On stability of continuous-time quantum filters

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On stability of continuous-time quantum filters

Hadis Amini, Mazyar Mirrahimi, Pierre Rouchon

To cite this version:

Hadis Amini, Mazyar Mirrahimi, Pierre Rouchon. On stability of continuous-time quantum filters.

50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) 2011,

Dec 2011, Orlando, United States. pp.6242 - 6247, �10.1109/CDC.2011.6160631�. �hal-00683468�

arXiv:1103.2706v2 [math.OC] 8 Aug 2011

On stability of continuous-time quantum filters ^∗

H. Amini

M. Mirrahimi

P. Rouchon

August 9, 2011

Abstract

We prove that the fidelity between the quantum state governed by a continuous time stochastic master equation driven by a Wiener process and its associated quantum-filter state is a sub-martingale. This result is a generalization to non-pure quantum states where fidelity does not coincide in general with a simple Frobenius inner product. This result implies the stability of such filtering process but does not necessarily ensure the asymptotic convergence of such quantum-filters.

1 Introduction

The quantum filtering theory provides a foundation of statistical inference in- spired in e.g. quantum optical systems. These systems are described by continuous- time quantum stochastic differential equations. These stochastic master equa- tions include the measurement back-action on the quantum-state. The quantum filtering theory has been developed by Davies in the 1960s [10, 11] and in its modern form by Belavkin in the 1980s [4, 5, 3].

To these stochastic master equations are attached so-called quantum filters providing, from the real-time measurements, estimations of the quantum states.

Robustness and convergence of such estimation process has been investigated in many papers. For example, sufficient convergence conditions, related to ob- servability issues, are given in [20] and [19]. As far as we know, general and verifiable necessary and sufficient convergence conditions do not exist yet. For links between quantum filtering and observers design on cones see [6]. In this paper, we generalize a stability result for pure states (see, e.g., [12]) to arbitrary

∗This work was partially supported by the ”Agence Nationale de la Recherche” (ANR), Projet Blanc CQUID number 06-3-13957 and Projet Jeunes Chercheurs EPOQ2 number ANR- 09-JCJC-0070.

†Mines ParisTech, Centre Automatique et Syst`emes, Math´ematiques et Syst`emes, 60 Bd Saint Michel, 75272 Paris cedex 06, France,[email protected]

‡INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay cedex, France,[email protected]

§Mines ParisTech, Centre Automatique et Syst`emes, Math´ematiques et Syst`emes, 60 Bd Saint Michel, 75272 Paris cedex 06, France,[email protected]

mixed quantum states. More precisely, we prove that the fidelity between the quantum state (that could be a mixed state) and its associated quantum-filter state is a sub-martingale: this means that in average, the estimated state tends to be closer to the system state. This does not imply its asymptotic convergence for large times. To prove such convergence, more specific analysis depending on the precise structure of the Hamiltonian, Lindbladian and measurement opera- tors defining the system model is required. This paper can also be seen as an extension to continuous-time evolution of [18].

This paper is organized as follows. In section 2, we introduce the non linear stochastic master equations driven by Wiener processes and providing the evo- lutions of the quantum state and of the quantum-filter state and we state the main result (Theorem 2.1). Section 3 is devoted to the proof of this result: firstly we consider an approximation via stochastic master equations driven by Pois- son processes (diffusion approximation); secondly, we prove the sub-martingale property via a time discretization. In final section, we suggest some possible extensions of this work.

2 Main result

We will 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 the evolution of the real stateρ∈ D is described by the following stochastic master equation (cf. [3, 7, 22])

dρ_t=−~ⁱ[H, ρt]dt+L(ρt)dt+ Λ(ρt) dW_t, (1) where

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

• H =H^† is a Hermitian operator which describes the action of external forces on the system ;

• dW_tis the Wiener process which is the following innovation dW_t=dy_t−Tr (L+L^†)ρ_t

dt , (2)

whereytis a continuous semi-martingale with quadratic variationhy, yit= t (which is the observation process obtained from the system) and L is an arbitrary matrix which determines the measurement process (typically the coupling to the probe field for quantum optic systems) ;

• the super-operatorLis the Lindblad operator, L(ρ) :=−¹₂{L^†L, ρ}+LρL^†, where the notation{A, B} refers toAB+BA;

• the super-operator Λ is defined by

Λ(ρ) :=Lρ+ρL^†−Tr (L+L^†)ρ ρ.

All the developments remain valid whenH andLare deterministic time-varying matrices. For clarity sake, we do not recall below such possible time dependence.

The evolution of the quantum filter of stateρbt∈ D is described by the fol- lowing stochastic master equation which depends on the time-continuous mea- surementytdepending on the true quantum stateρtvia (2) (see, e.g., [1]):

dρb_t=−~ⁱ[H,bρ_t]dt+L(ρb_t)dt+ Λ(ρb_t) dy_t− Tr (L+L^†)ρb_t dt

. (3) Replacingdy_t by its value given in (2), we obtain

dρb_t=−~ⁱ[H,ρb_t]dt+L(ρb_t)dt+ Λ(ρb_t)dW_t + Λ(bρt)

Tr (L+L^†)ρt

−Tr (L+L^†)ρbt dt.

A usual measurement of the difference between two quantum states ρand σ, is given by the fidelity, a real number between 0 and 1. More precisely, the fidelity betweenρandσinDis given by (see [16, chapter 9] for more details)

F(ρ, σ) = Tr²q√ρσ√ρ

. (4)

HereF(ρ, σ) = 1 meansρ=σ, andF(ρ, σ) = 0 means that the support ofρand σ are orthogonal. F(ρ, σ) coincides with their inner product Tr (ρσ) when at least one of the statesρorσis pure (i.e., orthogonal projector of rank one). It is well known that the stochastic master equations (1) and (3) leave the domain Dpositively invariant. This results form the fact that, using Ito rules, we have

ρt+dt =

I−iH

~ dt−¹₂L^†Ldt+Ldyt

ρt

I−iH

~dt−¹₂L^†Ldt+Ldyt

†

Tr I−iH

~ dt−¹₂L^†Ldt+Ldyt

ρt

I−iH

~dt−¹₂L^†Ldt+Ldyt

† (5) and

b

ρt+dt =

I−iH

~ dt−¹₂L^†Ldt+Ldyt

ρbt

I−iH

~dt−¹₂L^†Ldt+Ldyt

†

Tr I−iH

~ dt−¹₂L^†Ldt+Ldyt

ρbt

I−iH

~dt−¹₂L^†Ldt+Ldyt

† (6) wheredy_t= Tr (L+L^†)ρ_t

dt+dW_t.

These alternative formulations imply then directly that, as soon as,ρ0 and b

ρ0 belong toD,ρ_tandρb_tremain inDfor allt≥0. Therefore the expression of fidelity given by (4) is well defined.

We are now in position to state the main result of this paper.

Theorem 2.1. Consider the Markov processes (ρt,ρb_t) satisfying the stochastic master Equations (1) and (3) respectively with ρ0, ρb0 in D. Then the fidelity F(ρt,ρb_t), defined in Equation (4), is a submartingale, i.e. E(F(ρt,ρb_t)|(ρs,ρb_s))≥ F(ρs,ρb_s),for allt≥s.

We recall that the above theorem generalize the results of [12] to arbitrary purity of the real states and quantum filter. Ifρ0 is pure, thenρ_tremains pure for allt >0. In this case,F(ρ_t,ρb_t) coincides with Tr (ρ_tρb_t). It is proved in [12]

that this Frobenius inner product is a sub-martingale for any initial value ofρb_t:

d

dtE(Tr (ρtρbt)) ≥0. The main idea of the proof in [12] consists in using Itˆo’s formula to reduce the theorem to showing thatE(Tr (dρtρbt+ρtdρbt+dρtdρbt))≥ 0, and then using the shift invariance of the operator L in the dynamics (1) and (3) and choosing an appropriate value.

In the absence of any information on the purity of the real states and the quantum filter, the fidelity is given by (4), and the application of Itˆo’s formula for the above expression becomes much more involved. In particular, the calculation of the cross derivatives was so complicated that it became hopeless to proceed this way. As the proof presented in the next section shows, we had to choose an undirect way to approach the theorem which allowed us to avoid the heavy calculations based on second order derivative ofF.

3 Proof of Theorem 2.1

We proceed in two steps.

• In the first step, we describe briefly how we obtain the stochastic master equations (1) and (3) as the limits of the stochastic master equations with Poisson processes using the diffusive limits inspired from the physical homodyne detection model [2, 23].

• In the second step, we show that the fidelity between the real state and the quantum filter which are the solutions of stochastic master equations with Poisson processes is a submartingale.

Step 1. Take α >0 a large real number and consider the evolution of the quantum stateρ^α_t described by the following stochastic master equation derived from homodyne detection scheme (see section 6.4 of [8] or [2], [23]) for more physical details):

dρ^α_t =−~ⁱ[H, ρ^α_t]dt−¹4Λα(ρ^α_t)dt+ Υα(ρ^α_t)dN1 (7)

−¹4Λ−α(ρ^α_t)dt+ Υ−α(ρ^α_t)dN2, where the super-operators Υαis defined as follows

Υα(ρ) := (L+α)ρ(L^†+α) Tr ((L+α)ρ(L^†+α))−ρ,

and the super-operator Λαis defined by

Λα(ρ) := (L+α)ρ+ρ(L^†+α)−Tr (L+L^†+ 2α)ρ ρ.

The super-operators Λ−α and Υ−α are just obtained with replacing αby

−αin the expressions given in above.

The two processesdN1 anddN2 are defined by

dN1:=N_t+dt¹ −N_t¹ and dN2:=N_t+dt² −N_t²

whereN¹andN²are two Poisson processes. dN1anddN2take value 1 by prob- abilities ¹2Tr (L^†+α)(L+α)ρ^αt

dt and ¹2Tr (L^†−α)(L−α)ρ^αt

dt, respectively, and take value 0 by the complementary probabilities.

Similarly, the following stochastic master equation describes the infinitesimal evolution of associated quantum filter of stateρb^α_t (see [1]):

dρbtα=−~ⁱ[H,ρbtα]dt−¹₄Λα(ρbtα)dt+ Υα(ρbtα)dN1 (8)

−¹₄Λ−α(ρbtα)dt+ Υ−α(ρbtα)dN2.

The following diffusive limit is obtained by the central limit theorem when αtends to +∞for the semi-martingale processes applied todNq,q= 1,2, (see [15] or [14] for more details)

dNq

−→ hlaw ^dN_dt^qidt+ q

h^dN_dt^qidWq, (9) where the notationhAirefers to the mean value of A. Here

hdN1i= ¹₂Tr (L^†+α)(L+α)ρ^α_t

dtand hdN2i= ¹₂Tr (L^†−α)(L−α)ρ^α_t dt anddW1 anddW2 are two independent Wiener processes and the convergence in (9) is in law.

The stochastic master Equations (1) and (3) are obtained by replacing the processes dN_q for q ∈ {1,2} by their limits given in (9) in the master equa- tions (7) and (8) and taking the limit whenαgoes to +∞and keeping only the lowest ordered terms in α⁻¹. Such a result is usually called diffusion approxi- mation (see e.g [9]).

Notice thatdW appearing in the stochastic master equations (1) and (3) is given in terms of its independent constituents by

dW = q1

2 dW1+dW2

, and is thus itself a standard Wiener process.

The following theorem from [17] justifies the diffusion approximation de- scribed above.

Theorem 3.1 (Pellegrini-Petruccione [17]). The solutions of the stochastic master Equations (7) and (8) converge in law, when α → +∞, to the solu- tions of the stochastic master Equations (1) and (3), respectively.

Step 2. We now prove that the fidelity between two arbitrary solutions of the stochastic master Equations (7) and (8) is a submartingale.

Proposition 3.1.Consider the Markov process(ρ^α,ρb^α)which satisfy the stochas- tic master Equations (7) and (8). Then the fidelity defined in Equation (4) is a submartingale, i.e., for allt≥s, we have

E(F(ρ^α_t,ρb^α_t)|(ρ^α_s,ρb^α_s))≥F(ρ^α_s,ρb^α_s).

Proof. We consider approximations of the time-continuous Markov processes (7) and (8) by discrete-time Markov processesξk andξbk:

ξk+1= ^MµkξkM

† µk

Tr(^MµkξkM_µk^† ) and ξbk+1 = ^MµkξbkM

† µk

Tr(^MµkξbkM_µk^† ), (10) where

• k∈ {0,· · · , n} for a fixed largen;

• initial conditionξ0=ρ^α_s andξb0=ρb^α_s;

• µkis a random variable taking valuesµ∈ {0,1,2}with probabilityPµ,k = Tr MµξkM_µ^†

;

• The operatorsM0, M1 andM2 are defined as follows

M0:= 1−¹4(L^†+α)(L+α)ǫn−¹4(L^†−α)(L−α)ǫn−~ⁱHǫ_n; M1:= (L+α)q

1 2ǫn; and

M2:= (L−α)q

1 2ǫn; withǫ_n= ^t−s_n .

In the following lemma, we show that ξn and ξbn correspond to the Euler- Maruyama time discretization. Since (7) and (8) depend smoothly onρ^α_t and b

ρ^α_t,ξ_n andξb_n converge in law towardsρ^α_t andρb^α_t whenn7→+∞.

Lemma 3.1. The processes ξ_k andξb_k correspond up to second order terms in ǫ_n, to the Euler-Maruyama discretization scheme of (7)and (8)on [s, t].

Proof. we regard the three following possible cases which arrive in according to the different values of µ_k . In each case, we show that ξ_k and ξb_k for k ∈ {0,· · · , n}are the numerical solutions of the dynamics (7) and (8) respectively, with the following partition s≤ s+ǫ_n ≤ · · · ≤ s+ (n−1)ǫn ≤ t,where the uniform step lengthǫ_n is ^t−s_n .

Case 1. We first consider the case whereµk = 0 which arrives with proba- bilityP0,k= Tr

M0ξ_kM₀^†

.Note that

M0ξkM₀^†=ξk−¹₄{(L^†+α)(L+α), ξk}ǫn

−¹4{(L^†−α)(L−α), ξk}ǫ_n

−~ⁱ[H, ξk]ǫn+O(ǫ²_n).

Therefore Tr

M0ξkM₀^†

= 1−¹₂Tr (L^†+α)(L+α)ξk

ǫn

−¹₂Tr (L^†−α)(L−α)ξ_k

ǫ_n+O((ǫ_n)²) and

Tr

M0ξ_kM₀^{† −1}≈1 +¹

2Tr (L^†+α)(L+α)ξk

ǫ_n+

1

2Tr (L^†−α)(L−α)ξk

ǫ_n+O((ǫn)²).

Therefore, we find the following dynamics

ξk+1 ≈ξk−¹4{(L^†+α)(L+α), ξk}ǫn

−¹4{(L^†−α)(L−α), ξk}ǫ_n +¹

2Tr (L^†+α)(L+α)ξk

ξkǫn

+¹

2Tr (L^†−α)(L−α)ξk

ξkǫn+O(ǫ²_n).

This can also be written as follows

ξk+1−ξk≈ −¹₄Λα(ξk)ǫn−¹₄Λ−α(ξk)ǫn+O(ǫ²_n). (11) Obviously, this dynamics in the first order ofǫ_nis equivalent to the dynamics of the numerical solution of the stochastic master Equation (7) with the partition s≤s+ǫn≤ · · · ≤s+ (n−1)ǫn≤t,when

N_s+(k+1)ǫ¹ _n−N_s+kǫ¹ _n= 0 and N_s+(k+1)ǫ² _n−N_s+kǫ² _n= 0, which arrives with probability

1−¹₂Tr (L+α)(L^†+α)ξ_k ǫ_n

· · ·

· · · 1−¹₂Tr (L−α)(L^†−α)ξk

ǫn

. This probability, in the first order ofǫ_n is equal to Tr

M0ξ_kM₀^† .

Case 2. The second case corresponds toµ_k = 1 which arrives with proba- bility Tr

M1ξkM₁^†

. We find the following dynamics ξk+1=_Tr((L+α)ξ^{(L+α)ξk(L†+α)}

k(L^†+α)) = Υ[L+α]ξk+ξk.

We observe that the numerical solution of the stochastic master Equation (7) follows also the same dynamics when

N_s+(k+1)ǫ¹

n−N_s+kǫ¹ _n= 1 and N_s+(k+1)ǫ²

n−N_s+kǫ² _n= 0, which arrives with probability

1

2Tr (L+α)(L^†+α)ξk

ǫn

1−¹₂Tr (L−α)(L^†−α)ξk

ǫn

. This is equal to Tr

M1ξkM₁^†

,in the first order ofǫn.

Case 3. Now we consider the last caseµ_k= 2 which arrives with probability Tr

M2ξ_kM₂^†

.Therefore, we have ξk+1= _{Tr((L−α)ξ}^{(L−α)ξk(L†−α)}

k(L^†−α)) = Υ−α(ξk) +ξ_k.

Which can also be written by the stochastic master equation (7) with takingξ_k as the numerical solution and

N_s+(k+1)ǫ¹ _n−N_s+kǫ¹ _n= 0 and N_s+(k+1)ǫ² _n−N_s+kǫ² _n= 1, which arrives with probability

1−¹₂Tr (L+α)(L^†+α)ξk

ǫn

1

2Tr (L−α)(L^†−α)ξk

ǫN

. Where in the first order ofǫ_n,this probability is equal to Tr

M2ξ_kM₂^† . Remark that, if we neglect the terms in the order of ǫ²_n,The probability of N_s+(k+1)ǫ¹

n−N_s+kǫ¹ _n = 1 and N_s+(k+1)ǫ²

n−N_s+kǫ² _n = 1 is negligible. Now it is clear thatξk and similarlyξbk are respectively the numerical solutions of the stochastic master Equations (7) and (8) obtained by Euler-Maruyama method.

As the right hand side of the stochastic master Equations (7) and (8) are smooth with respect toρ andρ, we can use the result of [13, Theorem 1] to concludeb the convergence in law ofξn andξbn toρ^α_t andρb^α_t for largen.

Now we notice that

M₀^†M0+M₁^†M1+M₂^†M2=I+O(ǫ²_n) :=A, TakeMf_r:= (√

A)⁻¹M_rforr= 0,1,2 which satisfy necessarily Mf0

†Mf0+Mf1

†Mf1+Mf2

†Mf2=I. (12)

Now we define the following Markov processesχ_k andχb_k by χk+1= ^{MgµkχkMgµk}

†

Tr

Mg_µkχkMg_µk^† (13)

and

b

χk+1= ^{MgµkχbkMgµk}

†

Tr

Mg_µkχbkMg_µk^†, (14) where

• k∈ {0,· · · , n} for a fixed largen;

• χ0=ρ^α_s andχb0=ρb^α_s;

• µ_kis a random variable taking valuesµ∈ {0,1,2}with probabilityP_µ,k = Tr

Mf_µχ_kMf_µ^† .

Clearlyχkandχbkcan also be seen as the numerical solutions of the stochas- tic master Equations (7) and (8), since (√

A)⁻¹ =I− O(ǫ²_n), therefore in the first order ofǫn, the solutions ξk and ξbk are equal to χk and χbk, respectively.

But, the advantage of usingχk andχbkinstead ofξk andξbkis that the operators Mfr are Kraus operators since they satisfy Equality (12). Thus we can apply Theorem 1 in [18], which proves thatF(χk,χbk) is a sub-martingale.

Theorem 3.2([18]). Consider the Markov chain (χk,χbk) satisfying (13) and (14).

ThenF(χk,χbk) is a sub-martingale: E(F(χk+1,χbk+1)|(χk,χbk))≥F(χk,χbk).

Thus we have

E(F(χn,χbn)| χ0,χb0)≥F(χ0,χb0) =F(ρ^α_s,ρb^α_s) Therefore by Lemma 3.1, we have necessarily

E(F(ρ^α_t,ρb^α_t)|ρ^α_s,ρb^α_s))≥F(ρ^α_s,ρb^α_s),

for all t ≥ s, since we have (convergence in law) ρ^α_t = limn−→∞χ_n, ρb^α_t = limn−→∞χb_n, χ0=ρ^α_s andχb0=ρb^α_s.

We now apply Theorem 3.1 and we use the fact that the function F is bounded by one and continuous with respect toρandρ:b

E(F(ρt,ρbt)|(ρs,ρbs))≥F(ρs,ρbs), for allt≥s,which ends the proof of Theorem 2.1.

4 Numerical Test

In this section, we test the result of Theorem 2.1 through numerical simulations.

Considering the two-level system of [21], we take the following Hamiltonian and measurement operators:

H =σ_y =

0 −i i 0

L=σ_z= 1 0

0 −1

.

The simulations of figure 1 illustrates the fidelity for 500 random trajectories starting at

ρ0= ₁

2 1 1 4 4

1 2

and ρb0= ₁

3 0 0 ²₃

.

In particular, we note that both initial states are mixed ones. As it can be seen the average fidelity is monotonically increasing. Here, the fidelity converges to one indicating the convergence of the filter towards the physical state. An interesting direction here is to characterize the situations where this convergence is ensured.

Here in order to simulate the Equations (1) and (3), we have considered the alternative formulations (5) and (6) and the resulting discretization scheme (k∈Nand time step 0< dt≪1)

ρ(k+1)dt= ^{Mkρ(kdt)M†k}

Tr(^Mkρ(kdt)M^†_k), ρb(k+1)dt= ^{Mkbρ(kdt)M†k}

Tr(^Mkρb(kdt)M^†_k), whereMk=I−^iH~dt−¹2L^†Ldt+Ldy(kdt)anddy_(kdt)= Tr (L+L^†)ρ_(kdt)

dt+

dW_(kdt). For eachk, the Wiener incrementdW_(kdt)is a centered Gaussian ran- dom variable of standard deviation√

dt. The major interest of such discretiza- tion is to guaranty that, ifρ0,ρb0∈ D, thenρk andρbk also remain inDfor any k≥0.

5 Concluding remarks

The fact that the fidelity between the real quantum state and the quantum- filter state increases in average remains valid for more general stochastic master equations where other Lindbald terms are added toL(ρ) appearing in (1). In this case the dynamics (1) and (3) become

dρ_t = −~ⁱ[H, ρt] dt +

m^′

X

ν=1

L^′ν(ρt)dt + Xm

µ=1

Lµ(ρt)dt + Xm

µ=1

Λµ(ρt)dW_t^µ

dρbt=−~ⁱ[H,ρbt]dt+

m^′

X

ν=1

L^′ν(ρbt)dt+ Xm

µ=1

Lµ(ρbt)dt

+ Xm

µ=1

Λµ(ρbt)

dy_t^µ− Tr (Lµ+L^†_µ)ρbt

dt

.

0 0.5 1 1.5 2 2.5 3 0.9

0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

Fidelity

Time

Figure 1: The average fidelity between the Markov processesρandρ,b over 500 realizations, timetfrom 0 toT = 3 with discretization time stepdt= 10⁻⁴. wheredW_t^µ are independent Wiener processes,

Lµ(ρ) :=−¹₂{L^†_µLµ, ρ}+LµρL^†_µ, L^′ν(ρ) :=−¹₂{L^′_ν^†L^′_ν, ρ}+L^′_νρL^′_ν^†, and Λµ(ρ) :=L_µρ+ρL^†_µ−Tr (Lµ+L^†_µ)ρ

ρ.

Here m, m^′ ≥1, and (L^′_ν)1≤ν≤m^′ and (Lµ)1≤µ≤m are arbitrary operators.

The special case considered here corresponds tom= 1 andm^′= 1 withL1=L andL^′₁= 0.The formulations analogue to (5) and (6) read then

ρt+dt= (I−dMt)ρt(I−dM_t^†) +Pm^′

ν=1L^′_νρtL^′_ν^†dt Tr

(I−dM_t)ρt(I−dM_t^†) +Pm^′

ν=1L^′_νρ_tL^′_ν^†dt and

b

ρt+dt= (I−dM_t)ρb_t(I−dM_t^†) +Pm^′

ν=1L^′_νρb_tL^′_ν^†dt Tr

(I−dM_t)ρb_t(I−dM_t^†) +Pm^′

ν=1L^′_νρb_tL^′_ν^†dt where, denotingdy^µ_t = Tr (Lµ+L^†_µ)ρt

dt+dW_t^µ,

dMt= ^iH~dt+¹

2 m^′

X

ν=1

L^′_ν^†L^′_νdt+¹

2

Xm

µ=1

Lµ†

Lµdt− Xm

µ=1

Lµdy^µ_t.

For this general case, the proof of Theorem 2.1 should follow the same lines:

first step still relies on Theorem 3.1; second step relies now on [18, Theorem 2].

Acknowledgements. The authors thank C. Pellegrini, L. Zambotti and M.

Gubinelli for stimulating discussions and useful suggestions during the prepara- tion of the paper.

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