(Lb r+b rL† Tr (L +L†)b r b r)(dY Tr (L +L†)b r dt). respectively with initial states r 0 , b r 0 in D. Then, the fidelity F(rt ,b rt ), is a submartingale, i.e. E(F(rt ,b rt )|(rs,b rs)) F(rs,b rs), for all t s.

Numerical test Continuous homodyne measurement of a single qubit H = sy = ✓ 0 i i 0 ◆ L = sz = ✓ 1 0 0 1 ◆ . 0 0.5 1 1.5 2 2.5 3 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 Fidelity Time The average fidelity between the Markov processes r and b r, over 500 realizations. Here the initial states are r 0 = ✓ 1 2 1 4 1 4 1 2 ◆ and b r 0 = ✓ 1 3 0 0 2 3 ◆ .

Sketch of proof Proof : We proceed in two steps. I In the first step, we describe how we obtain the stochastic master equations with Wiener processes as the limits of the stochastic master equations with Poisson processes. Then, we use a theorem by C. Pellegrini and F. Petruccione, 2009, to show the convergence of the solution of SMEs driven by Poisson processes towards the ones driven by Wiener processes. I In the second step, we show that the fidelity between the real state and the quantum filter which are the solutions of stochastic master equations with Poisson processes is a submartingale. This can be done by discretizing SMEs driven by Poisson processes and using the stability results for discrete-time quantum non-linear Markov chain established by Rouchon, 2009.

Further directions I The considered fidelity has the potential to be used as a control Lyapunov function ; I The fact that the fidelity between the real quantum state and the quantum-filter state increases in average remains valid for more general stochastic master equations where other Lindblad terms are added to L(r) ; I The other extension of the problem is to consider the stochastic master equations driven by the Wiener processes and the jump processes at the same time and prove the stability of the quantum filter in this case (following section) ; I Characterize the situations where the asymptotic convergence of such quantum filter is ensured.

Outline Background and motivation Introduction to continuous-time filtering in classical and quantum cases Stability of continuous-time quantum filters driven by a Wiener process Design and stability of filters driven by both Poisson and Wiener processes with imperfections

Example : The closed-loop QED experiment 8 7 Sampling time (⇠ 100 µs) long enough for feedback computations. 7. Courtesy of Igor Dotsenko 8. C. Sayrin et al., Nature, 1-September 2011

Imperfections in LKB experiment Consider the ideal open-loop dynamics : rk+1 = 8 > > > > > > < > > > > > > : Mg rk M† g Tr ⇣ Mg rk M† g ⌘ yk = g with probability pg,k = Tr ⇣ Mg rk M† g ⌘ Me rk M† e Tr ⇣ Me rk M† e ⌘ yk = e with probability pe,k = Tr ⇣ Me rk M† e ⌘ Measurement Kraus operators Mg = cos ⇣ f 0N+fR 2 ⌘ and Me = sin ⇣ f 0N+fR 2 ⌘ : M† g Mg +M† e Me = 1 1 with N = a†a = diag(0,1,2,...) the photon number operator. Imperfections in LKB experiment include : I Denote the detector efficiency by hd = 0.8; I The result of the measurement (atom in the state g or e) can be inter-changed : the fault rate hf = 0.1; I The measurement pulses can be empty of atom : the occupancy rate ha = 0.4.

Markov chain with imperfections Bayes’ rule : P(A|B) = P(B|A)P(A) P(B|A)P(A)+P(B|Ac)P(Ac) . I Atom is detected in |gi : I Either the atom is in the state |ei and the detector detects it in the state |gi which arrives with the probability Pf g = hf Pe hf Pe +(1 hf )Pg , where Pg = Tr ⇣ Mg rM† g ⌘ and Pe = Tr ⇣ Me rM† e ⌘ . I Or the atom is really in the state |gi, this happens with probability 1 Pf g. Therefore, rk+1 = Pf g Me(rk )+(1 Pf g)Mg(rk ) = hf Me rk M† e +(1 hf )Mg rk M† g hf pe +(1 hf )Pg .

I Atom is detected in |ei : The conditional evolution of the density matrix is given by rk+1 = hf Mg rk M† g +(1 hf )Me rk M† e hf pg +(1 hf )Pe . I No atom is detected : I Either the pulse has been empty, which arrives with probability P na given below, P na = 1 ha ha(1 hd )+(1 ha) = 1 ha 1 ha hd . I or there has been an atom which has not been detected by the detector, this arrives with probability 1 P na. Then rk+1 = P na rk +(1 P na)(Mg rk M† g +Me rk M† e ) = (1 ha)rk +ha(1 hd )(Mg rk M† g +Me rk M† e ) 1 ha hd .

General modeI : the new ”observable” state b r I Take a set of Kraus operators Mq attached to the ideal detections for q 2 {1,.··· ,m} : Âm q=1 M† q Mq = 1 1, for some m 2 N. I Assume that the real sensors provide an outcome µ0 2 {1,··· ,m0}, for some m0 2 N. I Suppose that we know the correlation between the events µ = q and µ0 = p which is given through the stochastic matrix h 2 Rm0⇥m : hp,q = P(µ0 = p|µ = q), with hp,q 0 and for each q, Âm0 p=1 hp,q = 1.

General model : the new ”observable” state b r I Assume Mq;k denotes the Kraus operator corresponding to the k th ideal measurement for q 2 {1,...,m}. We have E(rk+1 |rk ,µk = q) = Mq;k (rk ) := Mq;k rk M† q;k Tr ⇣ Mq;k rk M† q;k ⌘, with P(µk = q|rk ) = Tr ⇣ Mq;k rk M† q;k ⌘ . I Let hk be the stochastic matrix at step k. The optimal estimate is defined as b rk = E rk |(r 0,µ0 0 ,··· ,µ0 k 1 ) .

General model : the new ”observable” state b r Theorem ( Somaraju et al., 2012.) The optimal estimate b rk satisfies the following recursive equation b rk+1 = Âm q=1 hpk ,q Mq;k b rk M† q;k Tr ⇣ Âm q=1 hpk ,q Mq;k b rk M† q;k ⌘, if µ0 k = pk . Moreover P(µ0 k = pk |(r 0,µ0 0 ,··· ,µ0 k 1 )) = Tr m  q=1 hpk ,q Mq;k b rk M† q;k ! .

SDE driven by Poisson and/or Wiener processes drt = L(rt )dt + mW  n=1 ⇤n(rt )dWn t + mp  µ=1 gµ(rt ) ⇣ dNµ t Tr Cµ rt C† µ dt ⌘ , where I L(rt ) := i[H,rt ]+ÂmP µ=1 LP µ (rt )+ÂmW n=1 LW n (rt ), where LP µ (r) := 1 2 {C† µ Cµ,r}+Cµ rC† µ, LW n (r) := 1 2 {L† n Ln,r}+Ln rL† n, and gµ(r) := Cµ rC† µ Tr(Cµ rC† µ ) r, ⇤n(r) := Ln r+rL† n Tr ⇣ (Ln +L† n)r ⌘ r. I Detector click µ is related to the Poisson process dNµ t = Nµ(t +dt) Nµ(t) = 1 and happens with probability Tr Cµ rC† µ dt; I Continuous detector n refers to the Wiener process dWn t by dyn t = dWn t +Tr ⇣ (Ln +L† n)rt ⌘ dt.

Heuristic approach : Jump processes Perfect measurements I Jump case : For any µ 6= 0, define the following jump operators Mµ = p dtCµ. The evolution is given by rt+dt = Mµ rt M† µ Tr ⇣ Mµ rt M† µ ⌘ = Cµ rt C† µ Tr ⇣ Cµ rt C† µ ⌘, which happens with probability Tr Cµ rt C† µ dt. I Non jump case : For µ = 0, we have m  µ=1 M† µ Mµ +M† 0 M 0 = I. we find M 0 = I 1 2 Âm µ=1 C† µ Cµ dt iH dt.

I We find rt+dt = 8 > > > < > > > : Cµ rt C† µ Tr(Cµ rt C† µ ) with probability Tr ⇣ Cµ rt C† µ ⌘ dt rt Âm µ=1 1 2 ⇣ C† µCµ rt +rt C† µCµ ⌘ dt +Âm µ=1 Tr ⇣ Cµ rt C† µ ⌘ rt dt i[H,rt ]dt, with probability 1 Âm µ=1 Tr ⇣ Cµ rt C† µ ⌘ dt. I Now define m Poisson processes : dNµ t = Nµ(t +dt) Nµ(t), which takes one with probability Tr Cµ rt C† µ dt and zero with probability 1 Tr Cµ rt C† µ dt. Hence, drt = L(rt )dt + m  µ=1 ⌥µ(rt ) dNµ t Tr Cµ rt C† µ dt .

Imperfect measurements. I Set the ideal outcome µ 2 {0,··· ,m} and the real outcomes µ0 2 {0,··· ,m0}; I The Kraus operators attached to the ideal outcome µ are given by Mµ = p dtCµ and M 0 = I 1 2 Âm µ=1 C† µ Cµ dt iH dt; I Take h as the stochastic matrix describing the correlation matrix between the events µ and µ0; I Suppose that h is well known with the following values : I for any µ0 6= 0, hµ0,0 = ¯ hµ0 dt and h 0,0 = 1 Âm0 µ0=1 ¯ hµ0 dt, with ¯ hµ0 0; I for any µ 6= 0, h 0,µ = 1 Âm0 µ0=1 hµ0,µ, where 0  hµ0,µ  1 and h 0,µ 0. I We have for any µ0 2 {0,1,··· ,m0} : ˆ rt+dt = Âm µ=0 hµ0,µ Mµˆ rt M† µ Tr ⇣ Âm µ=0 hµ0,µ Mµˆ rt M† µ ⌘, with probability Tr m  µ=0 hµ0,µ Mµˆ rt M† µ ! .

I Finally, we find ˆ rt+dt = 8 < : L0(b rt ) with probability 1 Âm0 µ0=1 ¯ hµ0 dt Âm µ=1 Âm0 µ0=1 hµ0,µTr ⇣ C† µb rt Cµ ⌘ dt, Lµ0 (b rt ) with probability ¯ hµ0 dt +Âm µ=1 hµ0,µTr ⇣ Cµb rt C† µ ⌘ dt, where L0(b rt ) = b rt 1 2 m  µ=1 {C† µ Cµ,b rt }dt i[H,b rt ]dt + m  µ=1 1 m0  µ0=1 hµ0,µ ! Cµb rt C† µ dt + m  µ=1 m0  µ0=1 hµ0,µTr ⇣ Cµb rt C† µ ⌘ b rt dt and Lµ0 (b rt ) = ¯ hµ0 b rt +Âm µ=1 hµ0,µ Cµb rt C† µ ¯ hµ0 +Âm µ=1 hµ0,µTr ⇣ Cµb rt C† µ ⌘.

I Now introduce m0 Poisson processes b Nµ0 t and the process d b Nµ0 t := b Nµ0 t+dt b Nµ0 t which takes one with probability ¯ hµ0 dt +Âm µ=1 hµ0,µTr Cµb rt C† µ dt and zero with the complementary probability ; I Hence, we have db rt = L(b rt )dt + m0  µ0=1 b ⌥µ0 (b rt ) d b Nµ0 t ¯ hµ0 dt m  µ=1 hµ0,µTr ⇣ Cµb rt C† µ ⌘ dt ! , with L(b rt ) = i[H,b rt ] 1 2 Âm µ=1 {C† µ Cµ,b rt }+Âm µ=1 Cµb rt C† µ and b ⌥µ0 (r) = ¯ h µ0 r+Âm µ=1 h µ0,µ Cµ rC† µ ¯ h µ0 +Âm µ=1 h µ0,µTr(Cµ rC† µ ) r.

SDE driven by both Poisson and/or Wiener processes including imperfections Some notations : I Imperfection model for the Poisson processes dNµ t : I real outcomes µ0 2 {0,1,··· ,m0 P}, I ideal outcomes µ 2 {1,··· ,mP}, I define (m0 P +1)⇥mP left stochastic matrix hP = (hP µ0,µ )0µ0m0 P ,1µmP , I define a positive vector ¯ hP = (¯ hP µ0 )1µ0m0 P in Rm0 P + . I Imperfection model for the diffusion processes dWn t : I m0 W real continuous signals yn0 t with n0 2 {1,··· ,m0 W }, I mW ideal continuous signals yn t with n 2 {1,··· ,mW }, I m0 W ⇥mW correlation matrix hW = (hW n0,n)1n0m0 W ,1nmW , with 0  hW n0,n  1 and Âm0 W n0=1 hW n0,n  1.

SDE driven by both Poisson and/or Wiener processes including imperfections Theorem (Amini, Pellegrini, and Rouchon, 2014) db r = L(b rt )dt + m0 W Â n0=1 q ¯ hW n0 b ⇤n0 (b rt )d b Wn0 t + m0 P Â µ0=1 b gµ0 (b rt ) ⇣ d b Nµ0 t ¯ hP µ0 dt mP Â µ=1 hP µ0,µTr ⇣ Cµb rt C† µ ⌘ dt ⌘ , I ¯ hW n0 = ÂmW n=1 hW n0,n, b gµ0 (r) := ¯ hP µ0 r+ÂmP µ=1 hP µ0,µ Cµ rC† µ ¯ hP µ0 +ÂmP µ=1 hP µ0,µ Tr(Cµ rC† µ ) r, b ⇤n0 (r) = b Ln0 r+rb L† n0 Tr ⇣ (b Ln0 +b L† n0 )r ⌘ r, b Ln0 := (ÂmW n=1 hW n0,n Ln)/¯ hW n0 , I the jump detector corresponds to b Nµ0 (t) : d b Nµ0 t = b Nµ0 (t +dt) b Nµ0 (t) = 1 happens with probability ¯ hP µ0 +ÂmP µ=1 hP µ0,µTr ⇣ Cµ rC† µ ⌘ , I the continuous detector n0 refers to b yn0 t and d b Wn0 t : db yn0 t = d b Wn0 t + q ¯ hW n0 Tr ⇣ (b Ln0 +b L† n0 )b rt ⌘ dt.

Stability results The estimate filter b re has the following form db re = L(b re t )dt + m0 W Â n0=1 q ¯ hW n0 b ⇤n0 (b re t ) ✓ db yn0 t q ¯ hW n0 Tr ⇣ (b Ln0 +b L† n0 )b re t ⌘ dt ◆ + m0 P Â µ0=1 b gµ0 (b re t ) ⇣ d b Nµ0 t ¯ hP µ0 dt mP Â µ=1 hP µ0,µTr ⇣ Cµb re t C† µ ⌘ dt ⌘ , Theorem (Amini, Pellegrini and Rouchon, 2014) The fidelity F(b rt ,b re t ) is a (Ft ) submartingale, where Ft = s{(b rt,b re t)|t  t}. In particular, we have, E(F(b rt,b re t)|Ft ) = E(F(b rt,b re t)|(b rt ,b re t )) F(b rt ,b re t ), for all t t.

Concluding remarks I In ”Amini, Pellegrini, Rouchon, Russian Journal of Mathematical Physics, 2014”, we have shown rigorously, by applying quantum repeated measurement approach introduced by Attal and Pautrat, and by Gough, that imperfect discrete-time Markov chain converges to the continuous-time dynamics driven by both Poisson and Wiener processes. I We have shown the stability of the filters taking into account measurement imperfections using the stability result for discrete-time filters which take into account imperfections. I Difficult issue : Characterize the situations when we can conclude the convergence of the filters. I Open problem : Filtering when the observation processes are non-commutative is an open problem. In ”Amini, Miao, Pan, James, and Mabuchi, On the generalization of linear least mean squares estimations to quantum systems with non-commutative outputs, EPJ Quantum Technology, 2015”, we extend Kalman filtering to non-commutative outputs.

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