Quantum Repeated Measurements, Continuous Time Limit

Quantum Repeated Measurements, Continuous Time Limit Clément Pellegrini Institut de Mathématiques de Toulouse, Laboratoire de Statistique et Probabilité, Université Paul Sabatier Autrans/ July 2013 Clément Pellegrini (IMT) Quantum Trajectories, SSE 1 / 48

Plan I) Discrete time Model: Quantum Repeated measurement H. Maasen: ergodic properties, purification properties. L. Bouten, R. Van Handel: discrete quantum filtering M. Bauer, D. Bernard, T. Benoist: large time behaviour, rate of convergence, quantum nondemolition measurement P. Rouchon and all: control (direct collaboration with the serge Haroche team at LKB) M. Merkli: quantum measurement of scattered particles S. Attal, N. Guillotin-Plantard, C. Sabot: Central Limit Theorem for OQRW II) Continuous Time model: Stochastic Schrödinger Equations, Stochastic Master Equations. Using linear stochastic master equations and change of measure (A.Barchielli, M.Gregoratti...) Quantum Filtering Theory based on Quantum stochastic differential equation (V.P.Belavkin...) Positive Operator Valued Measure and Instruments (E.B Davies...) From discrete to continuous time model (adapting the results of S.Attal-Y.Pautrat...) Clément Pellegrini (IMT) Quantum Trajectories, SSE 2 / 48

Plan III) Continuous Time Limit Continuous time limit of Quantum Repeated Measurements (P., also M. Bauer, D. Bernard, T. Benoist...) IV) Estimation, Temperature Estimation: work in progress with P. Rouchon And H. Amini Temperature (with I. Nechita and S. Attal) Clément Pellegrini (IMT) Quantum Trajectories, SSE 3 / 48

I) Discrete Time Model QUANTUM REPEATED MEASUREMENT and DISCRETE QUANTUM TRAJECTORIES Clément Pellegrini (IMT) Quantum Trajectories, SSE 4 / 48

1st interaction S, ρ 0 H 1... H 2... H 3...... H k... Clément Pellegrini (IMT) Quantum Trajectories, SSE 5 / 48

1st measurement S H 1... H 2... H 3...... H k Measurement Clément Pellegrini (IMT) Quantum Trajectories, SSE 6 / 48

2nd interaction S, ρ 1 H 1 H 2... H 3...... H k... Clément Pellegrini (IMT) Quantum Trajectories, SSE 7 / 48

2nd measurement S H 1 H 2... H 3...... H k... Measurement and so on discrete quantum trajectory (ρ n ) Clément Pellegrini (IMT) Quantum Trajectories, SSE 8 / 48

Proposition Let A be an observable of the form A = p λ i P i. i=0 Then there exists a probability space (Ω,C,P), where the the discrete quantum trajectory (ρ k ), describing the quantum repeated measurement of A, is a Markov chain. More precisely if ρ k = θ is a state on H 0, then ρ k+1 takes the values L i (θ) Tr[L i (θ)], i {0,...,p} where L i (θ) = Tr H [(I P i ) U(θ β)u (I P i )]. Each state appears with probability p i (θ) = Tr[L i (θ)]. Clément Pellegrini (IMT) Quantum Trajectories, SSE 9 / 48

The previous result can be summarized by writing the following evolution equation: ρ k+1 = p i=0 L i (ρ k ) Tr[L i (ρ k )] 1k+1 i, with L i (θ) = Tr H [(I P i ) U(θ β)u (I P i )]. Remark: The operator U depends on the time interaction τ U = e iτhtot. Questions What gives the limit τ goes to 0? What is the limit evolution? Clément Pellegrini (IMT) Quantum Trajectories, SSE 10 / 48

II) Stochastic Master Equations Continuous Quantum Trajectories Clément Pellegrini (IMT) Quantum Trajectories, SSE 11 / 48

Introduction Let us start with model where only one measurement apparatus is concerned Evolution of H 0 without measurement : Master Equation in Lindblad form dρ t = L(ρ t )dt. Effect of measurement = perturbation of this ode under the form of stochastic differential equations Stochastic Master Equations Clément Pellegrini (IMT) Quantum Trajectories, SSE 12 / 48

Diffusive Equation Diffusive Equation dρ t = L(ρ t )dt +[Cρ t +ρ t C Tr[ρ t (C +C )]ρ t ]dw t 1 The process (W t ) is a standard Brownian motion. 2 C is an arbitrary operator appearing in the Lindblad operator, L. Often this equation appears on the following form dρ t = L(ρ t )dt +[Cρ t +ρ t C Tr[ρ t (C +C )]ρ t ](dy t Tr[ρ t (C +C )]dt), where dy t = dw t +Tr[ρ t (C +C )]dt The process (y t ) represents the measurement process recorded by the measurement apparatus (Homodyne/Heterodyne detection in Quantum Optics). Clément Pellegrini (IMT) Quantum Trajectories, SSE 13 / 48

Jump Equation Jump Equation [ ] J(ρt ) ( dρ t = L(ρ t )dt + Tr[J(ρ t )] ρ t dñ t Tr [ J(ρ t ) ] dt) 1 The process (Ñ t ) is a counting process of stochastic intensity 2 J(ρ) = CρC. t t 0 Tr[J(ρ s )]ds. The process (Ñ t ) represents the number of photon detected up to time t by a photo detector. Clément Pellegrini (IMT) Quantum Trajectories, SSE 14 / 48

First fact: de[ρ(t)] = L(E[ρ(t)])dt. That is (E[ρ(t)]) reproduces the solution of the Lindblad master equation In the previous cases if L(ρ) = i[h,ρ] 1 2 {C C,ρ}+CρC and if at time 0 ρ 0 = ψ 0 ψ 0 then there exists ψ t such that ρ(t) = ψ t ψ t, t The equation satisfied by ψ t is called a Stochastic Schrödinger Equation In general we have ρ(t) = ρ (t) and Tr[ρ(t)] = 1, then if there is a solution and if the initial condition is a density matrix then the solution is self-adjoint and of trace 1. What is very difficult to show is that the solution is positive. Clément Pellegrini (IMT) Quantum Trajectories, SSE 15 / 48

Non Lipschitz coefficients One can use a truncature method We show that this equation preserves the property of being a state. Here we can use the approximation procedure to show the positivity of the solution Clément Pellegrini (IMT) Quantum Trajectories, SSE 16 / 48 Existence and uniqueness: the diffusive case Theorem Let (Ω,F,P) a probability space where (W t ) is a standard Brownian motion. the equation dρ t = L(ρ t )dt ] + [Cρ t +ρ t C Tr[ρ t (C +C )]ρ t dw t admits a unique solution (ρ t ) with values in the set of states of H 0.

Processus de comptage (Ñ t )? Jump equation dρ t = L(ρ t )dt + [ ] J(ρt ) Tr[J(ρ t )] ρ t (dñ t Tr[J(ρ t )]dt) Recall: (Ñ t ) is a counting process of intensity t 0 Tr[J(ρ s)]ds. Questions What is the meaning of this equation? How can we define (Ñ t ) and (ρ t )? Clément Pellegrini (IMT) Quantum Trajectories, SSE 17 / 48

Process-solution de [ ] dρ t = L(ρ t )dt + J(ρt ) Tr[J(ρ t )] ρ t (dñ t Tr[J(ρ t )]dt) Definition Let (Ω,F,F t,p) be a probability space. A process-solution of the jump equation is a couple (ρ t,ñ t ) such that t ] ρ t = ρ 0 + [L(ρ s ) J(ρ s )+Tr[J(ρ s )]ρ s ds 0 t [ ] J(ρs ) + Tr[J(ρ s )] ρ s dñ s a.s and such that is a F t -martingale. 0 t Ñ t Tr[J(ρ s )]ds 0 Clément Pellegrini (IMT) Quantum Trajectories, SSE 18 / 48

Existence and uniqueness Theorem Let (Ω,F,F t,p) be a probability space equipped with a random Poisson measure µ on R + R whose the intensity measure is the Lebesgue measure ds dx. The SDE ρ t = ρ 0 + t + 0 R t 0 ] [L(ρ s ) J(ρ s )+Tr[J(ρ s )]ρ s ds ] 1 0<x<tr[J(ρs )]µ(ds,dx) [ J(ρs ) Tr[J(ρ s )] ρ s admits a unique solution (ρ t ). The process (Ñ t ) defined by Ñ t = t 0 R 1 0<x<tr[J(ρs )]µ(ds,dx) is a counting process of intensity t 0 Tr[J(ρ s )]ds. Clément Pellegrini (IMT) Quantum Trajectories, SSE 19 / 48

Generalisation One can generalise ρ t = ρ 0 + + q i=0 t 0 L(ρ s )ds + R l i=0 t 0 h i (ρ s )dw i (s) g i (ρ s )1 0<x<vi (ρ s )[µ i (dx,ds) dxds], where (W t = (W 0 (t),...,w p (t)) are a p-dimensional Brownian motion and µ i are p +1 random measure of intensity ds dx. All the processes are independent. Remark The functions h i et g i are given by h i (ρ) = C i ρ+ρci Tr[ρ(C i +Ci )]ρ J i (ρ) g i (ρ) = Tr[J i (ρ)] ρ Clément Pellegrini (IMT) Quantum Trajectories, SSE 20 / 48

III) Convergence Result FROM DISCRETE TO CONTINUOUS QUANTUM TRAJECTORIES Clément Pellegrini (IMT) Quantum Trajectories, SSE 21 / 48

Back to the discrete setup Setup 1 Case H 0 = H = C 2 2 An observable of H is of the form A = λ 0 P 0 +λ 1 P 1. 3 The discrete stochastic Schrödinger equation ρ k+1 = L 0(ρ k ) p 0 (ρ k ) 1k+1 0 + L 1(ρ k ) p 1 (ρ k ) 1k+1 1. Let us introduce X k+1 = 1k+1 1 p 1 (ρ k ) p0 (ρ k )p 1 (ρ k ). In terms of X k+1, we get ρ k+1 = L 0 (ρ k )+L 1 (ρ k )+ [ ] p0 p1 L 0 (ρ k )+ L 1 (ρ k ) X k+1. p 1 p 0 Clément Pellegrini (IMT) Quantum Trajectories, SSE 22 / 48

Asymptotic assumptions Recall that (ρ k ) is defined through the quantity L i (ρ) = Tr H [(I P i ) U(n)(ρ β)u (n) (I P i )] U(n) = e i 1 n Htot Naturally the asymptotic assumptions are going to appear in U(n). Now, we fix a basis {Ω 0,Ω 1 } The reference state of the chain will be β = Ω 0 Ω 0. Clément Pellegrini (IMT) Quantum Trajectories, SSE 23 / 48

Expression de U(n) Let us write U(n) as U(n) = ( U 0 0 (n) U 1 0 (n) U 0 1 (n) U1 1 (n) where the U ij (n) are operators H 0. S. Attal-Y. Pautrat: From repeated to continuous quantum interactions, Annales Henri Poincaré In the previous article, the authors gives a precise description of the asymptotic conditions that we need to impose to U ij (n) = in order to obtain a non-trivial limit for the quantum repeated interactions model (interms of quantum stochastic calculus). In our context, we naturally adopt their conditions and we need U0 0 (n) = I + 1 ( ih 0 1 n 2 C C U1(n) 0 = 1 ( ) 1 C + n n ) ) + ( 1 n Clément Pellegrini (IMT) Quantum Trajectories, SSE 24 / 48 )

Limit evolution in the case of a diagonal A If A is diagonal in {Ω 0,Ω 1 }. For example A = 1 Ω 0 Ω 0 +0 Ω 1 Ω 1, then we have L 0 (ρ) = U0 0 ρ(u0 0 ) = ρ+ 1 [ ( ih 0 1 n 2 C C)ρ+ρ( ih 0 1 ] 2 C C) L 1 (ρ) = U 0 1ρ(U 0 1) = 1 n CρC The transition probabilities satisfies p 0 (ρ k ) = 1 1 ( ) 1 n Tr[J(ρ k)]+ n p 1 (ρ k ) = 1 ( ) 1 n Tr[J(ρ k)]+. n Clément Pellegrini (IMT) Quantum Trajectories, SSE 25 / 48

Limit evolution in the case of a diagonal A If A is diagonal in {Ω 0,Ω 1 }. then we have ρ k+1 = ρ k + 1 n [L(ρ k)+ (1)] ( ) J(ρ k ) + Tr[J(ρ k )] ρ k + (1) (11 k+1 p 1 (ρ k )). The transition probabilities satisfies p 0 (ρ k ) = 1 1 ( ) 1 n Tr[J(ρ k)]+ n p 1 (ρ k ) = 1 ( ) 1 n Tr[J(ρ k)]+. n Clément Pellegrini (IMT) Quantum Trajectories, SSE 26 / 48

In case of a non diagonal A If A = λ 0 P 0 +λ 1 P 1 de H is not diagonal in {Ω 0,Ω 1 }. For example A = Ω 0 Ω 1 + Ω 1 Ω 0 we get the following asymptotic expression L 0 (ρ) = 1 ( U 0 2 0 ρ(u0 0 ) +U0 0 ρ(u0 1 ) +U1 0 ρ(u0 0 ) +U1 0 ρ(u0 1 ) ) (1) = 1 ( ρ+ 1 (Cρ+ρC )+ 1 ) 2 n n L(ρ) (2) L 1 (ρ) = 1 ( U 0 2 0 ρ(u0) 0 U0ρ(U 0 1) 0 U1ρ(U 0 0) 0 +U1ρ(U 0 1) 0 ) Here the probabilities are p 0 (ρ k ) = 1 2 + 1 ] [Tr[ρ k (C +C )]+ (1) n p 1 (ρ k ) = 1 p 0 (ρ k ). Clément Pellegrini (IMT) Quantum Trajectories, SSE 27 / 48

In case of a non diagonal A If A = λ 0 P 0 +λ 1 P 1 de H is not diagonal in {Ω 0,Ω 1 }. For example A = Ω 0 Ω 1 + Ω 1 Ω 0 we get the following asymptotic expression ρ k+1 = ρ k + 1 n [L(ρ k)+ (1)] + 1 ] [Cρ k +ρ k C Tr[ρ k (C +C )]ρ k + (1) X k+1. n Here the probabilities are p 0 (ρ k ) = ξ + 1 [ ] ν Tr[ρ k (C +C )]+ (1) n p 1 (ρ k ) = 1 p 0 (ρ k ). Clément Pellegrini (IMT) Quantum Trajectories, SSE 28 / 48

Convergence to the diffusive case From the previous description, we put [nt] 1 ρ [nt] = ρ 0 + Putting k=0 [nt] 1 1 n [L(ρ k)+ (1)]+ k=0 1 n [H(ρ k )+ (1)]X k+1. ρ n (t) = ρ [nt], V n (t) = [nt] n, W n(t) = 1 [nt] 1 X k+1. n k=0 We have that (ρ n (t)) satisfies ρ n (t) = ρ 0 + t 0 L(ρ n (s ))dv n (s)+ t 0 H(ρ n (s ))dw n (s)+ε n (t). Clément Pellegrini (IMT) Quantum Trajectories, SSE 29 / 48

Convergence - Kurtz-Protter Theorem The process (W n (t),v n (t),ε n (t)) converge in distribution to (W t,v t,0) where (W t ) is a standard Brownian motion and V t = t for all t. Moreover, we have [ ] supe [W n (t),w n (t)] < n Then the process (ρ n (t)) satisfying t t ρ n (t) = ρ 0 + L(ρ n (s ))dv n (s)+ H(ρ n (s ))dw n (s)+ε n (t) 0 0 converge in ditribution to (ρ t ) the unique solution of t t ρ t = ρ 0 + L(ρ s )ds + H(ρ s )dw s. 0 0 Clément Pellegrini (IMT) Quantum Trajectories, SSE 30 / 48

The jump case Again [nt] 1 1 [ ] ρ [nt] = ρ 0 + L(ρ k ) J(ρ k )+Tr[J(ρ k )]ρ k + (1) n + [nt] 1 k=0 k=0 ( J(ρ k ) Tr[J(ρ k )] ρ k + (1) ) 1 k+1 1. Again, we put ρ n (t) = ρ [nt], V n (t) = [nt] [nt] 1 n, N n(t) = 1 k+1 1. k=0 We get the discrete SDE ρ n (t) = ρ 0 + t 0 Θ(ρ n (s ))dv n (s)+ t 0 Φ(ρ n (s ))dn n (s)+ε n (t). Clément Pellegrini (IMT) Quantum Trajectories, SSE 31 / 48

Convergence Kurtz-Protter? In the jump case, we can not directly show that (N n (t)) converge in distribution to the process (Ñ t ). Method: 1 Coupling. 2 Comparison with a Euler scheme. Theorem The process (ρ n (t)) defined from the quantum repeated measurement of a diagonal observable converge in distribution to (ρ t ) solution of the jump equation. Clément Pellegrini (IMT) Quantum Trajectories, SSE 32 / 48

General case How can we compare the two method? How can we show that ρ k+1 = p i=0 L i (ρ k ) Tr[L i (ρ k )] 1k+1 i converges to ρ t = ρ 0 + + q i=0 t 0 L(ρ s )ds + R l i=0 t 0 h i (ρ s )dw i (s) g i (ρ s )1 0<x<vi (ρ s )[µ i (dx,ds) dxds]. Martingale problem. Clément Pellegrini (IMT) Quantum Trajectories, SSE 33 / 48

General case The discrete model can be describe by the transition Kernel p Π n (ρ,µ) = p i (ρ)δ (n) L i (ρ)/tr[l (n) (µ), (3) i (ρ)] i=0 We can then define the Markov generator A n f(ρ) = n (f(µ) f(ρ))π n (ρ,dµ) = n p i=0 ( ) f(l (n) i (ρ)/tr[l (n) i (ρ)]) f(ρ) p i (ρ). (4) Defining ρ n (t) = ρ [nt] and F n t = σ(ρ n (s),s t) is a (F n k/n ) martingale k 1 1 f(ρ n (k/n)) f(ρ 0 ) n A nf(ρ n (j/n)) (5) Clément Pellegrini (IMT) Quantum Trajectories, SSE 34 / 48 j=0

General case You compute the limit of A n denoted by A In distribution, there exists a unique Markov process ρ(t) such that t f(ρ(t) f(ρ(0)) Af(ρ(s))ds 0 is a martingale with respect to the natural filtration of (ρ(t)) After identifying A, one can show that this Markov generator is the same as the one of the solution of the generalization of the stochastic master equation. This gives the expected convergence in distribution Clément Pellegrini (IMT) Quantum Trajectories, SSE 35 / 48

IV) Estimation, Temperature Clément Pellegrini (IMT) Quantum Trajectories, SSE 36 / 48

Work in progress The problem of estimation concerns similar models where we do not know the initial state. Nevertheless we have access to the results of the measurement (for example the value 0 or 1). If ρ design the true initial state (that we do not know), we know that 0 appears with probability p 0 (ρ) and 1 with probability p 1 (ρ). Now let ρ an arbitrary state, conditionally to the result of the measurement, we put ρ 1 (i) = L i( ρ) Tr[L i ( ρ)] depending on wether we observe 0 or 1. Clément Pellegrini (IMT) Quantum Trajectories, SSE 37 / 48

Estimation Clément Pellegrini (IMT) Quantum Trajectories, SSE 38 / 48

Work in progress What can we expect? We can expect that the distance between ρ 1 and ρ 1 is smaller than the one between ρ and ρ. Roughly speaking this means that knowing the result of the measurement allows us to estimate the true quantum trajectory. Within the previous procedure we can produce a random sequence ρ k whose transition probability are given by the one of the true quantum trajectory. Clément Pellegrini (IMT) Quantum Trajectories, SSE 39 / 48

Work in progress Within the previous procedure we can produce a random sequence ρ k whose transition probability are given by the one of the true quantum trajectory. In order to evaluate if the true and the estimate quantum trajectory get closer we use the fidelity distance ρµ F(ρ,µ) = Tr[ ρ] 2 We have F(ρ,µ) = F(µ,ρ) and if for example ρ = ψ ψ F(ρ,µ) = 1 if and only if ρ = µ. F(ρ,µ) = Tr[ρµ] Clément Pellegrini (IMT) Quantum Trajectories, SSE 40 / 48

Work in progress We have the following result E[F( ρ k+1,ρ k+1 ) ( ρ k,ρ k )] F( ρ k,ρ k ), This means that F( ρ k,ρ k ) is a sub-martingale. It is not a trivial result since there exist distance where this property is not satisfied. Problem: What do we have to impose on the system to have lim k F( ρ k,ρ k ) = 1 Similar result for continuous time models (the discrete approach is really useful to show the sub-martingale result). Clément Pellegrini (IMT) Quantum Trajectories, SSE 41 / 48

Work in progress Similar result for continuous time models. p dρ(t) = L(ρ(t ))dt + H i (ρ(t ))(dy i (t) Tr[(C i +Ci )ρ(t )]dt) + ρ(0) = ρ 0 i=p+1 i=0 n ( ) Ji (ρ(t )) v i (ρ(t )) ρ(t ) (dn i (t) v i (ρ(t ))dt). d ρ(t) = L( ρ(t ))dt + + ρ(0) = ρ 0, i=p+1 p i=0 H i ( ρ(t ))(dy i (t) Tr[(C i +C i ) ρ(t )]dt) n ( ) Ji ( ρ(t )) v i ( ρ(t )) ρ(t ) (dn i (t) v i ( ρ(t ))dt). Clément Pellegrini (IMT) Quantum Trajectories, SSE 42 / 48

Work in progress The fidelity is still a sub-martingale Almost impossible to estimate the term df( ρ(t),ρ(t)) = dtr[ ρ(t)ρ(t) 2 ρ(t)] Use of the sub-martingale property for the discrete time result and the convergence result Clément Pellegrini (IMT) Quantum Trajectories, SSE 43 / 48

Temperature Clément Pellegrini (IMT) Quantum Trajectories, SSE 44 / 48

Temperature The state of the environment β = Ω 0 Ω 0 represents the vacuum state. Considering such a state is crucial in the approach of Attal-Pautrat. In the work of Attal-Joye, they consider a modelization of a heat bath by taking ( ) β = e βh β0 0 =, Z 0 β 1 where β is the inverse of a temperature. this is the usual Gibbs state. Using a GNS representation and adapting the work of Attal-Pautrat they manage to derive Quantum Langevin Equation for heat bath. Clément Pellegrini (IMT) Quantum Trajectories, SSE 45 / 48

Temperature What are the limit equation in the context of QRM. Following the guideline of Attal-Joye we use the GNS representation and we adapt the general result (with multiple noise). Surprisingly (for me) no jumping processes remain. For example in the case of two results, in the case of the diagonal observable, the limit equations is deterministic: just the Lindblad master equation (with temperature parameter) In the case of non diagonal the Wiener process remains Clément Pellegrini (IMT) Quantum Trajectories, SSE 46 / 48

Temperature Alternative: ( from a physical ) point of view, you can consider that the β0 0 Gibbs state, is either 0 β 1 ( ) 1 0, 0 0 with probability β 0 or ( 0 0 0 1 with probability β 1. One can consider the evolution without measurement, the random aspect disappears at the limit and we recover the Lindblad equation (as if the random aspect is averaged) But with measurement, in the diagonal case it appears two different jumps and in the non diagonal case two different Wiener processes. Clément Pellegrini (IMT) Quantum Trajectories, SSE 47 / 48 ),

THANK YOU Clément Pellegrini (IMT) Quantum Trajectories, SSE 48 / 48