Parameter estimation from measurements along quantum trajectories
Pierre Six, Philippe Campagne-Ibarcq, Landry Bretheau, Benjamin Huard, Pierre Rouchon
aa r X i v : . [ m a t h . O C ] M a r Parameter estimation from measurements along quantum trajectories ∗ P. Six † Ph. Campagne-Ibarcq ‡ L. Bretheau ‡ B. Huard ‡ P. Rouchon † Abstract
The dynamics of many open quantum systemsare described by stochastic master equations. In thediscrete-time case, we recall the structure of the derivedquantum filter governing the evolution of the density op-erator conditioned to the measurement outcomes. Wethen describe the structure of the corresponding par-ticle quantum filters for estimating constant parameterand we prove their stability. In the continuous-time (dif-fusive) case, we propose a new formulation of theseparticle quantum filters. The interest of this new for-mulation is first to prove stability, and also to providean e ffi cient algorithm preserving, for any discretizationstep-size, positivity of the quantum states and parameterclassical probabilities. This algorithm is tested on ex-perimental data to estimate the detection e ffi ciency for asuperconducting qubit whose fluorescence field is mea-sured using a heterodyne detector.
1. Introduction
Parameter estimation in hidden Markov models isa well established subject (see, e.g., [7]). Twenty yearsago Mabuchi [15] has proposed maximum likelihoodmethods to estimate Hamiltonian parameters. Later on,Gambetta and Wiseman [11] have given a first formu-lation of particle filtering techniques for classical pa-rameter estimation in open quantum systems. This for-mulation has been analyzed in [8] via an embedding inthe standard quantum filtering formalism. Recently Ne-gretti and Mølmer [16] have exploited this embedding ∗ This work has been partly funded by the Idex PSL * under thegrant ANR-10-IDEX-0001-02 PSL *, by the Emergences program ofVille de Paris under the grant Qumotel and by the Projet Blanc ANR-2011-BS01-017-01 EMAQS. † Centre Automatique et Syst`emes, Mines-ParisTech, PSL Re-search University. 60, bd Saint-Michel 75006 Paris. ‡ Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure-PSL Re-search University, CNRS, Universit´e Pierre et Marie Curie-SorbonneUniversit´es, Universit´e Paris Diderot-Sorbonne Paris Cit´e, 24 rueLhomond, 75231 Paris Cedex 05, France to derive the general equations of a particle quantum fil-ter for systems governed by stochastic master equationsdriven by Wiener processes (di ff usive case). In thesecontributions, realistic simulations illustrate the interestof such filters for the estimation of continuous param-eters. In [14], similar filters are used for purely dis-crete parameters in order to discriminate between dif-ferent topologies of quantum networks. The Bayesianparameter estimation used in the measurement-basedfeedback experiment reported in [4] is in fact a spe-cial case of particle quantum filtering when the quantumstates remain diagonal in the energy-level basis, reduceto populations and classical probabilities.The contribution of this paper is twofold: with the-orem 2, we show that particle quantum filters are al-ways stable processes; with lemma 2, we propose andjustify a new positivity preserving formulation in thedi ff usive case. This formulation is shown to provide ane ffi cient algorithm for precisely estimating the detectione ffi ciency from experimental heterodyne measurementsof the fluorescence field that is emitted by a supercon-ducting qubit [5]. The statistics of the measurement out-comes generated by this system cannot be described byclassical probabilities since the density operators at var-ious times do not commute. As far as we know, thisis the first time that a particle quantum filter is appliedto an experiment [6] whose measurement statistics areruled by non-commutative quantum probabilities.Section 2 is devoted to the discrete-time formula-tion. The specific structure of Markov models describ-ing open-quantum systems is presented. Then particlequantum filters are detailed and shown to be always sta-ble (theorem 2). Finally, the link with MaxLike ap-proach and the case of multiple measurement recordsare addressed. In section 3, a positivity preserving for-mulation of particle quantum filters is proposed for dif-fusive systems. The mathematical justifications of thisformulation is given in lemma 2. In section 4, the nu-merical algorithm underlying lemma 2 is applied on ex-perimental data from which the detection e ffi ciency isestimated and compared to an existing calibration pro-ocol.
2. Discrete-time formulation
In the sequel, H is the finite-dimensional Hilbertspace of the system and expectation values are denotedby the symbol E ( . ). In this section, time is indexed bythe integer k = , , . . . The measurement outcome at k isdenoted by y k . It corresponds to a classical output sig-nal. We limit ourselves to the case where each y k cantake a finite set of values y k ∈ { , . . ., m } , m being a pos-itive integer (for continuous values of y , see section 3).We denote by ρ k the density operator at time-step k (anHermitian operator on H such that Tr (cid:0) ρ k (cid:1) = ρ k ≥ k knowing the initial condition ρ and the past outcomes y , . . . , y k . According to the law of quantum mechan-ics, ρ k is related to ρ k − via the following Markov pro-cess (see, e.g., [22]) corresponding to a Davies instru-ment [9] in a discrete context: ρ k = K y k ( ρ k − )Tr (cid:16) K y k ( ρ k − ) (cid:17) (1)where the super-operator ρ K y ( ρ ) depends on y , islinear and completely positive. It admits the follow-ing Kraus representation K y ( ρ ) = P µ M y µ ρ ( M y µ ) † wherethe operators on H , ( M y µ ), satisfy P µ, y ( M y µ ) † M y µ = I H with I H the identity operator. Moreover the probability P (cid:0) y k = y | ρ , y , . . . , y k − (cid:1) to detect y k knowing the pastoutcomes and the initial state ρ , depends only on ρ k − (Markov property) and is given by P (cid:16) y k = y (cid:12)(cid:12)(cid:12) ρ k − (cid:17) = Tr (cid:16) K y ( ρ k − ) (cid:17) . Notice that E (cid:16) ρ k (cid:12)(cid:12)(cid:12) ρ k − (cid:17) = K ( ρ k − ) where K ( ρ ) = P y K y ( ρ ) = P µ, y M y µ ρ ( M y µ ) † is a Kraus map (a quantumchannel) since it is not only completely positive but alsotrace preserving: Tr (cid:0) K ( ρ ) (cid:1) = Tr (cid:0) ρ (cid:1) . In the sequel, K y iscalled a partial Kraus map since it is not trace preservingin general: Tr (cid:16) K y ( ρ ) (cid:17) ≤ Tr (cid:0) ρ (cid:1) . See, e.g., [10, 2] for a de-tailed construction of such K y based on positive opera-tor value measures (POVM) and left stochastic matricesmodeling measurement uncertainties and decoherence.Now, we consider that the partial Kraus maps( K y ) y = ,..., m can depend on time k , ( K y , k ), and on somephysical parameters, grouped in the scalar or vectorialtime-invariant p , ( K py , k ), whose exact value p may notbe known with a su ffi cient precision, and whose esti-mation is the subject of this paper. Here, we considerthe case where the only reliable resource of information is some independent series of measurement outcomes,( y k ) k = ,..., T , associated to a quantum trajectory of dura-tion T . Starting from the exact quantum state ρ andthe exact parameter value p , the exact quantum statetrajectory ( ρ k ) k = ,..., T is given by the following Markovprocess: ρ k = K py k , k ( ρ k − )Tr (cid:18) K py k , k ( ρ k − ) (cid:19) (2)with the following probability of outcome y k knowing ρ k − and p : P (cid:16) y k = y (cid:12)(cid:12)(cid:12) ρ k − , p (cid:17) = Tr (cid:18) K py , k ( ρ k − ) (cid:19) . The parameter estimation method described in [11,8, 16] for continuous-time quantum trajectories admitsthe following discrete-time formulation. When the ex-act parameter value p and the initial state ρ are un-known, one can still resort to the approximate filter cor-responding to its a priori estimate value p , with partialKraus maps K py k , k , an initial guess for ρ and followingstates ρ pk satisfying ρ pk = K pyk , k ( ρ pk − )Tr (cid:18) K pyk , k ( ρ pk − ) (cid:19) . Here, the mea-surement outcomes ( y k ) k = ,..., T correspond to the hiddenstate Markov chain defined in (2) and involving the ac-tual value p of the parameter.Assume that the initial information of the true pa-rameter value p is that it can take only two di ff erentvalues a or b . This initial uncertainty on the value of p can be taken into account by using an extended den-sity operator, denoted ξ = diag( ξ a , ξ b ), block diagonal,where the first block ξ a corresponds to p = a , and thesecond block ξ b to p = b . The evolution of each block isthen handled with the corresponding partial Kraus maps( K ay , k ) and ( K by , k ) forming extended partial Kraus maps Ξ y , k = diag (cid:16) K ay , k , K by , k (cid:17) between block diagonal densityoperators on the Hilbert space H × H : Ξ y , k : ξ diag( K ay , k ( ξ a ) , K by , k ( ξ b )) . (3)The associated extended quantum filter reads: ξ k = Ξ y k , k ( ξ k − )Tr (cid:16) Ξ y k , k ( ξ k − ) (cid:17) . (4)For p ∈ { a , b } , the probability that p = p at step k knowing the initial quantum state ρ and initial pa-rameter probability ( π a , π b ) reads π pk = Tr (cid:16) ξ pk (cid:17) . In-deed, π ak + π bk = (cid:0) ξ (cid:1) = Tr (cid:0) ξ a (cid:1) + Tr (cid:16) ξ b (cid:17) = ξ = diag( π a ρ , π b ρ ). If the initial information onhe parameter value is only its belonging to { a , b } , then π a = π b = / ξ = diag( ξ a , ξ b ) itself, we decom-pose its terms into products of probabilities π p and den-sity operators ρ p = ξ p /π p . Then Eq. (4) reads ρ pk = K pyk , k ( ρ pk − )Tr (cid:18) K pyk , k ( ρ pk − ) (cid:19) π pk = Tr (cid:18) K pyk , k ( ρ pk − ) (cid:19) π pk − P p ′∈{ a , b } Tr (cid:18) K p ′ yk , k ( ρ p ′ k − ) (cid:19) π p ′ k − (5)for p ∈ { a , b } . In the sequel, we will identify the filterstate ξ with ( ρ a , ρ b , π a , π b ).We have the following stability result based on [19,21] and relying on the fidelity F ( ρ, ρ ′ ) ∈ [0 ,
1] betweentwo density operators ρ and ρ ′ defined here as the squareof the usual fidelity function used in quantum informa-tion [17]: F ( ρ, ρ ′ ) = Tr q √ ρρ ′ √ ρ ! . Theorem 1.
Take an arbitrary initial quantum state ρ and a parameter value p. Consider the quantumMarkov process (2) producing the measurement recordy k , k ≥ . Assume that the constant parameter p canonly take two di ff erent values, a and b. Consider theparticle (quantum) filter (5) initialized with ρ a = ρ b = ρ ( ρ any density operator) and ( π a , π b ) ∈ [0 , with π a + π b = . Then F ( ρ, ρ p ) and π p F ( ρ, ρ p ) are sub-martingales of the Markov process (2) and (5) of state ( ρ, ρ a , ρ b , π a , π b ) : When ρ = ρ , we have ρ p ≡ ρ , F ( ρ, ρ p ) =
1. Thus π p is a sub-martingale E (cid:18) π pk (cid:12)(cid:12)(cid:12) ρ k − , ξ k − (cid:19) ≥ π pk − This means that, in practice, the component of π associ-ated to the true value of the parameter tends to increase. Proof.
The fact that F ( ρ, ρ p ) is a sub-martingale is adirect consequence of [21, theorem IV.1]: ( ρ, ρ p ) is thestate of the following quantum Markov chain ρ k = K py k , k ( ρ k − )Tr (cid:18) K py k , k ( ρ k − ) (cid:19) , ρ pk = K py k , k ( ρ pk − )Tr (cid:18) K py k , k ( ρ pk − ) (cid:19) with initial state ( ρ , ρ ) and measurement outcome y k whose probability P (cid:16) y k = y (cid:12)(cid:12)(cid:12) ρ k − (cid:17) = Tr (cid:18) K py , k ( ρ k − ) (cid:19) de-pends only on ρ k − .For instance, assume that p = a . Denote by ξ the state of the quantum filter (4) initialized with ξ = diag( ρ , ξ ≡ ( ρ,
0) and thus ( ξ, ξ ) is solution ofthe extended Markov chain ξ k = Ξ y k , k ( ξ k − )Tr (cid:16) Ξ y k , k ( ξ k − ) (cid:17) , ξ k = Ξ y k , k ( ξ k − )Tr (cid:16) Ξ y k , k ( ξ k − ) (cid:17) with measurement outcome y k of probability P (cid:16) y k = y (cid:12)(cid:12)(cid:12) ξ k − (cid:17) = Tr (cid:16) Ξ y , k ( ξ k − ) (cid:17) depending onlyon ξ k − . Thus according to [21, theorem IV.1], F ( ξ, ξ )is a sub-martingale. Due to the block structure of ξ = diag( ρ,
0) and ξ = diag( π a ρ a , π b ρ b ), we have F ( ξ, ξ ) = π a F ( ρ, ρ a ). (cid:3) Extension of theorem 1 to an arbitrary number r ofparameter values is given below, the proof being verysimilar and not detailed here. Theorem 2.
Take an arbitrary initial quantum state ρ and parameter value p. Consider the quantum Markovprocess (2) producing the measurement record y k , k ≥ .Assume that the parameter p belongs to a set of r di ff er-ent values ( p l ) l = ,..., r . Take, for l = , . . ., r, the particlequantum filter ρ p l k = K plyk , k ( ρ plk − ) Tr (cid:18) K plyk , k ( ρ plk − ) (cid:19) π p l k = Tr (cid:18) K plyk , k ( ρ plk − ) (cid:19) π plk − P rj = Tr (cid:18) K pjyk , k ( ρ pjk − ) (cid:19) π pjk − initialized with ρ p l = ρ ( ρ any density operator) and ( π p , . . . , π p r ) ∈ [0 , r with P j π p j = .Then F ( ρ, ρ p ) and π p F ( ρ, ρ p ) are sub-martingalesof the Markov process driven by (2) and of state ( ρ, ρ p , . . . , ρ p r , π p , . . ., π p r ) : Extension to a continuum of values for p of suchparticle quantum filters and of the above stability resultcan be done without major di ffi culties. Assume that the initial density operator is wellknown: ρ = ρ . It is possible to choose as an es-timation of p , among a or b , the value p that max-imises the probability π pk after a certain amount oftime k . This method is actually a maximum-likelihood based technique. The multiplicative increment attime k for π ak is Tr (cid:18) K ay k , k ( ρ ak − ) (cid:19) , which is equal to P (cid:18) y k (cid:12)(cid:12)(cid:12)(cid:12) ρ , y , . . . , y k − , p = a (cid:19) . From this observation, wededuce that π ak = π a C k × k Y l = P (cid:18) y l (cid:12)(cid:12)(cid:12)(cid:12) ρ , y , . . . , y l − , p = a (cid:19) , here C k is a normalization factor to ensure π ak + π bk = y l ) l ≤ k is the probability of the measurement out-comes ( y l ) l ≤ k − times the probability of y k conditionallyto all prior measurements, one gets π ak = π a C k × P (cid:18) y , . . . , y k (cid:12)(cid:12)(cid:12)(cid:12) ρ , p = a (cid:19) , and similarly π bk = π b C k × P (cid:18) y , . . . , y k (cid:12)(cid:12)(cid:12)(cid:12) ρ , p = b (cid:19) . Choosing as an estimate the value a or b whose asso-ciated component of π tends towards 1 thus amounts tochoosing the parameter value that maximises the prob-ability of the measurement outcomes ( y , . . . , y T ). Such particle quantum filtering techniques ex-tend without di ffi culties to N records (indexed by n ∈{ , . . . N } ) of measurement outcomes, ( y ( n ) k ) k = ,..., T n withpossibly di ff erent lengths T n and initial conditions ρ ( n )0 .This extension consists in a concatenation of the N records into a single record (¯ y k ) k = ,..., T with T = P Nn = T n and(¯ y k ) k = ,..., T = (cid:18) y (1)1 , . . . , y (1) T , y (2)1 , . . ., y (2) T , . . . , y ( N )1 , . . . , y ( N ) T N (cid:19) This record can be associated to a single quantum tra-jectory of length T of form (2). First initialize at ρ (1)0 .Then for each k equal to T + . . . + T n − , ρ k + is reset to ρ ( n )0 . This can be done by applying a reset Kraus map K ρ ( n )0 after the computation of ρ k + relying on outcome y ( n − T n − and before using the outcome y ( n )1 . For any den-sity operator σ , it is simple to construct via its spec-tral decomposition, a Kraus map K σ such that, for alldensity operator ρ , K σ ( ρ ) = σ . With this trick (¯ y k ) isassociated to an e ff ective single quantum trajectory ofthe form (2) where the partial Kraus maps K py , k depende ff ectively on the time step k because of adding thesereset Kraus maps.For the particle quantum filter that is described intheorem 2 and associated to the record (¯ y k ) , each ρ ( p l ) k isreset in a similar way at each time step k = T + . . . + T n − contrarily to the parameter probability π ( p l ) k that is notreset.
3. Continuous-time formulation ff usive stochastic master equations For a mathematical and precise description of suchdi ff usive models, see [3]. We just recall here thestochastic master equation governing the time evolutionof the density operator t ρ t d ρ t = (cid:18) − i [ H , ρ t ] + m X ν = D ν ( ρ t ) (cid:19) d t + m X ν = √ η ν (cid:18) L ν ρ t + ρ t L † ν − Tr (cid:16) L ν ρ t + ρ t L † ν (cid:17) ρ t (cid:19) d W ν t (6)where H is the Hamiltonian, an Hermitian operator on H (¯ h = ν ∈ { , . . . , m } , • D ν is the Lindblad super-operator D ν ( ρ ) = L ν ρ L † ν − ( L † ν L ν ρ + ρ L † ν L ν ); • L ν is an operator on H , which is not necessarilyHermitian and which is associated to the measure-ment / decoherence channel ν ; • η ν ∈ [0 ,
1] is the detection e ffi ciency ( η ν = η ν > • W ν t is a Wiener process (independent of the otherWiener processes W µ , ν t ) describing the quantumfluctuations of the continuous output signal t y ν t .It is related to ρ t byd y ν t = √ η ν Tr (cid:16) L ν ρ t + ρ t L † ν (cid:17) d t + d W ν t . (7) We introduce here another formulation of (6) thatmimics the discrete-time formulation (2). This formu-lation is inspired of subsection 4.3.3 of [12], subsec-tion entitled ”Physical interpretation of the master equa-tion”. In (6), d ρ t stands for ρ t + d t − ρ t . It can thus bewritten as ρ t + d t = ρ t + (cid:18) − i [ H , ρ t ] + m X ν = D ν ( ρ t ) (cid:19) d t + m X ν = √ η ν (cid:18) L ν ρ t + ρ t L † ν − Tr (cid:16) L ν ρ t + ρ t L † ν (cid:17) ρ t (cid:19) d W ν t i.e., ρ t + d t is an algebraic expression involving ρ t , d t andd W ν t . With this form, it is not obvious that ρ t + d t remains density operator if ρ t is a density operator. The fol-lowing lemma provides another formulation based onIt¯o calculus showing directly that ρ t + d t remains a den-sity operator. In [20], similar formulations are proposedwithout the mathematical justifications given below andare tested in realistic simulations of measurement-basedfeedback scheme. Lemma 1.
Consider the stochastic di ff erential equa-tion (6) with an initial condition ρ , which is a non-negative Hermitian operator of trace one. Then it alsoreads: ρ t + d t = K dy t , dt ( ρ t ) Tr (cid:16) K dy t , dt ( ρ t ) (cid:17) , where dy t stands for ( dy t , . . . , dy mt ) , and where K ∆ y , ∆ t isa partial Kraus map depending on ∆ y ∈ R m and ∆ t > given by K ∆ y , ∆ t ( ρ ) = M ∆ y , ∆ t ρ M † ∆ y , ∆ t + m X ν = (1 − η ν ) ∆ t L ν ρ L † ν and M ∆ y , ∆ t is the following operator on H M ∆ y , ∆ t = I H − iH + m X ν = L † ν L ν / ∆ t + m X ν = √ η ν ∆ y ν L ν Proof.
Assume that m =
1. Then,d ρ t = (cid:18) − i [ H , ρ t ] + L ρ t L † − ( L † L ρ t + ρ t L † L ) (cid:19) d t + √ η (cid:18) L ρ t + ρ t L † − Tr (cid:16) L ρ t + ρ t L † (cid:17) ρ t (cid:19) d W t . (8)Using It¯o rules, d y t = d t . Hence, we have K d y t , d t ( ρ t ) = ρ t + √ η ( L ρ t + ρ t L † ) d y t + (cid:16) − i [ H , ρ t ] + L ρ t L † − ( L † L ρ t + ρ t L † L ) (cid:17) d t . Thus Tr (cid:16) K d y t , d t ( ρ t ) (cid:17) = + √ η Tr (cid:16) L ρ t + ρ t L † (cid:17) d y t and1Tr (cid:16) K d y t , d t ( ρ t ) (cid:17) = − √ η Tr (cid:16) L ρ t + ρ t L † (cid:17) d y t + η Tr (cid:16) L ρ t + ρ t L † (cid:17) d t . We get K d y t , d t ( ρ t )Tr (cid:16) K d y t , d t ( ρ t ) (cid:17) − ρ t = √ η (cid:18) L ρ t + ρ t L † − Tr (cid:16) L ρ t + ρ t L † (cid:17) ρ t (cid:19) d y t + (cid:16) − i [ H , ρ t ] + L ρ t L † − ( L † L ρ t + ρ t L † L ) (cid:17) d t − η Tr (cid:16) L ρ t + ρ t L † (cid:17) (cid:18) L ρ t + ρ t L † − Tr (cid:16) L ρ t + ρ t L † (cid:17) ρ t (cid:19) d t . One recognizes (8) since d y t − √ η Tr (cid:16) L ρ t + ρ t L † (cid:17) d t = d W t . For m >
1, the computations are similar and notdetailed here. (cid:3)
Assume the system dynamics depends on a con-stant parameter p appearing either in the SME (6)and / or in the output maps (7). As in section 2, assumethat p can take a finite number r of values p , . . . , p r .Denote by ρ pt the quantum state associated to p :d ρ pt = L p ( ρ pt ) d t + m X ν = M p ( ρ pt ) d W ν t (9)where the super-operators L p ( ρ ) = − i [ H p , ρ ] + m X ν = L p ν ρ ( L p ν ) † −
12 (( L p ν ) † L p ν ρ + ρ ( L p ν ) † L p ν )and M p ( ρ ) = q η p ν (cid:18) L p ν ρ + ρ ( L p ν ) † − Tr (cid:16) L p ν ρ + ρ ( L p ν ) † (cid:17) ρ (cid:19) depend on p since the operators L p ν and the e ffi ciencies η p ν could depend on p . The m outputs that are associatedto the parameter p then read: dy ν t = C p ν ( ρ pt )d t + d W ν t (10)for ν = , . . . , m , and where: C p ν ( ρ ) = q η p ν Tr (cid:16) L p ν ρ + ρ ( L p ν ) † (cid:17) . With these notations, the particle quantum filter in-troduced in [11] and further developed and analyzedin [8, 16] reads as follows. For each l ∈ { , . . . , r } , ρ p l t is governed by the quantum filter:d ρ p l t = L p l ( ρ p l t ) d t + m X ν = M p l ( ρ p l t ) (cid:16) dy ν t − C p l ν ( ρ p l t )d t (cid:17) , (11)and the parameter probability π p l t is governed by:d π p l t = π p l t m X ν = (cid:16) C p l ν ( ρ p l t ) − C ν t (cid:17) (cid:16) dy ν t − C ν t d t (cid:17) , (12)where C ν t = P rj = π p j t C p j ν ( ρ p j t ).Here again, the lemma below provides another for-mulation of this particle quantum filter that mimics thediscrete-time setting of theorem 2. emma 2. For each l ∈ { , . . . , r } , the particle quantumfilter (11) and (12) can be formulated as follows: ρ p l t + d t = K pldyt , dt ( ρ plt ) Tr (cid:18) K pldyt , dt ( ρ plt ) (cid:19) π p l t + d t = Tr (cid:18) K pldyt , dt ( ρ plt ) (cid:19) π plt P rj = Tr (cid:18) K pjdyt , dt ( ρ pjt ) (cid:19) π pjt where dy t stands for ( dy t , . . . , dy mt ) and where K p ∆ y , ∆ t is apartial Kraus map depending on p, ∆ y ∈ R m and ∆ t > given by: K p ∆ y , ∆ t ( ρ ) = M p ∆ y , ∆ t ρ (cid:18) M p ∆ y , ∆ t (cid:19) † + m X ν = (1 − η p ν ) ∆ t L p ν ρ ( L p ν ) † , and M p ∆ y , ∆ t is the following operator on H :M p ∆ y , ∆ t = I H − iH p + m X ν = ( L p ν ) † L p ν / ∆ t + m X ν = q η p ν ∆ y ν L p ν . The proof is very similar to the proof of lemma 1.It relies on simple but slightly tedious computations ex-ploiting It¯o calculus. Due to space limitation, this proofis not detailed here. This lemma, combined with themathematical machineries exploited in [1], opens theway to an extension to the di ff usive case of theorem 2.
4. An experimental validation
The estimation of the detection e ffi ciency is con-ducted on a superconducting qubit whose fluorescencefield is measured using a heterodyne detector [18, 13].For the detailed physics of this experiment, see [5, 6].The Hilbert space H is C . The system dynamics is de-scribed by a stochastic master equation of the form (6),with m = η = η = η is the total e ffi ciency of the het-erodyne measurement of the fluorescence signal; η = L = q T X − iY , L = iL , L = q T φ Z where X , Y and Z are the usual Pauli matrices [17].The time constants T = . µ s and T φ = µ s aredetermined independently using Rabi or Ramsey pro-tocols, which is not the case of η . Using a calibrationof the average resonance fluorescence signal, the mea-sured vacuum noise fluctuations provide a first estima-tion of η = . ± . η , we have mea-sured N = × quantum trajectories of 10 µ s, startingfrom the same known initial state ρ = I H + X . The sam-pling time ∆ t is equal to 0 . µ s. For each trajectory, π t ( i ) η =0.1 η =0.26 η =0.4 Figure 1: First estimation, with pattern values η = . η = .
26 close to η , and η = . η ≈ .
26 and discard 0 .
10 and 0 . t k = k ∆ t , k ∈ { , . . ., } ,corresponds to the two quadratures of the fluorescencefield ∆ y k = y k ∆ t − y k − ∆ t and ∆ y k = y k ∆ t − y k − ∆ t . Fromlemma 2, we derive a simple recursive algorithm where(d y t ) and d t are replaced by ( ∆ y k ) and ∆ t . Moreover, asexplained in subsection 2.4, the 3 × quantum trajec-tories are concatenated into a single one.The estimation is done by taking some pattern val-ues η , η , ..., η r , assuming that the real value η is su ffi -ciently close to one of them. We begin with a first esti-mation that keeps a big interval between each possiblevalue η i of η , in order to validate our estimation scheme.We then sharpen this estimation by reducing the inter-vals between each value η i , until arriving to a level ofaccuracy after which no distinct discrimination can beperformed. The results are given at figures 1, 2 and 3.They give the following refinement of the initial cali-bration: η = . ± . π η i k and the Y-axisdisplays these probabilities.
5. Conclusion
We have shown that particle quantum filtering isalways a stable process. We have proposed an origi-nal positivity preserving formulation for systems gov-erned by di ff usive stochastic master equation. A firstvalidation on experimental data confirms the interest ofthe resulting parameter algorithm. This positivity pre-serving algorithm appears to be robust enough to copewith sampling time of more than 2% of the characteris-tic time attached to the measurement. The convergence π t ( i ) η =0.22 η =0.24 η =0.26 η =0.28 Figure 2: Second estimation, realized with more nar-row intervals between each pattern values. We noticethat η is actually closer to 0 .
24 than 0 .
26, the calibratedvalue, and that the number of trajectories required forthe discrimination has drastically increased to 1 × . π t ( i ) η =0.23 η =0.235 η =0.24 η =0.245 η =0.25 Figure 3: Last estimation, with very narrow intervals.We use all the trajectories available, i.e. 3 × tra-jectories. Filter does not converge to a distinct choicebetween 0 .
240 and 0 . Acknowledgment
The authors thank Michel Brune, Igor Dotsenkoand Jean-Michel Raimond for useful discussions onquantum filtering and parameter estimation in thediscrete-time case.
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