Hadis Amini
email: nhamini@stanford.edu

Clément Pellegrini
email: clement.pellegrini@math.univ-toulouse.fr.

Pierre Rouchon
email: pierre.rouchon@mines-paristech.fr.

Stability of continuous-time quantum filters with measurement imperfections

Keywords: Quantum filtering, stability, continuous-time stochastic master equations, Wiener process, Poisson process, quantum trajectories, measurement errors

The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modeled by a left stochastic matrix.

Introduction

Since the work of Davies [START_REF] Davies | Quantum theory of open systems[END_REF], the time evolution of the state (density operator) ρ t at time t of an observed quantum system can be described by a Stochastic Master Equation (SME) taking into account the back-action of the measurements on ρ t . Such SMEs (see [START_REF] Barchielli | Quantum Trajectories and Measurements in Continuous Time: the Diffusive Case[END_REF] for a modern exposure) have been the starting point of the seminal contributions of Belavkin to quantum filtering and control [START_REF] Belavkin | Quantum filtering of Markov signals with white quantum noise[END_REF][START_REF] Belavkin | Quantum stochastic calculus and quantum nonlinear filtering[END_REF][START_REF] Barchielli | Measurements continuous in time and a posteriori states in quantum mechanics[END_REF][START_REF] Belavkin | Eventum mechanics of quantum trajectories: Continual measurements, quantum predictions and feedback control[END_REF]. Quantum filters are used to get an estimate ρ e t of ρ t based on an initial guess ρ e 0 of ρ 0 and on the measurement outcomes between 0 and t. Quantum filtering is related to quantum trajectories [START_REF] Dalibard | Wave-function approach to dissipative processes in quantum optics[END_REF][START_REF] Carmichael | An Open Systems Approach to Quantum Optics[END_REF] and their original motivations for Monte-Carlo simulations. Roughly speaking, quantum filtering replaces the random numbers used at each time-step of a Monte-Carlo simulation by the measurement outcomes to update the estimate ρ e t . An important practical issue deals with the convergence: does ρ e t converge towards ρ t when t tends to +∞, even if ρ e 0 = ρ 0 ? Few convergence results are available up to now, except the sufficient conditions established in [START_REF] Van Handel | Filtering, Stability, and Robustness[END_REF][START_REF] Van Handel | The stability of quantum Markov filters[END_REF] for diffusive SMEs. Particular results for quantum non demolition indirect measurements have been established in [START_REF] Benoist | Repeated quantum non-demolition measurements: Convergence and continuous time limit[END_REF][START_REF] Benoist | Large time behavior and convergence rate for quantum filters under standard non demolition conditions[END_REF] As far as we know, general and checkable necessary and sufficient convergence conditions do not exist yet.

In this paper we do not investigate directly convergence issues. We focus on stability issues to ensure the absence of time divergence. We prove in Theorem 5 that for a large class of continuous-time SMEs driven simultaneously by Wiener and Poisson processes, associated quantum filters are stable: the fidelity between ρ and its estimate ρ e , dt driven by the the Poisson processes N µ (t) with dN µ (t

) = θ µ + ν η µ,ν Tr V ν ρ t V † ν dt,
where the detection imperfections are modeled through the parameters θ µ ≥ 0 and η µ,ν ≥ 0 with µ η µ,ν ≤ 1.

The proof of Theorem 5 is based on discrete-time approximations of continuous-time SMEs. Such approximations have already been investigated in [START_REF] Attal | From repeated to continuous quantum interactions[END_REF][START_REF] Gough | Stochastic Schrödinger equations as limit of discrete filtering[END_REF][START_REF] Pellegrini | Existence, uniqueness and approximation for stochastic Schrödinger equation: the poisson case[END_REF][START_REF] Pellegrini | Existence, uniqueness and approximation of a stochastic Schrödinger equation: the diffusive case[END_REF]. They rely on indirect measurements, originally introduced in [START_REF] Braginsky | Quantum Measurement[END_REF] and well explained with suggestive physical systems in [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities and Photons[END_REF][START_REF] Wiseman | Quantum Measurement and Control[END_REF]. For discrete-time SMEs, it is proved in [START_REF] Rouchon | Fidelity is a sub-martingale for discrete-time quantum filters[END_REF][START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF] that the fidelity between the quantum state and its estimate is always a submartingale. Theorem 5 is obtained by passing to the limit from discrete to continuous time. The fidelity between the quantum state and its estimate remains a submartingale. The obtained continuous-time SMEs are slightly more general than the ones usually encountered in the literature. Such SMEs could be of some interest to derive quantum filters taking into account a larger class of incompleteness and errors in measurements and jump detections.

This paper is structured in two main sections. Section 2 is devoted to Theorem 2, a restrictive version of Theorem 5 to the diffusive SMEs with perfect measurements. For this simplified but representative case, the discrete-time approximation is presented and the passage to the continuous-time limit is detailed during the proof of Proposition 1. Section 3 is devoted to Theorem 5 and fully exploits the tools and methods developed in Section 2. We consider SMEs driven simultaneously by Poisson and Wiener processes. We recall first the structure, described in [START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF] and based on a left stochastic matrix, of discrete-time SMEs associated to imperfect measurements. We apply on the discrete-time approximations such left stochastic matrix modeling of imperfections and errors. Then we take, thanks to Theorem 4, the limit to get the continuous-time SMEs and its associated quantum filters with imperfections. Their structures are more general than the usual ones encountered in the literature. This leads to Theorem 5 ensuring the stability of the obtained quantum filters. Section 4 is a short conclusion proposing some connection with Petz characterization of monotone metrics on matrix spaces. Some intermediate and partial results related to Theorems 5 can be founded in [START_REF] Amini | On stability of continuous-time quantumfilters[END_REF][START_REF] Amini | Stabilization of Discrete-Time Quantum Systems and Stability of Continuous-Time Quantum Filters[END_REF].

Perfect measurements

Let us start this section by presenting the jump-diffusive SMEs describing the evolutions of quantum state ρ t and its estimate ρ e t .

Continuous-time filters

We consider quantum systems of finite dimensions 1 < N < ∞. The state space of such a system is given by the set of density matrices

D := {ρ ∈ C N ×N | ρ = ρ † , Tr (ρ) = 1, ρ ≥ 0}.
Formally a real quantum trajectory ρ ∈ D in Schrödinger picture can be described by the following SME (cf. [START_REF] Belavkin | Quantum stochastic calculus and quantum nonlinear filtering[END_REF][START_REF] Belavkin | Eventum mechanics of quantum trajectories: Continual measurements, quantum predictions and feedback control[END_REF][START_REF] Barchielli | Quantum Trajectories and Measurements in Continuous Time: the Diffusive Case[END_REF])

dρ t =   -i[H, ρ t ] + ξ L ξ (ρ t )   dt+ ν Λ ν (ρ t )dW ν (t)+ µ Υ µ (ρ t ) dN µ (t) -Tr V µ ρ t V † µ dt , (2)
where

• the notation [A, B] refers to AB -BA;

• H = H † is a Hermitian operator corresponding to the total Hamiltonian of the system;

• dN µ are the Poisson processes with µ ∈ {1, • • • , m P } and dW ν are the Wiener processes with ν

∈ {m P + 1, • • • , m P + m W }; • The Lindblad superoperator L ξ (ξ ∈ {1, . . . , m W + m P }) is defined by L ξ (ρ) := V ξ ρV † ξ - 1 2 (V † ξ V ξ ρ + ρV † ξ V ξ ), (3)
where V ξ is an arbitrary matrix which determines the measurement process (typically the coupling to the probe field for quantum optic systems);

• The superoperators Λ ν and Υ µ are defined respectively by

Λ ν (ρ) := V ν ρ + ρV † ν -Tr (V ν + V † ν )ρ ρ and Υ µ (ρ) := V µ ρV † µ Tr V µ ρV † µ -ρ;
• The measurement outcomes are dN µ and dy ν where

N µ (t) - t 0 Tr V µ ρ s V † µ ds is a martingale and dy ν (t) = dW ν (t) + Tr V ν ρ t V † ν dt. (4)
All the developments remain valid when H and V ξ are deterministic time-varying matrices.

For clarity sake, we do not recall below such possible time dependence.

In this paper, the notation ρ e corresponds to the estimate filter associated to the filter ρ. This estimate filter is provided from the measurement outcomes dy ν and dN µ and depends on the real quantum trajectory ρ via the measurement outcomes (4). It has the following expression derived from [START_REF] Amini | On stability of continuous-time quantumfilters[END_REF] where dW ν (t) is replaced by dy ν (t) -Tr V ν ρ e t V † ν dt (see e.g., [START_REF] Belavkin | Eventum mechanics of quantum trajectories: Continual measurements, quantum predictions and feedback control[END_REF][START_REF] Barchielli | Quantum Trajectories and Measurements in Continuous Time: the Diffusive Case[END_REF]):

dρ e t =   -i[H, ρ e t ] + ξ L ξ (ρ e t )   dt + ν Λ ν (ρ e t ) dy ν (t) -Tr V ν ρ e t V † ν dt + µ Υ µ (ρ e t ) dN µ (t) -Tr V µ ρ e t V † µ dt . (5)
When ρ e 0 = ρ 0 , ρ e t and ρ t do not coincide in general. However, we will see that the fidelity between ρ e t and ρ t is a sub-martingale.

Discrete-time filters

First let us briefly remind the model of quantum repeated indirect measurement approach.

The physics underlying such approach is well explained in [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities and Photons[END_REF][START_REF] Wiseman | Quantum Measurement and Control[END_REF]. In [START_REF] Attal | From repeated to continuous quantum interactions[END_REF][START_REF] Gough | Stochastic Schrödinger equations as limit of discrete filtering[END_REF][START_REF] Pellegrini | Markov chains approximation of jump-diffusion stochastic master equations[END_REF], it was rigorously shown that such discrete-time approximations associated to the real state converges to the continuous model described in Equation (2).

Quantum repeated measurement approach

We consider the setup of quantum repeated interaction of H (describing the Hilbert space of the system state) with an infinite chain k K k (describing the Hilbert space of the environment) with K k = K for all k. More precisely, the first copy K 1 = K interacts with H during a time δ and then disappear. Next, the second copy K 2 comes to interact with H and so on.

This setting is coupled with indirect measurement, that is, after each interaction between H and K, a measurement of an observable of K is performed.

Here, we assume that H = C N and the environment (meter system) is composed of m P + m W qubits, so K = (C 2 ) ⊗m P +m W . Also, take |ψ as the initial state of the system and |G as the initial state of the qubits' meter which is defined as the ground state of the meter system: all qubits in the ground state |g . As a result, the initial state of the system coupled to its environment is described by |ψ ⊗ |G .

For δ = 1/n with n large, assume that the Schrödinger evolution between time 0 to time 1/n is given by

H tot = H ⊗ I + √ n ξ iV ξ ⊗ σ + ξ -iV † ξ ⊗ σ -ξ , (6)
where

• H is the Hamiltonian of the system used in (2) and the operators V ξ are those appearing in the Lindblad superoperators (3)

• σ + ξ = (|e g|) ξ and σ -ξ = σ † + ξ = (|g e|) ξ where the notation (A) ξ is an operator K defined by (A) ξ = 1≤i<ξ I ⊗ A ⊗ ξ<i≤m W +m P I. Note that σ -ξ |G = 0.
In the sequel, symbol ⊗ will be remove for compact formulae.

Take the observables

X ν = (|g e|) ν + (|e g|) ν for ν ∈ {m P + 1, • • • , m W + m P } and Z µ = (|e e|) µ -(|g g|) µ for µ ∈ {1, • • • , m P }.
For n large, the measurement of all qubits at final time t = 1/n according to the observables X ν and Z µ , yields an approximation for the Wiener and Poisson processes, respectively. This results from a development versus 1/n of the measurement operators associated to the associated discrete-time stochastic evolution.

We have e -iHtot/n = I -i n H tot -1 2n 2 H 2 tot + O(1/n 3/2 ), where I is the identity operator. Now replacing H tot by its expression given in [START_REF] Benoist | Repeated quantum non-demolition measurements: Convergence and continuous time limit[END_REF], we find

e -iHtot/n ≈ I + 1 √ n ξ V ξ σ + ξ -V † ξ σ -ξ -i n H -1 2n ξ V † ξ V ξ P g ξ + V ξ V † ξ P e ξ ,
where ≈ means up to O(1/n 3/2 ) terms. Here, P g ξ = (|g g|) ξ and P e ξ = (|e e|) ξ . Note that

P g ξ |G = |G
and P e ξ |G = 0.

Thus the system coupled to its environment evolves as follows

e -iHtot/n |ψ ⊗ |G ≈   I -i n H -1 2n ξ V † ξ V ξ   |ψ ⊗ |G + 1 √ n ξ V ξ |ψ ⊗ |G ξ .
Let us consider the qubits measurements. The measurement outcomes for X ν are stored in x ν-m P ∈ {-1, 1}, x = (x ν-m P ) ∈ {-1, 1} m W , and those of Z µ in z µ ∈ {0, 1}, z = (z µ ) ∈ {0, 1} m P , as follows (all these observables commute):

• if during the measure of X ν , the corresponding qubit collapses to (|g + |e )/ √ 2 (resp. (|g -|e )/ √ 2) set x ν-m P = +1 (resp. x ν-m P = -1).

• if during the measure of Z µ , the corresponding qubit collapses to |g (resp.|e ), set z µ = 0 (resp. z µ = 1).

The probability to get, for two different µ and µ ′ , z µ = z µ ′ = 1, is in order of O(1/n 3/2 ). Thus for z, we need only consider the following cases: either all z µ are equal to 0 or only a single one is equal to 1, the other ones being 0.

Consider the measurement outcomes z = (z µ ) and x = (x ν-m P ). The associated wave packet collapse of e -iHtot/n |ψ ⊗ |G yields the un-normalized state M x,z |ψ ⊗ |x, z , where the measurement operator is denoted by M x,z and where |x, z is the normalized state of the qubits characterized by X ν |x, z = x ν-m P |x, z and Z µ |x, z = (2z µ -1) |x, z . We have to consider two situations: either z µ = 0 denoted by z = 0 or µ z µ = 1. When z µ = 0 for all µ, some computations yield

M x,0 ≈ ν 1 √ 2   I -i n H -1 2n ξ V † ξ V ξ + 1 √ n ν x ν-m P V ν   . (7)
When z µ = 1 and z µ ′ = 0 for all µ ′ = µ, similar computations give

M x,µ ≈ ν 1 √ 2 V µ √ n (8)
where we have denoted M x,z by M x,µ for such z.

For an arbitrary state ρ at t = 0, not necessarily pure as ρ = |ψ ψ|, the state ρ 1 at time t = 1/n is given by

ρ 1 = M x,z ρM † x,z p x,z (ρ) , (9)
which happens with probability p x,z (ρ) = Tr M x,z ρM † x,z . The expression of ρ 1 is obtained by neglecting the terms of orders strictly greater than 1 versus 1/n. Some usual calculations yield

M x,0 ρM † x,0 = c W ρ + 1 √ n ν x ν-m P (V ν ρ + ρV † ν ) + 1 n -i[H, ρ] - µ {ρ,V † µ Vµ} 2 + ν L ν (ρ) (10)
with probability

p x,0 (ρ) = c W 1 + 1 √ n ν x ν-m P Tr (V ν + V † ν )ρ -1 n µ Tr ρV † µ V µ . ( 11
)
The notation {A, B} used in [START_REF] Benoist | Large time behavior and convergence rate for quantum filters under standard non demolition conditions[END_REF] corresponds to AB + BA and c W = ( 1 2 ) m W . Similarly, we have

M x,µ ρM † x,µ = c W n V µ ρV † µ , with probability p x,µ (ρ) = c W n Tr V µ ρV † µ . (12)
Now we can obtain the asymptotic description of the transition for all possible observations. Indeed, the expression of the whole quantum trajectory ρ k can be obtained by replacing ρ 1 by ρ k+1 and ρ by ρ k in Equation ( 9). As a conclusion, we find

ρ k+1 = M x,µ ρ k M † x,µ Tr M x,µ ρ k M † x,µ with probability p x,µ (ρ k ) (13) (14)
where p x,µ (ρ k ) is given by [START_REF] Carmichael | An Open Systems Approach to Quantum Optics[END_REF][START_REF] Braginsky | Quantum Measurement[END_REF].

Stability with respect to initial condition

Consider the Markov chain [START_REF] Dalibard | Wave-function approach to dissipative processes in quantum optics[END_REF]. Assume that we do not know precisely the initial state ρ 0 and we have at our disposal an estimate ρ e 0 . Assume also that we know the measurement result at step k

(x k , µ k ) ∈ {-1, 1} m W × {0, 1}.
It is then natural to consider the following recursive update of our estimation ρ e k+1 using the knowledge of measurement result at step k and the previous estimate ρ e k (see e.g., [START_REF] Wiseman | Quantum Measurement and Control[END_REF]):

ρ e k+1 = M x k ,µ k ρ e k M † x k ,µ k Tr M x k ,µ k ρ e k M † x k ,µ k (15)
Note that the probability p x k ,µ k (ρ k ) given by [START_REF] Braginsky | Quantum Measurement[END_REF][START_REF] Carmichael | An Open Systems Approach to Quantum Optics[END_REF] to get (x k , µ k ) depends on the hidden state ρ k and not on ρ e k .

Remark 1. Let us stress that the above description is not always valid. Indeed, the normalization Tr M x k ,µ k ρ e k M † x k ,µ k can vanish and the formula (15) is then not defined. This problem does not appear when describing the true evolution since the normalization describing the true state corresponds to the probability of apparition (then if this vanishes this means that the corresponding state can not appear). This problem of non-definition for the discrete-time estimate filter is related to the problem underlined in the definition of [START_REF] Barchielli | Quantum Trajectories and Measurements in Continuous Time: the Diffusive Case[END_REF]. In general, this question has been taken into account in [START_REF] Rouchon | Fidelity is a sub-martingale for discrete-time quantum filters[END_REF][START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF] for the discrete-time filter. In our context focusing on asymptotic evolution, such a problem will not appear.

Theorem 1 ([21]

). Consider any arbitrary Markov chain (ρ k , ρ e k ) satisfying respectively Equations ( 13) and ( 15):

(ρ k+1 , ρ e k+1 ) = Mx,µρ k M † x,µ Tr Mx,µρ k M † x,µ , Mx,µρ e k M † x,µ Tr Mx,µρ e k M † x,µ with probability p x,µ (ρ k )
where p x,µ (ρ k ) is given by ( 12) and [START_REF] Braginsky | Quantum Measurement[END_REF]. Then the fidelity (F (ρ k , ρ e k )) defined in Equation ( 1) is a (F k ) submartingale where

F k = σ{(ρ l , ρ e l )|l ≤ k)}.
In particular, we have

E (F (ρ l , ρ e l )|F k ) = E (F (ρ l , ρ e l )|(ρ k , ρ e k )) ≥ F (ρ k , ρ e k ),
for all l > k.

Stability of continuous-time filters

From the description of the quantum trajectory (ρ k ) and its associated quantum filter (ρ e k ) with the time parameter n, we can define the associated continuous-time stochastic processes denoted by (ρ n (t)) and (ρ e n (t)) with

ρ n (t) = ρ [nt] , ρ e n (t) = ρ e [nt]
.

It is clear that (ρ n , ρ e n ) is a Markov process as (ρ k , ρ e k ) is a Markov chain. Before announcing the main result of this section, first let us show the following proposition.

Proposition 1. The Markov process (ρ n (t), ρ e n (t)) with ρ n and ρ e n satisfying respectively Equations ( 13) and (15) converges in distribution, when n goes to infinity, to the Markov process (ρ t , ρ e t ) with ρ t and ρ e t satisfying respectively Equations (2) and (5).

Proof. The approach to prove this proposition is usual in probability theory: we show the convergence of the generator associated to the Markov process (ρ n , ρ e n ) towards the one associated to the Markov process (ρ t , ρ e t ). Also, we need to prove the tightness property of the sequence (ρ n , ρ e n ). The tightness of the sequence (ρ n (t), ρ e n (t)) is guaranteed if, for any T > 0, exists M > 0 such that for all 0

≤ t 1 ≤ t 2 ≤ T and ∀t ∈ [t 1 , t 2 ], E (ρ n (t 2 ), ρ e n (t 2 )) -(ρ n (t), ρ e n (t)) 2 (ρ n (t), ρ e n (t)) -(ρ n (t 1 ), ρ e n (t 1 )) 2 ≤ M (t 2 -t 1 ) 2 .
We do not develop the arguments for showing the tightness property. They can be obtained directly with the same arguments as [START_REF] Pellegrini | Markov chains approximation of jump-diffusion stochastic master equations[END_REF].

Thus here, we just focus on the convergence of the generators. Take A n and A as the generators associated respectively to the sequences (ρ n , ρ e n ) and (ρ, ρ e ). We have to prove that, for any

C 2 -real valued function (ρ, ρ e ) → f (ρ, ρ e ), lim n→+∞ sup (ρ,ρ e )∈D 2 |A n f (ρ, ρ e ) -Af (ρ, ρ e )| = 0.
We have the following expression for the generator Af (ρ, ρ e )

Af (ρ, ρ e ) = D (ρ,ρ e ) f.   -i[H, ρ] + ξ L ξ (ρ), -i[H, ρ e ] + ξ L ξ (ρ e ) + K(ρ, ρ e )   + 1 2 ν D 2 (ρ,ρ e ) f. (Λ ν (ρ), Λ ν (ρ e ); Λ ν (ρ), Λ ν (ρ e )) + µ Tr V µ ρV † µ f VµρV † µ Tr VµρV † µ , Vµρ e V † µ Tr Vµρ e V † µ -f (ρ, ρ e ) -D (ρ,ρ e ) f. (Υ µ (ρ), Υ µ (ρ e )) , (16)
where

K(ρ, ρ e ) := µ Υ µ (ρ e ) Tr V µ ρV † µ -Tr V µ ρ e V † µ + ν Λ ν (ρ e ) Tr V ν ρV † ν -Tr V ν ρ e V † ν
Now let us calculate the expression of A n . We have, up to O(n -1/2 ) terms,

A n f (ρ, ρ e ) ≈ x np x,0 (ρ) f M x,0 ρM † x,0 Tr M x,0 ρM † x,0 , M x,0 ρ e M † x,0 Tr M x,0 ρ e M † x,0 -f (ρ, ρ e ) + x,µ np x,µ (ρ) f Mx,µρM † x,µ Tr Mx,µρM † x,µ , Mx,µρ e M † x,µ Tr Mx,µρ e M † x,µ -f (ρ, ρ e ) .
Since

M x,0 ρM † x,0 p x,0 (ρ) = ρ + O(n -1/2
) and also, by using Taylor's formula, we have

f M x,0 ρM † x,0 p x,0 (ρ) , M x,0 ρ e M † x,0 p x,0 (ρ e ) -f (ρ, ρ e ) = D (ρ,ρ e ) f • M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M † x,0 p x,0 (ρ e ) -ρ e + 1 2 D 2 (ρ,ρ e ) f • M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M † x,0 p x,0 (ρ e ) -ρ e ; M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M †
x,0 p x,0 (ρ e ) ρ e +O(n -3/2 ).

Consequently,

n x p x,0 (ρ) f M x,0 ρM † x,0 p x,0 (ρ) , M x,0 ρ e M † x,0 p x,0 (ρ e ) -f (ρ, ρ e ) = nD (ρ,ρ e ) f • x M x,0 ρM † x,0 -p x,0 (ρ)ρ, x M x,0 ρ e M † x,0 -p x,0 (ρ e )ρ e p x,0(ρ) p x,0(ρ e ) + 1 2 x np x,0 (ρ)D 2 (ρ,ρ e ) f • M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M † x,0 p x,0 (ρ e ) -ρ e ; M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M † x,0 p x,0 (ρ e ) -ρ e + O(n -1/2 ), (17)
where according to [START_REF] Benoist | Large time behavior and convergence rate for quantum filters under standard non demolition conditions[END_REF] and [START_REF] Braginsky | Quantum Measurement[END_REF], up to O(n -1/2 ) terms, we have

n x M x,0 ρM † x,0 -p x,0 (ρ)ρ ≈ -i[H, ρ] + µ Tr ρV † µ V µ ρ - {ρ,V † µ Vµ} 2 + ν L ν (ρ) = -i[H, ρ] + ξ L ξ (ρ) - µ Υ µ (ρ)Tr V µ ρV † µ , (18)
since, for any ν, x x ν-m P Λ ν (ρ) = 0 and x c W = 1. Also, we have

M x,0 ρ e M † x,0 -p x,0 (ρ e )ρ e = c W   1 √ n ν x ν-m P Λ ν (ρ e ) + 1 n -i[H, ρ e ] + ξ L ξ (ρ e ) - µ Υ µ (ρ e )Tr V µ ρ e V † µ   .
Therefore, we find

n x M x,0 ρ e M † x,0 -p x,0 (ρ e )ρ e p x,0 (ρ) p x,0 (ρ e ) = -i[H, ρ e ] + ξ L ξ (ρ e ) - µ Υ µ (ρ e )Tr V µ ρ e V † µ + ν Λ ν (ρ e ) Tr V ν ρV † ν -Tr V ν ρ e V † ν + O(n -1 2 ) (19) since p x,0 (ρ) p x,0 (ρ e ) = 1 + 1 √ n ν x ν-m P Tr (V ν + V † ν )ρ -Tr (V ν + V † ν )ρ e + O(n -1/2
). Moreover, we have used the fact that

x ν c W x ν-m P Λ ν (ρ e ) ν ′ x ν ′ -m P Tr (V ν ′ + V † ν ′ )ρ -Tr (V ν ′ + V † ν ′ )ρ e = ν Λ ν (ρ e ) Tr (V ν + V † ν )ρ -Tr (V ν + V † ν )ρ e , since x 2 ν-m P = 1, x c W x 2 ν-m P = 1 and ∀ν = ν ′ : x c W x ν-m P x ν ′ -m P Λ ν (ρ e ) Tr (V ν ′ + V † ν ′ )ρ -Tr (V ν ′ + V † ν ′ )ρ e = 0.
Equation ( 19) can be rewritten as follows

n x M x,0 ρ e M † x,0 -p x,0 (ρ e )ρ e p x,0 (ρ) p x,0 (ρ e ) = -i[H, ρ e ] + ξ L ξ (ρ e ) + K(ρ, ρ e ) - µ Υ µ (ρ e )Tr V µ ρV † µ + O(n -1 2 ).
As a result, the first righthand side term in Equation ( 17) can be written as follows

nD (ρ,ρ e ) f • x M x,0 ρM † x,0 -p x,0 (ρ)ρ, x M x,0 ρ e M † x,0 -p x,0 (ρ e )ρ e p x,0(ρ) p x,0(ρ e ) = D (ρ,ρ e ) f.   -i[H, ρ] + ξ L ξ (ρ), -i[H, ρ e ] + ξ L ξ (ρ e ) + K(ρ, ρ e )   - µ Tr V µ ρV † µ D (ρ,ρ e ) f • Υ µ (ρ), Υ µ (ρ e ) + O(n -1/2 ).
Now let us calculate the second righthand side term in Equation [START_REF] Pellegrini | Existence, uniqueness and approximation for stochastic Schrödinger equation: the poisson case[END_REF]. To get the zero order

terms of x np x,0 (ρ)D 2 (ρ,ρ e ) f • M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M † x,0 p x,0 (ρ e ) -ρ e ; M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M †
x,0 p x,0 (ρ e )ρ e , we just need to combine the terms of order n -1/2 in M x,0 ρM † x,0 p x,0 (ρ) -ρ and in M x,0 ρM † x,0 -p x,0 (ρ)ρ. Since

M x,0 ρM † x,0 -p x,0 (ρ)ρ = c W √ n ν x ν-m P Λ ν (ρ) + O(n -1/2 )
and

M x,0 ρM † x,0 p x,0 (ρ) -ρ ≈ ρ + 1 √ n ν x ν-m P (V ν ρ + ρV † ν ) 1 + 1 √ n ν x ν-m P Tr (V ν + V † ν )ρ -ρ ≈ 1 √ n ν x ν-m P Λ ν (ρ) +O(n -1/2 ),
we get, up to O(n -1/2 ) terms

x np x,0 (ρ)D 2 (ρ,ρ e ) f • M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M † x,0 p x,0 (ρ e ) -ρ e ; M x,0 ρM † x,0 p x,0 (ρ) -ρ, M x,0 ρ e M † x,0 p x,0 (ρ e ) -ρ e ≈ x,ν,ν ′ c W x ν-m P x ν ′ D 2 (ρ,ρ e ) f •(Λ ν (ρ), Λ ν (ρ e ); Λ ν ′ (ρ), Λ ν ′ (ρ e )) = ν D 2 (ρ,ρ e ) f •(Λ ν (ρ), Λ ν (ρ e ); Λ ν (ρ), Λ ν (ρ e )) ,
where for the above equality, we have used the following facts

x 2 ν-m P = 1, x c W = 1 and ∀ν = ν ′ : x c W x ν-m P x ν ′ -m P D 2 ρ f (ρ)•(Λ ν (ρ), Λ ν ′ (ρ)) = 0.
Thus, we find

x p x,0 (ρ) f M x,0 ρM † x,0 p x,0 (ρ) , M x,0 ρ e M † x,0 p x,0 (ρ e ) -f (ρ, ρ e ) = D (ρ,ρ e ) f.   -i[H, ρ] + ξ L ξ (ρ), -i[H, ρ e ] + ξ L ξ (ρ e ) + K(ρ, ρ e )   - µ Tr V µ ρV † µ D (ρ,ρ e ) f • Υ µ (ρ), Υ µ (ρ e ) . + 1 2 ν D 2 ρ f (ρ) • (Λ ν (ρ), Λ ν (ρ e ); Λ ν (ρ), Λ ν (ρ e )) + O(n -1/2 ). ( 20
)
According to [START_REF] Carmichael | An Open Systems Approach to Quantum Optics[END_REF], we have also

x,µ np x,µ (ρ) f Mx,µρM † x,µ Tr Mx,µρM † x,µ , Mx,µρ e M † x,µ Tr Mx,µρ e M † x,µ -f (ρ, ρ e ) = µ Tr V µ ρV † µ f VµρV † µ Tr VµρV † µ , Vµρ e V † µ Tr Vµρ e V † µ -f (ρ, ρ e ) . (21)
Finally, by Equations ( 20) and ( 21), we find that

A n f (ρ, ρ e ) = Af (ρ, ρ e ) + O(n -1/2 ),
which finishes the proof of Proposition 1.

Now we are in the state to announce the main result of this section.

Theorem 2. Consider the Markov process (ρ t , ρ e t ) satisfying respectively Equations (2) and (5). Then the fidelity (F (ρ t , ρ e t )) defined in Equation (1) is a (F t )-submartingale, where F t = σ{(ρ τ , ρ e τ )|τ ≤ t}. In particular, we have,

E (F (ρ τ , ρ e τ )|F t ) = E (F (ρ τ , ρ e τ )|(ρ t , ρ e t )) ≥ F (ρ t , ρ e t )
, for all τ ≥ t.

Proof. By Theorem 1, we know that the fidelity F (ρ n , ρ e n ) is a submartingale with respect to the natural filtration of (ρ k , ρ e k ). In terms of (ρ n (t), ρ e n (t), it follows that

E (F (ρ n (τ ), ρ e n (τ ))|(ρ n (t), ρ e n (t))) ≥ F (ρ n (t), ρ e n (t)), (22)
for all τ ≥ t. In Proposition 1, we showed that (ρ n (t), ρ e n (t)) converges in distribution to (ρ t , ρ e t ). This implies that for any continuous real function f (ρ, ρ e ) we have ∀τ ≥ t,

lim n→+∞ E (f (ρ n (τ ), ρ e n (τ ))|(ρ n (t), ρ e n (t)) = E (f (ρ τ , ρ e τ )|(ρ t , ρ e t )) .
As F is continuous, the limit of [START_REF] Sayrin | Real-time quantum feedback prepares and stabilizes photon number states[END_REF] for n tending to ∞ yields E (F (ρ τ , ρ e τ )|(ρ t , ρ e t )) ≥ F (ρ t , ρ e t ) for all τ ≥ t. Since (ρ t , ρ e t ) is a Markov process, the result follows.

Imperfect measurements

By imperfect measurements, we mean both unread measurements performed by the environment (decoherence) and active measurements performed by non-ideal detectors. Starting from the discrete-time case experimentally used in [START_REF] Sayrin | Real-time quantum feedback prepares and stabilizes photon number states[END_REF] and detailed in [START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF], we derive the continuous-time optimal filters driven either by Poisson, Wiener processes or both of them. We obtain the SMEs and their associated quantum filters.

Discrete-time filters

The presentation here is very much inspired from [START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF] and adapted to [START_REF] Dalibard | Wave-function approach to dissipative processes in quantum optics[END_REF] for n large. The jump-events labelled by (x, z) are of type (x, µ) with µ ∈ {0, 1, • • • , m P }. The effectively measured events are labelled by s ∈ {1, . . . , m}, with m denoting the number of distinct experimental detector outcomes. Suppose that we know the correlation between the jumpevents (x, µ) and the experimental detection s. These correlations are modeled here by classical probabilities through a stochastic matrix η: η n s,(x,µ) that gives the probability of experimental detection s knowing that the effective jump-event is (x, µ). Since η n s,(x,µ) ≥ 0 and for each (x, µ), m s=1 η n s,(x,µ) = 1, the matrix η n = (η n s,(x,µ) ) is a left stochastic matrix. We assume the following asymptotic for η n ,

η n = η ∞ + η n + O(n -2 ), (23)
where η ∞ is a left stochastic matrix. We still denote by ρ the state associated to these experimental detections which is the best estimation of the system state knowing the initial state and all previous experimental detections. Following [START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF], it obeys to the following Markov process:

ρ k+1 = M n s (ρ k ) Tr (M n s (ρ k )) with probability p s (ρ k ) = Tr (M n s (ρ k )) ( 24
)
where M n s (ρ)

x,µ η n s,(x,µ) M x,µ ρM † x,µ , M x,0 and M x,µ are given respectively by Equations [START_REF] Belavkin | Quantum filtering of Markov signals with white quantum noise[END_REF] and ( 8) depending also on n.

Suppose that the initial state of dynamics ( 24) is not well known. Let ρ e 0 be an arbitrary initial estimate, the estimate discrete-time filter satisfies the following dynamics

ρ e k+1 = M n s k (ρ e k ) Tr M n s k (ρ e k ) ( 25
)
where s k corresponds to the experimental detection at time-step k.

Remark 2. For the same reason given in Remark 1, the above description is not always valid. Since the normalization Tr (M n s (ρ e k )) can vanish and the formula (25) is then not well defined. Such a problem will not be appeared when ρ e k is full rank. (When Tr (M n s (ρ e k )) = 0, we can still define the value of ρ e k+1 , see more details in [START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF].) Again, as we consider the asymptotic evolution, such a problem will not appear.

We now state a theorem ensuring the stability of such estimation procedure whatever the initial state ρ e 0 is. Theorem 3 ([21,[START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF]). Consider the Markov chain (ρ k , ρ e k ) satisfying ( 24) and ( 25):

(ρ k+1 , ρ e k+1 ) = M n s (ρ k ) Tr(M n s (ρ k )) , M n s (ρ e k ) Tr(M n s (ρ e k ))
with probability p s (ρ k ) = Tr (M n s (ρ k )) .

Then the fidelity (F (ρ k , ρ e k )) is a (F k )-submartingale where F k = σ{(ρ l , ρ e l )|l ≤ k)}. In particular, we have

E (F (ρ l , ρ e l )|F k ) = E (F (ρ l , ρ e l )|(ρ k , ρ e k )) ≥ F (ρ k , ρ e k ), for all l > k.
As far as we know, the continuous-time asymptotic versions of the discrete-time dynamics ( 24) and ( 25) associated to a left stochastic matrix η n with asymptotics [START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF], have not been established up to now. In the following, we derive such continuous-time SMEs which are the limits of these discrete-time dynamics and prove their stability.

Continuous-time filters as limit of discrete-time filters

Take n large and consider the piece-wise constant continuous-time stochastic processes denoted by ρ n (t) and ρ e n (t) with

ρ n (t) = ρ [nt] , ρ e n (t) = ρ e [nt] . (26)
where the discrete-time process (ρ k , ρ e k ) obeys to [START_REF] Van Handel | Filtering, Stability, and Robustness[END_REF] with k = [nt], the entire part of nt. It is clear that (ρ n (t), ρ e n (t)) is a Markov process, as (ρ k , ρ e k ) is the Markov chain of Theorem 3. Now suppose that ρ t and ρ e t be respectively the solutions of the continuous-time dynamics of the true filter and its estimate at time t. Let A n and A be respectively the Markov generators of (ρ n , ρ e n ) and (ρ, ρ e ). Then for all C 2 -real valued function f, we are looking for the continuous-time processes ρ and ρ e such that the following limit holds.

lim n→∞ sup (ρ,ρ e )∈D 2
|A n f (ρ, ρ e ) -Af (ρ, ρ e )| = 0.

We will see that such continuous-time limit depends essentially on the structure of η ∞ = lim n→∞ η n described in Lemma 1 and yields a generalization of the usual stochastic master equations driven by Wiener and Poisson processes. Lemma 1. Take the left stochastic matrix η ∞ s,(x,µ) defined by ( 23) with line index s ∈ {1, . . . , m} and column index (x, µ) ∈ {-1, 1} m W × {0, 1, . . . , m P }. Consider the following partition (S W , S P ) of {1, • • • , m} labeling the number of experimental detections:

S P = s ∈ {1, . . . , m} x∈{-1,1} m W η ∞ s,(x,0) = 0 , S W = s ∈ {1, . . . , m} x∈{-1,1} m W η ∞ s,(x,0) > 0 .
Then we have

• The η s,µ 's defined by η s,µ 2 -m W x∈{-1,1} m W η ∞ s,(x,µ) , (s, µ) ∈ S P × {1, . . . , m P }, (27) satisfy
∀(s, µ) ∈ S P × {1, . . . , m P }, 0 ≤ η s,µ ≤ 1 and ∀µ ∈ {1, . . . , m P }

s∈S P η s,µ ≤ 1. ( 28
)
• The singular values of the matrix E with entries

∀(s, ν) ∈ S W × {m P , . . . , m P + m W }, E s,ν 2 -m W /2 x x ν-m P η ∞ s,(x,0) x η ∞ s,(x,0) (29)
belong to [0, 1].

Proof. Inequality (28) is a direct consequence of S P definition, because for any (x, µ), s η ∞ s,(x,µ) = 1 and x belongs to a set of cardinal 2 m W .

The matrix E is well defined because, by definition of S W , the denominators in (29) are all strictly positive. The singular values of E do not exceed 1, if and only if, for any unitary vector z ∈ R m W , the Euclidian norm of Ez does not exceed 1. With z = (z j ) we have

Ez 2 = 2 -m W s∈S W x,j x j z j η ∞ s,(x,0) 2 x η ∞ s,(x,0) where (x, j) varies in {-1, 1} m W × {1, . . . , m W }. With ϑ s = x η ∞ s,(x,0) , ϑ s,x = η ∞ s,(x,0) ϑs and x|z = j x j z j we have Ez 2 = 2 -m W s∈S W ϑ s ( x x|z ϑ s,x ) 2 . By convexity of α → α 2 , we have ∀s ∈ S W , x ϑ s,x x|z 2 ≤ x ϑ s,x ( x|z ) 2 since x ϑ s,x = 1 and ϑ s,x ≥ 0. Thus Ez 2 ≤ 2 -m W s∈S W η ∞ s,(x,0) x ( x|z ) 2 = 2 -m W x ( x|z ) 2
since s η ∞ s,(x,0) = 1 for any x. We have

x ( x|z ) 2 = x,j,j ′ x j x j ′ z j z j ′ = j,j ′ z j z ′ j x x j x j ′ = j z 2 j x x 2 j = 2 m W j z 2 j since x in {-1, 1} m W implies that x x j x j ′ = 0 for j = j ′ . Thus Ez 2 ≤ 1 when z = 1.
Next theorem provides a generalization of usual SME driven by Wiener processes with detection errors to SME driven simultaneously by Wiener and Poisson processes with detections errors.

Theorem 4. Consider (ρ n (t), ρ e n (t)) defined by ( 26) and associated to [START_REF] Van Handel | Filtering, Stability, and Robustness[END_REF][START_REF] Van Handel | The stability of quantum Markov filters[END_REF] with a left stochastic matrix η n verifying [START_REF] Somaraju | Design and stability of discretetime quantum filters with measurement imperfections[END_REF]. Then, for n → +∞, the process (ρ n (t), ρ e n (t)) converges in distribution to the unique solutions of

dρ t = -i[H, ρ t ] dt +   ξ V ξ ρ t V † ξ - 1 2 (V † ξ V ξ ρ t + ρ t V † ξ V ξ )   dt + s∈S P   θ s ρ t + µ η s,µ V µ ρ t V † µ θ s + µ η s,µ Tr V µ ρ t V † µ -ρ t   dN s (t) -θ s + µ η s,µ Tr V µ ρ t V † µ dt + s∈S W η s ν c s,ν V ν ρ t + ρ t V † ν -Tr (V ν + V † ν )ρ t ρ t dW s (t) (30)
and

dρ e t = -i[H, ρ e t ] +   ξ V ξ ρ e t V † ξ - 1 2 (V † ξ V ξ ρ e t + ρ e t V † ξ V ξ )   dt + s∈S P   θ s ρ e t + µ η s,µ V µ ρ e t V † µ θ s + µ η s,µ Tr V µ ρ e t V † µ -ρ e t   dN s (t) -θ s + µ η s,µ Tr V µ ρ e t V † µ dt + s∈S W η s ν c s,ν V ν ρ e t + ρ e t V † ν -Tr (V ν + V † ν )
dN s (t) = θ s + µ η s,µ Tr V µ ρ t V † µ dt
with η s,µ given by (27) and θ s = 2 -m W x η s,(x,0) ≥ 0 ( η defined in ( 23)).

• s ∈ S W is related to the continuous signal y s related to the Wiener process dW s via

dy s (t) = dW s (t) + η s Tr ν c s,ν (V ν + V † ν )ρ t dt. (32)
The efficiencies η s belong to [0, 1] and correspond to the eigenvalues of EE † with matrix E defined in Lemma 1. The real coefficients c s,ν are given by the entries of the orthogonal matrix C appearing in the singular value decomposition of E = RDC with R and C orthogonal and D the rectangular diagonal matrix with diagonal entries √ η s :

∀s, s ′ ∈ S W , ν c s,ν c s ′ ,ν = δ s,s ′
The proof of this theorem is given in the next subsection. It admits the same structure as the proof of Proposition 1 but with sightly more complicated computations for the Markov generators. The stability of such quantum filters is ensured in the following theorem.

Theorem 5. Consider the Markov process (ρ t , ρ e t ), satisfying (30,31) where the positive integers m P , m W and m are arbitrary, where the m p + m W square matrices V ξ are arbitrary, where the partition (S W , S P ) of {1, . . . , m} is arbitrary, where ∀(s, µ) ∈ S P × {1, . . . , m P }, θ s ≥ 0, η s,µ ∈ [0, 1] and s ′ ∈S P η s ′ ,µ ≤ 1, where ∀(s, s ′ ) ∈ S W , η s ∈ [0, 1] and ν∈{m P +1,...,m P +m W } c s,ν c s ′ ,ν = δ s,s ′ .

Then, the fidelity F (ρ t , ρ e t ) defined in Equation ( 1) is a (F t )-submartingale, where F t = σ{(ρ τ , ρ e τ )|τ ≤ t}. In particular, we have,

E (F (ρ τ , ρ e τ )|F t ) = E (F (ρ τ , ρ e τ )|(ρ t , ρ e t )) ≥ F (ρ t , ρ e t ),
for all τ ≥ t.

Proof. The proof is the similar to the one given for Theorem 2. It relies on a direct application of Theorem 3 and Theorem 4. The assumptions made on the real coefficients θ s , η s,mu , η s and c s,ν implies the existence of a family of stochastic process (ρ n (t), ρ e n (t)) defined in (26) converging towards (ρ t , ρ e (t)) for n large. This implication relies on manipulations based on Lemma 1 and providing a family of stochastic matrices η n = η ∞ + η/n + O(1/n) such that these coefficients (θ s , η s,mu , η s , c s,ν ) are related to η ∞ and η according to Theorem 4. These manipulations are simple and not detailed here.

Proof of Theorem 4

The tightness property of (ρ n (t), ρ e n (t)) can be concluded by the similar argument appeared in [START_REF] Pellegrini | Markov chains approximation of jump-diffusion stochastic master equations[END_REF]. We give here the convergence proof of the Markov generators A n of (ρ n (t), ρ e n (t)) towards the Markov generator A of (ρ t , ρ e t ). We detail the convergence proof for S P = ∅ with S W = {1, . . . , m} and for S P = {1, . . . , m} with S W = ∅. The general case where both S P and S W are not empty is just a concatenation of the two previous ones .

-m W x η ∞ s,(x,0) , p+ s,ν 2 -m W x|x ν-m P =1 η ∞ s,(x,0) , p- s,ν 2 -m W x|x ν-m P =-1 η ∞ s,(x,0) . Then E s,ν = ν p+ s,ν -p - s,ν

ps

. Since R is an orthogonal matrix, dW = (dW s ) s∈{1,...,m} and R † dW define similar Wiener processes. Thus, with replacing dW by R † dW in (30,31), we have

dρ t = -i[H, ρ t ] dt + ξ L ξ (ρ t ) dt + s ν p+ s,ν -p- s,ν √ ps Λ ν (ρ t ) dW s (t), ( 33
)
dρ e t = -i[H, ρ e t ] dt + ξ L ξ (ρ e t ) dt + K(ρ t , ρ e t ) dt + s ν p+ s,ν -p - s,ν √ ps Λ ν (ρ e t )dW s (t), (34)
with

K(ρ, ρ e ) := s ν p+ s,ν -p - s,ν ps Tr (V ν + V † ν )(ρ -ρ e ) ν ′ (p + s,ν ′ -p- s,ν ′ )Λ ν ′ (ρ e ) .
The infinitesimal generator associated to the Markov process (ρ n , ρ e n ) satisfying dynamics ( 24) and [START_REF] Van Handel | The stability of quantum Markov filters[END_REF], can be written as follows

A n f (ρ, ρ e ) ≈ n s p s (ρ) f M n s (ρ) ps(ρ) , M n s (ρ e ) ps(ρ e ) -f (ρ, ρ e ) .
We can approximate the generator A n via the second order expansion of f around ρ and ρ e .

A n f (ρ, ρ e ) = nD (ρ,ρ e ) f • s M n s (ρ) -p s (ρ)ρ, s M n s (ρ e ) -p s (ρ e )ρ e p s (ρ) p s (ρ e ) + n s p s (ρ) 1 2 D 2 (ρ,ρ e ) f • M n s (ρ) ps(ρ) -ρ, M n s (ρ e ) ps(ρ e ) -ρ e ; M n s (ρ) ps(ρ) -ρ, M n s (ρ e ) ps(ρ e ) + O(n -1/2 ).
Let us first calculate the first term in above. By using [START_REF] Benoist | Large time behavior and convergence rate for quantum filters under standard non demolition conditions[END_REF], [START_REF] Braginsky | Quantum Measurement[END_REF] and [START_REF] Carmichael | An Open Systems Approach to Quantum Optics[END_REF], we find

M n s (ρ) -p s (ρ)ρ = x η n s,(x,0) M x,0 ρM † x,0 -p x,0 (ρ)ρ + x,µ η n s,(x,µ) M x,µ ρM † x,µ -p x,µ (ρ)ρ = x η n s,(x,0) c W √ n ν x ν-m P Λ ν (ρ) + x η n s,(x,0) c W n   -i[H, ρ] + ξ L ξ (ρ) - µ Υ µ (ρ)Tr V µ ρV † µ   + x,µ η n s,(x,µ) c W n Υ µ (ρ)Tr V µ ρV † µ . (35)
We have

s M n s (ρ) -p s (ρ)ρ = 1 n   -i[H, ρ] + ξ L ξ (ρ)   , (36)
since for all (x, 0) and (x, µ): s η n s,(x,0) = s η n s,(x,µ) = 1. Also note that for all ν :

x x ν-m P Λ ν (ρ) = 0.

Equation (35) can be rewritten as follows,

M n s (ρ) -p s (ρ)ρ = x η n s,(x,0) c W √ n ν x ν-m P Λ ν (ρ) + O(n -1/2 ) = ν p+ s,ν -p - s,ν √ n Λ ν (ρ) + O(n -1/2 ). ( 37
) Thus M n s (ρ) -p s (ρ)ρ = O(n -1/2
). Now using ( 11) and ( 12), we have

p s (ρ) = x η n s,(x,0) p x,0 (ρ) + x,µ η n s,(x,µ) p x,µ (ρ) = x η n s,(x,0) c W 1 + 1 √ n ν x ν-m P Tr (V ν + V † ν )ρ -1 n µ Tr ρV † µ V µ + x,µ η n s,(x,µ) c W n Tr V µ ρV † µ .
By equation in above, the approximate value of ps(ρ) ps(ρ e ) is given by

ps(ρ) ps(ρ e ) = 1 + c W √ nps x η n s,(x,0) ν x ν-m P Tr (V ν + V † ν )ρ -Tr (V ν + V † ν )ρ e + O(n -1/2 ) = 1 + 1 √ nps ν p+ s,ν -p- s,ν Tr (V ν + V † ν )ρ -Tr (V ν + V † ν )ρ e + O(n -1/2 ).
Finally, we find the following

ps D 2 (ρ,ρ e ) f • ν (p + s,ν -p- s,ν )Λ ν (ρ), ν (p + s,ν -p- s,ν )Λ ν (ρ e ); ν ′ (p + s,ν ′ -p- s,ν ′ )Λ ν ′ (ρ), ν ′ (p + s,ν ′ -p- s,ν ′ )Λ ν ′ (ρ e ) + O(n -1/2 ),
where we have used Equation (37) and the fact that we have to keep only the zero order terms of 1 ps(ρ) and ps(ρ) ps(ρ e ) 2 which is equal to 1 ps . We get finally the following expression for the generator A which is given as the limit of A n when n goes to infinity. The expression given in above corresponds well to the infinitesimal generator associated to the Markov process (ρ, ρ e ) satisfying dynamics (33) and (34).

Comparing this equation to the one given in (2) and ( 5) for perfect measurements, we observe that the Poisson processes have completely disappeared. It just remains the ensemble average µ L µ (ρ) in the deterministic part. Also, the original m W Wiener processes indexed by ν have been reorganized in m Wiener processes indexed by s. Now by using Equations (36), ( 41) and (42), we find the following expression for the limit of the generator A n expressed in (40)

In fact, we have

A n f (ρ, ρ e ) = Af (ρ, ρ e ) + O(n -1 2 ).

We observe that the expression of Af (ρ, ρ e ) corresponds well to the generator associated to the Markov process (ρ, ρ e ) satisfying SMEs (38) and (39). Now use the tightness property of the sequence (ρ n , ρ e n ) to conclude the convergence of the processes (ρ n , ρ e n ) towards (ρ, ρ e ). f (ρ, ρ e ) .

The limit for n large of each sum versus s ∈ S W and s ∈ S P is then obtained following exactly the same calculations as the ones given in subsections 3.3.1 and 3.3.2. Since (30,31) are just the concatenation of (33,34) for the Wiener part and (38,39) for the Poisson part, we get directly the convergence of the processes (ρ n (t), ρ e n (t)) towards (ρ t , ρ e t ).

Conclusion

For a large class of SMEs driven by Wiener and Poisson processes, Theorem 5 shows that Tr 2 √ ρρ e √ ρ between the quantum state ρ t and its estimate ρ e t is a submartingale. Thus the "metric" 1 -Tr 2

√ ρρ e √ ρ is a non-negative super-martingale that vanishes only when ρ = ρ e . A natural question is the following: do there exist other "metrics" D(ρ, ρ e ) that are super-martingales for such a large class of quantum systems? Any such "metric" D must be contractive for all Lindblad equations: in Theorem 5, there is no restriction on the degree of incompleteness of the measurements. Thus we can assume η s = 0, S W = {1, . . . , m} and S P = ∅. In this case, ρ and ρ e obey the same ordinary Lindblad differential equation

d dt ρ = -i[H, ρ] + ξ V ξ ρV † ξ - 1 2 (V † ξ V ξ ρ + ρV † ξ V ξ ),
where H and V ξ are arbitrary. In [START_REF] Petz | Monotone metrics on matrix spaces[END_REF], Petz has given, via the theory of operator monotone functions, a complete characterization of distance that are contractive for all Lindblad evolutions. Could we exploit Petz results to characterize "metrics" D(ρ, ρ e ) that are supermartingale for all the quantum filtering processes considered in this paper?

3. 3 . 1

 31 Proof of Theorem 4 when S P = ∅ and S W = {1, . . . , m} Consider the singular value decomposition of matrix E defined in Lemma 1: E = RDC with R ∈ O(m), C ∈ O(m W ) and D the rectangular diagonal matrix with diagonal formed by √ η s . Notice that the entries of C coincide with the c s,ν . Set ps 2

AfΛ

  (ρ, ρ e ) = D ρ f •   -i[H, ρ] + ξ L ξ (ρ), -i[H, ρ e ] + ξ L ξ (ρ e ) + K(ρ, ρ e ) ν ′ (ρ e ) .

3. 3 . 3

 33 Proof of Theorem 4 when S P = ∅ and S W = ∅ In this general case, the Markov generator of (ρ n , ρ e n ) is decomposed into two different sums:A n f (ρ, ρ e ) ≈ n s∈{1,...,m} p s (ρ) f M n s (ρ) ps(ρ) , M n s (ρ e ) ps(ρ e ) f (ρ, ρ e ) = n s∈S W p s (ρ) f M n s (ρ) ps(ρ) , M n s (ρ e ) ps(ρ e ) f (ρ, ρ e ) +n s∈S P p s (ρ) f M n s (ρ)ps(ρ) , M n s (ρ e ) ps(ρ e )

  P , S W ) of {1, . . . , m} defined in Lemma 1, with, in the above sums, µ ∈ {1, . . . , m P }, ν ∈ {m P + 1, . . . , m P + m W } and ξ ∈ {1, . . . , m P + m W }, and where• s ∈ S P is related to the Poisson process dN s (t) characterized by

			ρ e t ρ e t	× . . .
	. . . × dy s (t) -η s Tr	ν	c s,ν (V ν + V † ν )ρ e t	dt.	(31)
	with partition (S


Hélène Quashie

Dolorès Pourette

Olivier Rakotomalala

Frédérique Andriamaro

Traditional therapeutics, biomedicine and maternal health in Madagascar: paradoxes and power issues around the knowledge and practices of reninjaza

Keywords: Madagascar, traditional midwives, medical traditionalism, biomedicine, maternal health, politics

Public health development in Madagascar is not free from contradictory power dynamics surrounding medical knowledge, and often occurring between biomedicine and traditional healing. In the following paper, we will mainly focus on the therapeutic fields of maternal health, with mortality rates remaining a key public health issue in sub-Saharan Africa. We will examine the paradoxes, power struggles and contradictions affecting medical care during pregnancy and childbirth, with regard to specific knowledge and practices of particular traditional healers: reninjaza. The following study will show that institutional, social and political distinctions that fragment medical knowledge into mirroring areas, i.e. "traditional" and "modern" practices, feed power dynamics that obscure social realities at stake in maternal healthcare.

Introduction

To face up to the failings of preventive medicine and the deficiencies of hospital care, national and international policies gave a more prominent role to populations in Southern countries to solve their health issues in the late 1970s. These new strategies supported the notion of endogenous development based on existing local capacities [START_REF] Dozon | Ce que valoriser la médecine traditionnelle veut dire[END_REF]. Following the 1978 Alma-Ata Conference, the World Health Organisation recommendations included the actors of traditional medicine in primary healthcare programs. Many countries, especially in sub-Saharan Africa, redefined their health policies to combine biomedicine and traditional therapeutics, and therefore strengthened the construction of their national identity. Humanitarian 2 aid networks also contributed to promoting traditional medicine as they were committed to moving towards alternative notions of development (ibid.). The NGOs' programs were thus in line with sceptical and critical movements emerging from Northern countries that challenged over-medicalisation. Recognising alternative medicines to biomedical science thus became a transnational political, social and economic issue.

In Southern countries, however, the challenge of combining biomedical and traditional knowledge fuelled power struggles among practitioners, as well as the continuous redefinition of professional identities and the role of the various actors labelled as "traditional doctors", "traditional practitioners", "folk healers", etc. The specificity of what is referred to as "traditional medicine" lies in its constant state of (re)definition. Power issues and conflicts surrounding medical knowledge which divide the health arena in developing countries brought about multiscale interactions, reinforced by the absence of any real definition of traditional medicineexcept in a stereotypical and essentialised opposition to Western modernity [START_REF] Dozon | Ce que valoriser la médecine traditionnelle veut dire[END_REF]. These different changes were also grafted onto postcolonial identity politics that went beyond the scope of medicine. Nonetheless, professionals associated with the biomedical system continued to rule in the health arena and remained in charge of evaluating the capacities and skills of traditional healers. The legitimacy, career paths and socioprofessional recognition of these practitioners, along with the potential combination of their practices to those of biomedical health facilities, therefore continue to be constantly questioned and challenged in an increasingly complex way due to the globalised nature of these debates (Pordié & Simon, 2013).

Public health development in Madagascar is not free from contradictory power dynamics surrounding medical knowledge, which in many fields occur between biomedicine and traditional therapeutics5 [START_REF] Lefèvre | Les discours sur la médecine traditionnelle à Madagascar. Entre idéologie coloniale, salut de l'âme, raison économique et pouvoir biomédical[END_REF][START_REF] Didier | Les médecines traditionnelles entre politiques locales et décisions internationales : le cas de Madagascar[END_REF]. We will mainly focus on the field of maternal mortality, which remains one of the key public health issues in sub-Saharan Africa. The majority of women's deaths occur during the third term of pregnancy, during childbirth and immediately after delivery: they are mainly due to haemorrhage, hypertension, infections and obstructed labour [START_REF] Fournier | Le rôle des relais communautaires et des accoucheuses traditionnelles durant la période périnatale dans la région de Kayes au Mali[END_REF]. In the late 1990s, faced with the high maternal mortality rate, the Ministry of Public Health of Madagascar promoted, with the support of UN agencies and several NGOs, the involvement of traditional practitioners and reninjaza 6 in the biomedical maternal health system and developed, among other things, training courses for them. However, the mortality rate only dropped slightly 7 . To the national and international institutions involved, this result was a crucial argument to assert the failings of including traditional healers in the maternal healthcare system and to stop these training courses from 2007. The situation in 3 Madagascar is by no means unique: from the 1990s onward, several other sub-Saharan countries stopped offering training courses for "traditional birth attendants" [START_REF] Fournier | Le rôle des relais communautaires et des accoucheuses traditionnelles durant la période périnatale dans la région de Kayes au Mali[END_REF].

New directions in aid and development policies were therefore promoted in maternal healthcare. However, the fact that biomedical and traditional practices had existed alongside for several years impacted and shaped the Malagasy health arena in a significant way. It must be noted that the political importance of medical traditionalism in Madagascar goes back to the 1960s, a decade during which research on pharmacopoeia and inventories of medicinal plants and essential oils for therapeutic use flourished [START_REF] Debray | Médecine et pharmacopée traditionnelles à Madagascar[END_REF]. Even before the WHO recommendations, several French research organisations such as the Institut Pasteur in Madagascar or the ORSTOM (now the Institut de Recherche pour le Développement), alongside national institutions, took interest in the complex flora of Madagascar and its specific geological history as compared to the rest of the African continent [START_REF] Cortadellas | Les substances naturelles d'intérêt pharmacologique[END_REF]. From the very beginning, biomedical scientists from Madagascar and abroad were fascinated by the prevalence of endemic species on the island. The knowledge surrounding traditional uses of local plants was also associated with a broader identity politics: the WHO recommendations coincided with the formation of the Second Republic (1975), which promoted a national identity based on a wide process of linguistic "Malagasisation" and a visible trend towards "ancestral heritage" [START_REF] Rakotomalala | Transformations du politique et pluralité thérapeutique[END_REF]. As was the case in other sub-Saharan countries advocating a socialist regime [START_REF] Dozon | Ce que valoriser la médecine traditionnelle veut dire[END_REF], promoting Malagasy traditional medicine was a means of revitalising specific values within a broader nation-building political project. The Second Republic constituted a more marked postcolonial break with the Western world, especially France, and supported the development of an authentic Malagasy identity based on primitivism [START_REF] Amselle | Rétrovolutions. Essais sur les primitivismes contemporains[END_REF][START_REF] Amselle | Au nom des peuples : primitivismes et postcolonialismes[END_REF].

As for therapeutics, traditional healers now enjoy great prestige among populations. Patients' trajectories within the healthcare system show that these tradipractitioners are the first port of call [START_REF] Pourette | Etude qualitative MEDALI-Mission d'Etudes des Déterminants de l'Accès aux Méthodes de Lutte antipaludique et de leur Impact[END_REF]. However, as in other Southern countries, this often correlates with the patients' socio-economic backgrounds: if at all possible, they tend to seek out multiple therapies or take both traditional and biomedical remedies [START_REF] Dozon | Ce que valoriser la médecine traditionnelle veut dire[END_REF][START_REF] Olivier De Sardan | Anthropologie du développement : essai en socio-anthropologie du développement social[END_REF]. On the other hand, some traditional practitioners appear to have an influential status locally, especially among decision makers in rural areas [START_REF] Ramanantenasoa | Méthodes de diagnostic et traitements des maladies par les plantes médicinales chez un guérisseur dans le fokontany d'Antanetibe Ambohitrimanjaka[END_REF] but also in urban settings. As with other healthcare actors, many traditional practitioners' activities stand at the crossroads of the fields of medicine, politics, economics and religion [START_REF] Fassin | Pouvoir et maladie en Afrique[END_REF]. Their practice falls into several categories, as some are not only consulted for healthcare purposes, but also for their psychic abilities or knowledge of sorcery [START_REF] Rakotomalala | A coeur ouvert sur la sexualité Merina (Madagascar). Une anthropologie du non-dit[END_REF]. Once again, these two skills strongly correlate with positions of social power.

In the present paper, we will focus on the paradoxes and power struggles impacting on maternal healthcare with regard to the specific knowledge and practices of particular traditional healers: reninjaza. These female actors in the modern Malagasy "childbirth system" 8 are seldom mentioned: their occupation is regulated by the Public Health Code 9 and is mainly carried out by women, although there are a few male reninjaza, especially in the capital. Reninjaza occupy a unique place in the health arena: they play a key role and attend most deliveries, especially in rural communities, but also in urban areas [START_REF] Ravololomanga | Etre femme et mère à Madagascar[END_REF]. Reninjaza are therefore well recognised locally. However, they do not enjoy any medical, economic, religious or political prestige. Unlike in other Southern countries, they are not organised in associations either [START_REF] Boyer | Le mouvement des sages-femmes traditionnelles dans l'Amapá : Une mobilisation féminine exemplaire et les limites du développement local[END_REF]. As a result, their practices are not very highly valued by healthcare authorities and professionals, who accuse them, among other things, of indirectly contributing to the high maternal mortality rate. This lack of regard is not specific to Madagascar: similar views can be found elsewhere concerning the practices of "traditional midwives" [START_REF] Hancart-Petitet | Les derniers jours de Palavankodie. Transformations et effacement des pratiques des matrones en inde méridionale[END_REF]). We will nevertheless examine the contradictions and social issues that are developing at several levels around the role and status of reninjaza, which are strongly and commonly downplayed from a medical, economic and political perspective. Many of the contradictions within the organisation of the Malagasy maternal healthcare system complicate access to medical treatment for women during pregnancy and childbirth, along with relevant policies.

Our data was collected during several field works focusing on access to healthcare during pregnancy. Several exploratory missions and qualitative studies10 were conducted in 2012 and 2013 with healthcare professionals, families, reninjaza and traditional practitioners in two regions in Madagascar, in rural and urban areas (in the central region of Analamanga and the Vatovavy-Fitovinany region on the east coast). Semi-directive interviews were carried out with 60 women, 30 men, 45 reninjaza, 24 health workers and 9 representatives of health authorities, all in the rural and urban areas of Moramanga, Manakara and Antananarivo.

I. Social classes, medical power and the recognition of the role of reninjaza

The local social status held by reninjaza is primarily due to the fact that they facilitate so many deliveries, especially in rural areas. According to the latest Demographic and Health Study conducted in Madagascar (INSTAT, 2010), one in two women (77% in some regions) give birth with the help of a reninjaza (52% in rural areas and 16% in urban areas). But these figures are most likely higher. Moreover reninjaza are involved at several points in time during maternity. Before falling pregnant, some women may consult them for reasons of infertility, in which case they are administered massages and herbal teas such as tambavy, which is very well known on other islands in the Indian Ocean, especially in the Mascarene Islands [START_REF] Pourchez | Savoir des femmes : médecine traditionnelle et nature -Maurice, Réunion, Rodrigues. Savoirs locaux et autochtones[END_REF]. During early pregnancy, pregnant women use the services of reninjaza to confirm that they are expecting. Then, during the course of pregnancy, reninjaza administer massages to relieve lower back pains or to try to change the foetus's position, especially just before childbirth. Finally, they attend the delivery, mostly at their patients' homes in rural areas, at their surgery or patients' home in urban areas. They assist women throughout childbirth and after the delivery, as reninjaza also take care of newborn babies and look after the mothers' health11 daily for about a week after childbirth.

As in many countries where the "childbirth system" comprises traditional midwives or traditional birth attendants [START_REF] Akoto | Infirmiers privés, tradipraticiens, accoucheuses traditionnelles à la campagne et à la ville[END_REF][START_REF] Pourchez | Savoir des femmes : médecine traditionnelle et nature -Maurice, Réunion, Rodrigues. Savoirs locaux et autochtones[END_REF], reninjaza assert that their knowledge and know-how were passed down to them by their mother, or sometimes their grandmother or their aunt, who carried out the same role within their community. This means that they mainly gained this knowledge and these practices by accompanying members of their family and watching them at work. Some of them also mention an ancestral heritage and a "gift from God", which is interpreted as a blessing given to them as "chosen ones". As in other countries, reninjaza usually already have children themselves when they attend their first delivery (ibid.). Their skills are thus based on a non-theoretical body of knowledge and know-how, and are clearly in line with the political and institutional distinction drawn between biomedicine and medical traditionalism. Finally, they often start practising at the request of relatives or neighbours, when their mother, grandmother or aunt can no longer carry out this role or are not on hand to do so. Socially and locally, they are recognised for mastering skills that no one else can practise. This situation creates a social identity that reninjaza adopt over the course of their life history and medical practice, as they must comply with social demand.

The local and social status held by reninjaza is therefore based on the specific role they play within a population, by whom they are also valued for their fundamentally different attitudes to those of health workers. The behaviour of the latter often reflects the classic asymmetry around medical knowledge that exists between patients and practitioners, but also class dynamics that are especially at play with low-income pregnant women from rural and urban areas. Conflicting relationships between carers and patients, and the latter's misgivings regarding health facilities, are not specific to Madagascar or to the field of maternal health [START_REF] Jaffré | Une médecine inhospitalière. Les difficiles relations entre soignants et soignés dans cinq capitales d'Afrique de l'ouest[END_REF]. However, in this case, they encourage the use of reninjaza by pregnant women for care and treatments. In poorer rural and urban areas, reninjaza belong to the same low-income classes and to the same social circles as the pregnant and parturient women they attend to. They are neighbours or family members who sometimes have attended all births in the family. They are popular for being available at all times, for listening to pregnant women's needs, and for showing respect for the moment of childbirth while finding reassuring and comforting attitudes and words. The qualities associated with reninjaza are deemed to be lacking in health workers in biomedical infrastructures. Like the women they attend to, reninjaza have a limited level of education and therefore do not speak French. This also restricts their recognition among biomedical practitioners and contributes to keeping them at the lower end of the social and medical hierarchy. However, most reninjaza can write down the names of the parturient women and their children in Malagasy along with their birth dates in a notebook, which is helpful for recording births. Reninjaza play a key role in geographic and social situations where pregnant and poorly educated women cannot be treated in distant or expensive health facilities. By consulting reninjaza, they can also avoid the process of social prejudice they are subjected to when encountering health workers.

However, in these same social contexts, reninjaza owe most of their livelihood to agriculture or basket-makingdepending on the regionand not to their healthcare practice. This observation can be made in many other Southern countries, especially in rural areas [START_REF] Akoto | Infirmiers privés, tradipraticiens, accoucheuses traditionnelles à la campagne et à la ville[END_REF]. Reninjaza do not earn a living from their occupation, as pregnant women and their families either pay them in kind or can only offer small amounts (500 -5000 ariary). In many cases, too, they do not receive anything for their services. Yet most reninjaza we met hope to see their social status rewarded by financial compensations from families or grants from local authorities (e.g. donations in medical supplies such as alcohol, bandages, blades, candles, etc.). They stress they are under pressure within their social networks to monitor pregnancies and attend deliveries as they otherwise would endanger the lives of pregnant women and their children. Reninjaza sometimes see this responsibility as a constraint, because they cannot refuse to practise. But it also makes them realise what a key role they play and the social prestige they could enjoy, as with any other actor in the healthcare system.

The fact that there is no correlation between the constant calls upon reninjaza to practise and their ordinary social status within their communities is often due not only to a low socioeconomic background but also to their position at the lower end of the internal hierarchy of traditional medicine. Their practice is not at the crossroads of economic, religious, political and medical issues, which is what generally confers social prestige to healthcare actors, be they from biomedical or traditionalist spheres. For example, the protection of unborn children from misfortune and death during pregnancy is locally assigned to traditional practitioners. Despite the local status of reninjaza, the central role they play in the "childbirth system", and the fact that they are closest to pregnant women, their knowledge is not on a par with that of traditional practitioners. In the field of maternal health, the most respected sociopolitical power and medical identity associated with traditional knowledge are held by the healers who are only indirect parties to childbirth and whose practices generally relate to other medical issues. By extension, these paradoxes strengthen the distinction and hierarchy between various categories of actors in the medical field. They especially contribute to impeding any broader legitimacy or status for reninjaza. Health workers systematically devalue their remedies, for example, which they do not do with traditional practitioners.

The undermining of the status of reninjaza in political and medical spheres is particularly notable in Antananarivo, a city in which the intrinsic paradoxes of traditional medicine appear to be multiplied. The knowledge and practices of reninjaza are only valued if they are associated with the prestigious status of a traditional practitioner. This is how reninjaza in the capital (men and women) can build their social, political and medical legitimacy while strengthening hierarchies within traditional medicine. Some healers from the middle-income and wealthy classes decide to become reninjaza and call themselves "traditional birth attendants". However, they attend to far fewer women during pregnancy and childbirth than their counterparts from low-income backgrounds who offer their services to the lowest social classes. These traditional birth attendants are educated men and women who speak a bit of French, which is a source of social prestige in the capital. Some of them even studied at university. Finally, all of them are members or chairs of associations of traditional healers affiliated with the National Association of Traditional Practitioners of Madagascar (NATPM) 12 , which links their therapeutic work to the political field. Their delivery services cost from 5,000 to 150,000 ariary depending on the patients' social backgrounds, i.e. the capital's middle-class and bourgeoisie. These patients usually go to private clinics in the city but they use the services of these traditional birth attendants because of their good "reputation". Their renown arises from their well-diversified knowledge and practices as traditional practitioners, and from the fact that it is their main activity. As maternal health actors, they emphasise the value of the "gift" passed down from ancestors by whom they were chosen, and who revealed this know-how to them in a dream or through a voice telling them what to do. They also define themselves as descendants of a family of traditional practitioners, psychics or traditional midwives. They state that their "gifts" help them visualise the foetus's position and to reposition it for childbirth without having to use biomedical instruments. Thanks to their gift, they assert that they can also cure sterility, stop haemorrhaging resulting from childbirth, and cure any diseases that are or are not linked to pregnancy. The strong promotion of traditional knowledge and practices by these traditional healers from middle and wealthy social classes is structured around spiritual and primitivist notions which, from a postcolonial perspective, conflict with biomedical, and in a broader sense, Western techniques. As an example, two renowned traditional birth attendants we met claim they do not use vazaha13 supplies, including gloves, for delivery, but only their hands and eyes. The promotion of their knowledge asserts a specific political point of view that improves their reputation and, in turn, their socio-economic prestige.

Conversely, reninjaza from lower-income groups in the capital do not emphasise the unusual nature of their skills, although they practice more than traditional birth attendants: as outside the capital and in rural areas, their local recognition is easily built, especially with underprivileged groups. Because of their social background, these reninjaza have no ideological standpoint, a low socio-economic status and few demands or expectations in terms of biomedical supplies and resources. They are not involved in the political construction of medical knowledge, in which Western techniques mirror Malagasy magical and religious practices. Yet this postcolonial rhetoric supports the socio-economic status of traditional birth attendants and, in turn, their hierarchical and elitist position compared to reninjaza. For example, the traditional birth attendants we met visited training courses offered by the National Institute for Public and Community Health, by the Pharmacology Association of Madagascar, or by the Barefoot Doctors Association to complement their "gift", be vetted by the authorities and be able to work in massage or naturopathy surgeries. Their socio-professional, medical and political status is associated with scholarly knowledge of pharmacopoeia and facilitated by their belonging to the educated social classes. Their recognition overlaps with primitivist ideologies valuing an authentic Malagasy identity, which, in some ways, continue to support national development policies 14 .

II. Institutional contradictions and tensions around the status of reninjaza

Within the Malagasy Ministry of Public Health, the activities of reninjaza fall into two divisions: the Department of Pharmacopoeia and Traditional Medicine (DPTM), since the early 1990s (it succeeded the Department of Traditional Pharmacopoeia founded in 1986); and the Directorate of Maternal and Child Health and Reproduction (DMCHR). Since they come under the category of traditional healers, reninjaza fall within the remit of DPTM, which aims to promote and integrate positive aspects of pharmacopoeia and traditional medicine into healthcare provision. The Department was created with the support of biomedical professionals and scientists to advocate the involvement of reninjaza in "safe maternity" by training them and providing them with supplies. DPTM continues to value their practices and knowledge as ancestral skills and expertise. It works with the National Association of Traditional Practitioners of Madagascar (NATPM) and encourages reninjaza to become members of one of its affiliated associations. Through DPTM, the medical establishment therefore values the role and status of traditional practitioners and, by extension, the practices of reninjaza as any other traditional healers from a clinical and political perspective.

Conversely, professional representatives affiliated with the Directorate of Maternal and Child Health and Reproduction (DMCHR), which also advocates "safe maternity", are opposed to reninjaza attending deliveries. They challenge and even reject their activity as traditional healers: reninjaza are not deemed to possess the necessary technical skills to provide maternal care. These biomedical professionals see reninjaza as unfit to oversee pregnancy, childbirth and the subsequent afterbirth care. DMCHR was one of the bodies advocating the stopping of training courses for reninjaza in 2007. Its professionals now recommend that reninjaza systematically refer pregnant women to health facilities so that they receive at least four antenatal consultations during pregnancy, and for childbirth. As it is difficult or even impossible to access health facilities in some remote rural areas, it is accepted that reninjaza attend the delivery, provided there is no health facility within 5km. Reninjaza are therefore valued within the medical establishment as traditional healers who carry out "ancestral practices", but also underappreciated with regard to their skills. They face paradoxical lines of questioning that impact their practices. These two contradictory positions within the Ministry of Public Health contribute to the emergence of sociopolitical issues around medical knowledge, with outcomes directly impacting the treatments received by pregnant women in the Malagasy maternal healthcare system. This institutional ambivalence regarding the status and role of reninjaza crystallises more than anything the hierarchical relationships between biomedicine and traditional medicine.

Training courses for reninjaza, which were originally aimed at combining their practices and knowledge with those of the biomedical system in an attempt to reduce maternal mortality rates, is a concrete example of how they were discredited socially and medically by healthcare professionals. The organisation of these training courses shows, among other things, that the hierarchy between biomedicine and traditional medicine is based on and, in turn, reinforced by class dynamics that differentiate the varying categories of actors in maternal health. Training courses for reninjaza, especially in rural areas, were often organised by UNFPA: they were generally carried out in groups, were very short (never longer than one or two days) and were sometimes remunerated. New biomedical supplies were sometimes handed out, but these were often insufficient to be given out to all participants and swiftly ran out. Reninjaza were selected and appointed for these training courses by the doctors of the nearest primary healthcare center. Literacy was one of the requisite conditions, which meant that not all reninjaza could take part in these training courses. The selection process thus established a medical and professional hierarchy between health facility personnel and reninjaza which was reinforced by socio-economic inequalities and power relations. As to the input they received during the training courses, it essentially centred on raising awareness and focussed on medical hygiene recommendations and on the importance of referring pregnant women to healthcare centers for antenatal care and delivery. This meant that these training courses fundamentally did not change reninjaza practicesave for the fact that they sometimes granted them access to the biomedical supplies they needed -nor the communities' expectations towards them.

Beyond the socioprofessional and medical hierarchy imposed on reninjaza, the institutional and political opposition between biomedicine and traditional medicine seems out of line with social and therapeutic realities. For example, although they are classified as traditional healers relating to maternity, the practices of reninjaza do not conflict with those of biomedical professionals and do not claim to do so. On the contrary, reninjaza wish to integrate the tools and lessons obtained through collaboration with the biomedical system into their body of knowledge. From what they say, institutional representatives of biomedical professionals, on the other hand, tend to devalue the knowledge of reninjaza and ignore aspects of their practices that could give better access to healthcare to pregnant women. As we have seen, social ties between them and reninjaza, especially in rural areas and more generally in low-income social classes, in which maternal mortality rates are high, are key for antenatal care, childbirth and postnatal care. Yet health workers and their institutional representatives never mention conflicts or difficulties they face in their own relationships with pregnant women, despite the impact these have on the trajectories of their patients, who are then likely to turn to reninjaza in their communities.

The ambivalent attitude of the Ministry of Public Health relating to the role and status of reninjaza has also led to their occupation being labelled "official", "legal" or "illegal". The institutional organisation implemented to promote traditional medicine did not manage to avoid the development of any form of hierarchy with regard to biomedicine. By becoming members of NATPM, which is widely recommended by DPTM, reninjaza, as with all traditional practitioners, have the right to hold a membership card that enables them to practise "legally" and be "protected" if one of their patients dies or faces serious medical problems. The collaboration between DPTM and NATPM to create a register of "real" traditional healerstherefore of "real" and "fake" reninjazaand provide these medical actors with recognition and legitimacy through association membership, results in making non-affiliated reninjaza feel "illegal", (especially those practising in the capital). Not being a member of NATPM can be synonymous with illegal practice. Some reninjaza therefore become members of an association to appear as "lawabiding" citizens, out of fear of being sent to prison if one of their patients dies during childbirth. However, according to these reninjaza, codes and regulations of their associations constrain their occupation. First of all, they have to pay 30,000 ariary in registration fees, followed by an annual association membership fee of 5,000 to 10,000 ariary. They must also comply with the association's rules, which vary depending on the organisation (e.g. compliance with specific medical hygiene measures and professional confidentiality, giving up alcohol and cigarettes, never being found guilty of "misconduct"). These rules impact their practice and do not always take into account their perception and vision of their own role, nor the social ties that connect them to their patients. Some of these ethical codes also promote primitivism to value medical traditionalism in the health arena: a few associations, for example, recommend the avoidance of any medicines or "chemical products", preferring "natural" products only. This primitivism appears to be unique to the institutional and political organisation of traditional therapeutics in the capital and reflects class dynamics that conflict with the practices and perceptions of reninjaza from low-income and underprivileged social classes. The only advantage they derive from their membership with an association is the "legalisation" of their status in the eyes of the State: willingly or not, they apply a certain political and professional rationale to their activity. Outside the capital, in rural and urban areas, their status is under less pressure and strain from the State and NATPM. Only one of the surveyed reninjaza had heard of NATPM, since it had been mentioned by a traditional practitioner she had consulted for her own child. But she did not know what benefits she could expect from being a member of such a medical association. On the other hand, reninjaza who refer pregnant women to healthcare centers are given a more favourable introduction by professionals in the biomedical care system: they are presented as "legal" or "official". However, despite institutional and political processes establishing hierarchies between biomedicine and traditional medicine, instances of overlap and collaboration between these mirroring fields of medical knowledge are inevitable within maternal health facilities.

III. Ambiguities and realities in maternal healthcare provision

Professional and political discourse that distinguishes, opposes, excludes and establishes hierarchies between biomedical and traditional practices and knowledge does not hold up to therapeutic realities. An interview with a representative of Malagasy reproductive health professionals, conducted in a district health unit in the region Antananarivo, shows the discrepancies between Ministry of Public Health guidelines issued by DMCHR on the role of reninjaza and practical needs in maternal healthcare provision for pregnant women: "According to national policies here in Madagascar, reninjaza are no longer allowed to receive and attend delivery of women. They have to refer them to healthcare centers. (…) The Ministry of Public Health no longer accepts the attendance of reninjaza at deliveries. (…) Before I took up this post, it was permitted. However, this has not been the case for 7 years now because we have not reached our Millennium Development Goals, and because of maternal deaths occurring outside health facilities during childbirths and attended to by reninjaza. (…) But [primary healthcare center personnel] organise meetings because we cannot downplay the role of reninjaza, as health workers cannot cope with the workload by themselves. And there are always women who go to reninjaza to give birth and ask them for advice. So on the one hand, in the Ministry's view, reninjaza are no longer allowed to attend deliveries, but on the other hand, they still practice, so the heads of primary healthcare centers are charged with retraining reninjaza. (…) It's better to work together than ban them altogether, because that would cause harm. It's better to raise awareness among reninjaza and to retrain them periodically."

Health workers try to adapt their practices to daily situations, socio-economic background, and their own capacities to provide care to parturient women, as well as to their perception of the reninjaza with whom they can collaborate. In rural areas, for example, many women have no other option than to give birth on their own or seek assistance from a reninjaza. Primary healthcare center personnel must deal with this social reality. At different levels of maternal healthcare provision during pregnancy, various forms of collaboration around reninjaza practices and knowledge emerge, despite recurring contradictions with regard to their status and role. Administratively, health authorities, such as district health units or village authorities, list reninjaza, require them to provide records of childbirths they attend to and to fill in a register to draw up acts of birth. Some reninjaza keep notebooks that they hand over to the fokontany president of their community to perform these administrative formalities (in cases where reninjaza are not literate, the fokontany president fills out the register on their behalf). They also take care of registering births at the town hall, even in the capital. Due to the fact that the last national census dates back to 1993, births and deaths are difficult to monitor (especially in lowincome and underprivileged classes in urban and rural areas), and because the patronymic does not automatically establish genealogy in a lot of families, reninjaza appear to be key resource persons.

From a medical viewpoint, collaboration between health workers and reninjaza takes different forms in urban and rural areas. In Antananarivo, for example, instances of collaboration are few and far between according to the Institut Pasteur in Madagascar, even if complications do occur during childbirth. There are no "handover forms" to ensure communication between healthcare centers and reninjaza. Those who are not members of NATPM are deemed to be practising illegally. As for traditional birth attendants, although they are recognised by political institutions, associations and some members of Antananarivo's upper-class families, their practices are not combined with those of hospital and healthcare center personnel. On the other hand, in rural areas, most primary healthcare center professionals stated in the survey that they cannot attend all deliveries due to a lack of personnel, time, resources and space. They acknowledge that reninjaza play a key role in their own practices, although they lack the necessary medical training or skills. These health workers value several models of collaboration with the reninjaza they chose to work with, either because they have basic training or because they comply with some medical hygiene standards. Unlike in other Southern countries, the training courses that used to be offered to reninjaza are given little weight when establishing such collaborations. But these happen to be easier when reninjaza consider themselves at the lower end of a social and professional hierarchy that includes healthcare center personnel.

Collaboration can take various forms and is always initiated by primary healthcare center professionals with the approval or at the request of the local authorities. It can be structured around the accompaniment of pregnant women by reninjaza for antenatal consultations and during delivery, in line with DMCHR recommendations. When involved in care provision around childbirth, the role of reninjaza is limited to accompanying the patient: they are not tasked with any medical procedures. They are rather seen as a family member who has come to support the pregnant or parturient woman. Although this type of collaboration with reninjaza does not imply any therapeutic practice, it is considered necessary by authorities and health workers to improve maternal healthcare. According to this vision, the know-how of reninjaza is also considerably discredited: their practices during childbirth are deemed dangerous, inappropriate and too far removed from requisite medical and technical knowledge. For example, reninjaza are accused of waiting too long before delivery, hindering any possible identification of complications and timely referral of parturient women to primary healthcare centers. The remedies they use, such as herbal teas, as well as their failure to comply with some medical hygiene standards, are also seen as sources of infection. These points provide yet another argument for holding reninjaza involvement responsible for the sustained high rates of maternal mortality.

A second form of collaboration consists in giving reninjaza who often refer parturient women to primary healthcare centers a role of assistants or helpers. Health workers are grateful for some of their abilities, such as monitoring the perineum or washing the newborn baby, especially if they are busy attending to other patients. Some are happy to be able to supervise and control what reninjaza do to teach them "good practice". Once they have acquired the relevant techniques, they are sometimes delegated tasks. However, despite participative collaboration, primary healthcare center personnel still assert their authority over reninjaza and use control and pedagogy to uphold their medical legitimacy. Moreover, when health workers mention these forms of cooperation, they do not highlight their professional interest in the training of reninjaza or in women and newborn care, but more personal expectations, such as the opportunity to receive support, to leave the primary healthcare centers for some time, and to delegate childbirth. In this context, too, the role of the reninjaza is considered inferior to biomedical science. Once again, this hierarchy is based on class distinction and on the actors' positions on the social ladder. It must be stressed here that most reninjaza are farmers, something that accentuates their inferiorization by primary healthcare centers personnel.

In the Malagasy maternal healthcare system, combining the know-how of reninjaza with health workers' practices challenges power dynamics in the medical arena and, as in other countries, makes for weak and porous boundaries between biomedicine and traditional medicine [START_REF] Longuenesse | Santé, Médecine et Société dans le monde arabe[END_REF][START_REF] Benoist | Soigner au pluriel. Essais sur le pluralisme médical[END_REF]. Patients' health trajectories also build bridges between reninjaza and biomedical professionals [START_REF] Akoto | Infirmiers privés, tradipraticiens, accoucheuses traditionnelles à la campagne et à la ville[END_REF]. Many women decide to turn to a health facility during their pregnancy to access antenatal consultations and make sure they will receive treatment in the event of complications during childbirth. This practice is becoming a social norm. Likewise, some reninjaza ask women to have at least one consultation in a healthcare center, which maintains their collaboration with facility personnel. Finally, some midwives even ask their patients to consult reninjaza for massages to reposition the foetus before childbirth. Health workers do not see practices of reninjaza as scholarly or scientific, but some concede that they have a "gift". Coupled with a form of spirituality related to primitivism, these "gifts" are deemed to guarantee the effectiveness and success of some of the medical care that health workers cannot offer (such as turning the foetus if it is in the wrong position before childbirth). As we can see, although they are considered inferior to biomedical science, some therapeutic practices carried out by reninjaza are sometimes recommended, i.e. in cases where biomedical professionals are powerless. In this situation, care and treatments are "Malagasised", to use a common expression. This leads to a strongly ethnicised promotion of local medical traditionalism, and, in practice, to erasing boundaries between two therapeutic fields that are essentialised in mirroring positions.

Conclusion

In Madagascar, national and international health policies promoted the association of, and subsequently established hierarchies between, knowledge and practices from biomedicine on the one hand and traditional medicine on the other. Both processes aimed to improve maternal healthcare and reduce mortality rates during childbirth. However, neither of the two approaches has led to the desired outcomes, as they do not take into account the distinctions established between the various categories of medical actors. In practice, the socio-economic realities faced by pregnant women take precedence over political issues about the definition and institutionalisation of medical knowledge related to maternity. Health actors, on the other hand, seem to abstain from a certain reflexivity that would prioritise their patients' needs.

Pregnant women from low-income and underprivileged social classes in rural and urban areas widely choose to consult reninjaza, as the services they offer are cheap or even free. However, some women consulting reninjaza fear that there will be complications during labour and delivery, that they will be referred too late to a healthcare center or that they will have to undergo risky and potentially fatal emergency surgery. Some pregnant women therefore decide to save up to make sure they receive antenatal care from a healthcare facility or turn to midwives who attend home births. Reninjaza are fully aware of the limitations of their own knowledge, especially with regard to possible complications during childbirth. However, none of the reninjaza, or traditional practitioners fulfilling a similar function we met, mentioned any deaths of women during childbirth. This is surprising in light of the fact that maternal mortality rates lead to believe that many reninjaza would have witnessed the death of a patient at least once in their career. What they mentioned instead were deaths of newborn babies, fatalities that are treated as commonplace by women as much as by reninjaza. Their attitudes and perceptions of the realities of maternal and infant mortality therefore reinforce the derogatory discourse of biomedical practitioners towards them.

On the other hand, biomedical professionals never mention the difficulties they face with pregnant women, nor their perceptions of healthcare centers, or the conflicting and violent patient-carer relationships that emerge within these same facilities [START_REF] Mestre | Maladies et violences ordinaires dans un hôpital malgache[END_REF]). Yet they reinforce power dynamics based on socio-economic inequalities and on the classic asymmetry between patients and holders of medical knowledge. These aspects, which often complicate patient healthcare, are often overlooked by analyses of health workers' practices in health facilities [START_REF] Jaffré | Une médecine inhospitalière. Les difficiles relations entre soignants et soignés dans cinq capitales d'Afrique de l'ouest[END_REF]. Biomedical professionals tend to attribute deaths during childbirth to harmful practices by reninjaza, to the lack of (financial, materiel and human) resources in health facilities, and to many women's early and frequent pregnancies. However, even in the middle and privileged classes in urban areas, especially in the capital, some pregnant women turn to reninjaza, even though their social and geographical backgrounds are very different from those of women from low-income, underprivileged and rural classes. What these women have in common is their misgivings regarding how they will be received by health workers, the attitudes of some biomedical practitioners who are neither attentive to their needs, nor competent or even present, and regarding unnecessary surgery.

Maternal health in Madagascar would certainly benefit from reflection and changes based on patients' perceptions, experiences and practices, as well as on their trajectories within the healthcare system. For the essentialised, simplistic and stigmatising distinctionthat tends to divide medical knowledge between traditionalism on the one hand and "modernity" on the other fuels power issues that prevent from gaining a full understanding of the social realities at stake.

In Madagascar, traditional medicine is defined institutionally as the "whole body of knowledge and practices used for diagnosis, prevention and elimination of physical, mental and social imbalances, and based exclusively on practical experience and observations handed down from generation to generation, either orally or in writing, which are used to prevent and cure diseases, and alleviate suffering."(Public Health Code, Bill no. 2011-002).

The term literally means "mother of the child" and can be translated as "traditional midwife" ["matrone", in French], even though some men also carry out this position that is supposed to be for women only. More generally, the word reninjaza refers to traditional practitioners who provide healthcare during pregnancy, and during and after childbirth. This is the most commonly used term locally, whereas political and institutional texts use "traditional birth attendant".

The maternal mortality rate per 100,000 live births in Madagascar was 660 for the 1986-1992 period, 469 during the 1998-2000 period, and 498 during the 2002-2009 period. In 2009, the rate of stillbirths was estimated at 21 per 1000 live births(WHO, 2013).

Concept used to define childbirth as a biological, social and symbolic event(Jordan, 1978). Moreover, "it defines all actors, practices and perceptions mobilised around this event in a specific healthcare system (be it popular, traditional or biomedical)."(Hancart Petitet, 2011: 11).

Article 96 of Bill no. 2011-002 provides that the word reninjaza refers to "any person deemed fit to provide healthcare to women and their newborn babies before, during and after childbirth on the basis of prevalent concepts in her community."

These field works were carried out as part of a programme entitled "Risks and social challenges of reproduction in Madagascar" headed by the Institute of Research for Development and led by D. Pourette; of the project entitled "Patrimonialisation of knowledge and practices of traditional birth attendants in Madagascar" funded by the Institute of Research for Development (Regional Pilot Programme on Heritage, Resources and Governance); and of the project entitled "Determinants of use or non-use of antenatal and birthing health facilities in Moramanga" funded by the Institut Pasteur in Madagascar and led by P. Piola and D. Pourette.

For example, sovoka refers to cold air penetrating into the woman following childbirth, which can be fatal. To prevent sovoka, the mother must observe a rest period and stay warm for several weeks or as much as 3 months depending on the region. Reninjaza administer specific herbal teas during this period called mifana.

Association in charge of identifying and recruiting traditional practitioners in Madagascar, and which consists of several member associations.

Term that originally referred to "white" Europeans, but now refers to the Western world more generally.

In Madagascar, tourism, environmental resources, homeopathy and natural heritage are economic sectors that combine development dynamics and identity politics.