Quantum Repeated Measurements, Continuous Time Limit
|
|
- John Parks
- 5 years ago
- Views:
Transcription
1 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
2 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
3 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
4 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
5 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
6 I) Discrete Time Model QUANTUM REPEATED MEASUREMENT and DISCRETE QUANTUM TRAJECTORIES Clément Pellegrini (IMT) Quantum Trajectories, SSE 4 / 48
7 1st interaction S, ρ 0 H 1... H 2... H H k... Clément Pellegrini (IMT) Quantum Trajectories, SSE 5 / 48
8 1st measurement S H 1... H 2... H H k Measurement Clément Pellegrini (IMT) Quantum Trajectories, SSE 6 / 48
9 2nd interaction S, ρ 1 H 1 H 2... H H k... Clément Pellegrini (IMT) Quantum Trajectories, SSE 7 / 48
10 2nd measurement S H 1 H 2... H H k... Measurement and so on discrete quantum trajectory (ρ n ) Clément Pellegrini (IMT) Quantum Trajectories, SSE 8 / 48
11 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
12 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
13 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
14 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
15 II) Stochastic Master Equations Continuous Quantum Trajectories Clément Pellegrini (IMT) Quantum Trajectories, SSE 11 / 48
16 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
17 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
18 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
19 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
20 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
21 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.
22 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.
23 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
24 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
25 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
26 Generalisation One can generalise ρ t = ρ 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
27 III) Convergence Result FROM DISCRETE TO CONTINUOUS QUANTUM TRAJECTORIES Clément Pellegrini (IMT) Quantum Trajectories, SSE 21 / 48
28 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 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
29 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 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
30 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 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
31 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
32 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 )
33 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
34 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
35 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 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 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 ) = ] [Tr[ρ k (C +C )]+ (1) n p 1 (ρ k ) = 1 p 0 (ρ k ). Clément Pellegrini (IMT) Quantum Trajectories, SSE 27 / 48
36 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
37 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
38 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
39 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
40 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
41 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
42 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 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
43 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 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
44 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
45 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
46 IV) Estimation, Temperature Clément Pellegrini (IMT) Quantum Trajectories, SSE 36 / 48
47 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
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
49 Estimation Clément Pellegrini (IMT) Quantum Trajectories, SSE 38 / 48
50 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
51 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
52 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
53 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
54 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
55 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
56 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
57 Temperature Clément Pellegrini (IMT) Quantum Trajectories, SSE 44 / 48
58 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
59 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
60 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 ( 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 ),
61 THANK YOU Clément Pellegrini (IMT) Quantum Trajectories, SSE 48 / 48
Open quantum random walks: bi-stability and ballistic diffusion. Open quantum brownian motion
Open quantum random walks: bi-stability and ballistic diffusion Open quantum brownian motion with Michel Bauer and Antoine Tilloy Autrans, July 2013 Different regimes in «open quantum random walks»: Open
More informationProjection Filters. Chapter Filter projection in general
19 Chapter 2 Projection Filters Dynamics of open quantum systems take place in the space of density matrices, which can be a very high dimensional space, particularly when photon fields are involved. Strictly
More informationSimulation methods for stochastic models in chemistry
Simulation methods for stochastic models in chemistry David F. Anderson anderson@math.wisc.edu Department of Mathematics University of Wisconsin - Madison SIAM: Barcelona June 4th, 21 Overview 1. Notation
More informationOn the Goodness-of-Fit Tests for Some Continuous Time Processes
On the Goodness-of-Fit Tests for Some Continuous Time Processes Sergueï Dachian and Yury A. Kutoyants Laboratoire de Mathématiques, Université Blaise Pascal Laboratoire de Statistique et Processus, Université
More informationC.W. Gardiner. P. Zoller. Quantum Nois e. A Handbook of Markovian and Non-Markovia n Quantum Stochastic Method s with Applications to Quantum Optics
C.W. Gardiner P. Zoller Quantum Nois e A Handbook of Markovian and Non-Markovia n Quantum Stochastic Method s with Applications to Quantum Optics 1. A Historical Introduction 1 1.1 Heisenberg's Uncertainty
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationRepeated nondemolition measurements and wave function collapse
Tristan Benoist (ENS Paris) Repeated nondemolition measurements and wave function collapse Quantum nondemolition (QND) indirect measurements are often used in quantum optics in order to gain information
More informationLAN property for sde s with additive fractional noise and continuous time observation
LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday,
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationQuantum Optics Project II
Quantum Optics Project II Carey Phelps, Jun Yin and Tim Sweeney Department of Physics and Oregon Center for Optics 174 University of Oregon Eugene, Oregon 9743-174 15 May 7 1 Simulation of quantum-state
More informationLAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC
LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic
More informationOn the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family
More informationA geometric analysis of the Markovian evolution of open quantum systems
A geometric analysis of the Markovian evolution of open quantum systems University of Zaragoza, BIFI, Spain Joint work with J. F. Cariñena, J. Clemente-Gallardo, G. Marmo Martes Cuantico, June 13th, 2017
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationUniformly Uniformly-ergodic Markov chains and BSDEs
Uniformly Uniformly-ergodic Markov chains and BSDEs Samuel N. Cohen Mathematical Institute, University of Oxford (Based on joint work with Ying Hu, Robert Elliott, Lukas Szpruch) Centre Henri Lebesgue,
More informationEntanglement of Bipartite Quantum Systems driven by Repeated Interactions
Entanglement of Bipartite Quantum Systems driven by Repeated Interactions S Attal 1, J Descamps and C Pellegrini 3 1 Université de Lyon Université de Lyon 1, CNRS Institut Camille Jordan 1 av Claude Bernard
More informationFeedback and Time Optimal Control for Quantum Spin Systems. Kazufumi Ito
Feedback and Time Optimal Control for Quantum Spin Systems Kazufumi Ito Center for Research in Scientific Computation North Carolina State University Raleigh, North Carolina IMA, March, 29-1- NCSU CONTENTS:
More informationFree probability and quantum information
Free probability and quantum information Benoît Collins WPI-AIMR, Tohoku University & University of Ottawa Tokyo, Nov 8, 2013 Overview Overview Plan: 1. Quantum Information theory: the additivity problem
More informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 1, 216 Statistical properties of dynamical systems, ESI Vienna. David
More informationStochastic (intermittent) Spikes and Strong Noise Limit of SDEs.
Stochastic (intermittent) Spikes and Strong Noise Limit of SDEs. D. Bernard in collaboration with M. Bauer and (in part) A. Tilloy. IPAM-UCLA, Jan 2017. 1 Strong Noise Limit of (some) SDEs Stochastic differential
More informationEffective dynamics for the (overdamped) Langevin equation
Effective dynamics for the (overdamped) Langevin equation Frédéric Legoll ENPC and INRIA joint work with T. Lelièvre (ENPC and INRIA) Enumath conference, MS Numerical methods for molecular dynamics EnuMath
More informationStabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays
Stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays Pierre Rouchon, Mines-ParisTech, Centre Automatique et Systèmes, pierre.rouchon@mines-paristech.fr
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationfor all f satisfying E[ f(x) ] <.
. Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. An (H, H)-valued random variable X is independent of D if and only if P ({X Γ} D) = P {X Γ}P (D) for all Γ H and D D. Prove that if
More informationNon-Markovian Quantum Dynamics of Open Systems: Foundations and Perspectives
Non-Markovian Quantum Dynamics of Open Systems: Foundations and Perspectives Heinz-Peter Breuer Universität Freiburg QCCC Workshop, Aschau, October 2007 Contents Quantum Markov processes Non-Markovian
More informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 12, 215 Averaging and homogenization workshop, Luminy. Fast-slow systems
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
More informationFidelity is a Sub-Martingale for Discrete-Time Quantum Filters
Fidelity is a Sub-Martingale for Discrete-Time Quantum Filters Pierre Rouchon To cite this version: Pierre Rouchon. Fidelity is a Sub-Martingale for Discrete-Time Quantum Filters. IEEE Transactions on
More informationOpen Quantum Systems and Markov Processes II
Open Quantum Systems and Markov Processes II Theory of Quantum Optics (QIC 895) Sascha Agne sascha.agne@uwaterloo.ca July 20, 2015 Outline 1 1. Introduction to open quantum systems and master equations
More informationStabilization of Schrödinger cats in a cavity by reservoir engineering
Stabilization of Schrödinger cats in a cavity by reservoir engineering Pierre Rouchon Mines-ParisTech, Centre Automatique et Systèmes Mathématiques et Systèmes pierre.rouchon@mines-paristech.fr QUAINT,
More informationThe Wiener Itô Chaos Expansion
1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in
More informationThe Cameron-Martin-Girsanov (CMG) Theorem
The Cameron-Martin-Girsanov (CMG) Theorem There are many versions of the CMG Theorem. In some sense, there are many CMG Theorems. The first version appeared in ] in 944. Here we present a standard version,
More informationVolume comparison theorems without Jacobi fields
Volume comparison theorems without Jacobi fields Dominique Bakry Laboratoire de Statistique et Probabilités Université Paul Sabatier 118 route de Narbonne 31062 Toulouse, FRANCE Zhongmin Qian Mathematical
More informationarxiv: v2 [quant-ph] 1 Oct 2017
Stochastic thermodynamics of quantum maps with and without equilibrium Felipe Barra and Cristóbal Lledó Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago,
More informationTopics in fractional Brownian motion
Topics in fractional Brownian motion Esko Valkeila Spring School, Jena 25.3. 2011 We plan to discuss the following items during these lectures: Fractional Brownian motion and its properties. Topics in
More informationLet (Ω, F) be a measureable space. A filtration in discrete time is a sequence of. F s F t
2.2 Filtrations Let (Ω, F) be a measureable space. A filtration in discrete time is a sequence of σ algebras {F t } such that F t F and F t F t+1 for all t = 0, 1,.... In continuous time, the second condition
More information22.51 Quantum Theory of Radiation Interactions
.5 Quantum Theory of Radiation Interactions Mid-Term Exam October 3, Name:.................. In this mid-term we will study the dynamics of an atomic clock, which is one of the applications of David Wineland
More informationOperator norm convergence for sequence of matrices and application to QIT
Operator norm convergence for sequence of matrices and application to QIT Benoît Collins University of Ottawa & AIMR, Tohoku University Cambridge, INI, October 15, 2013 Overview Overview Plan: 1. Norm
More informationLAN property for ergodic jump-diffusion processes with discrete observations
LAN property for ergodic jump-diffusion processes with discrete observations Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Arturo Kohatsu-Higa (Ritsumeikan University, Japan) &
More informationHandbook of Stochastic Methods
C. W. Gardiner Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Third Edition With 30 Figures Springer Contents 1. A Historical Introduction 1 1.1 Motivation I 1.2 Some Historical
More informationarxiv: v1 [math-ph] 27 Sep 2018
Central limit theorems for open quantum random walks on the crystal lattices arxiv:809.045v [math-ph] 27 Sep 208 Chul Ki Ko, Norio Konno, Etsuo Segawa, and Hyun Jae Yoo Abstract We consider the open quantum
More informationSemiclassical analysis and a new result for Poisson - Lévy excursion measures
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol 3 8, Paper no 45, pages 83 36 Journal URL http://wwwmathwashingtonedu/~ejpecp/ Semiclassical analysis and a new result for Poisson - Lévy
More informationBrownian Motion and lorentzian manifolds
The case of Jürgen Angst Institut de Recherche Mathématique Avancée Université Louis Pasteur, Strasbourg École d été de Probabilités de Saint-Flour juillet 2008, Saint-Flour 1 Construction of the diffusion,
More informationarxiv:cond-mat/ v3 15 Jan 2001
Continuous quantum measurement of two coupled quantum dots using a point contact: A quantum trajectory approach Hsi-Sheng Goan 1, G. J. Milburn 1, H. M. Wiseman, and He Bi Sun 1 1 Center for Quantum Computer
More informationStochastic Numerical Analysis
Stochastic Numerical Analysis Prof. Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Stoch. NA, Lecture 3 p. 1 Multi-dimensional SDEs So far we have considered scalar SDEs
More informationOn pathwise stochastic integration
On pathwise stochastic integration Rafa l Marcin Lochowski Afican Institute for Mathematical Sciences, Warsaw School of Economics UWC seminar Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic
More informationRepresenting Gaussian Processes with Martingales
Representing Gaussian Processes with Martingales with Application to MLE of Langevin Equation Tommi Sottinen University of Vaasa Based on ongoing joint work with Lauri Viitasaari, University of Saarland
More informationOn Stochastic Adaptive Control & its Applications. Bozenna Pasik-Duncan University of Kansas, USA
On Stochastic Adaptive Control & its Applications Bozenna Pasik-Duncan University of Kansas, USA ASEAS Workshop, AFOSR, 23-24 March, 2009 1. Motivation: Work in the 1970's 2. Adaptive Control of Continuous
More informationQuantum Hydrodynamics models derived from the entropy principle
1 Quantum Hydrodynamics models derived from the entropy principle P. Degond (1), Ch. Ringhofer (2) (1) MIP, CNRS and Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex, France degond@mip.ups-tlse.fr
More informationSome functional (Hölderian) limit theorems and their applications (II)
Some functional (Hölderian) limit theorems and their applications (II) Alfredas Račkauskas Vilnius University Outils Statistiques et Probabilistes pour la Finance Université de Rouen June 1 5, Rouen (Rouen
More informationTyler Hofmeister. University of Calgary Mathematical and Computational Finance Laboratory
JUMP PROCESSES GENERALIZING STOCHASTIC INTEGRALS WITH JUMPS Tyler Hofmeister University of Calgary Mathematical and Computational Finance Laboratory Overview 1. General Method 2. Poisson Processes 3. Diffusion
More informationConvergence of Feller Processes
Chapter 15 Convergence of Feller Processes This chapter looks at the convergence of sequences of Feller processes to a iting process. Section 15.1 lays some ground work concerning weak convergence of processes
More informationMean-field dual of cooperative reproduction
The mean-field dual of systems with cooperative reproduction joint with Tibor Mach (Prague) A. Sturm (Göttingen) Friday, July 6th, 2018 Poisson construction of Markov processes Let (X t ) t 0 be a continuous-time
More informationAlternative Characterizations of Markov Processes
Chapter 10 Alternative Characterizations of Markov Processes This lecture introduces two ways of characterizing Markov processes other than through their transition probabilities. Section 10.1 describes
More informationDynamical Collapse in Quantum Theory
Dynamical Collapse in Quantum Theory Lajos Diósi HAS Wigner Research Center for Physics Budapest August 2, 2012 Dynamical Collapse in Quantum Theory August 2, 2012 1 / 28 Outline 1 Statistical Interpretation:
More informationNon equilibrium thermodynamic transformations. Giovanni Jona-Lasinio
Non equilibrium thermodynamic transformations Giovanni Jona-Lasinio Kyoto, July 29, 2013 1. PRELIMINARIES 2. RARE FLUCTUATIONS 3. THERMODYNAMIC TRANSFORMATIONS 1. PRELIMINARIES Over the last ten years,
More informationNonlinear representation, backward SDEs, and application to the Principal-Agent problem
Nonlinear representation, backward SDEs, and application to the Principal-Agent problem Ecole Polytechnique, France April 4, 218 Outline The Principal-Agent problem Formulation 1 The Principal-Agent problem
More informationHigher order weak approximations of stochastic differential equations with and without jumps
Higher order weak approximations of stochastic differential equations with and without jumps Hideyuki TANAKA Graduate School of Science and Engineering, Ritsumeikan University Rough Path Analysis and Related
More informationInterest Rate Models:
1/17 Interest Rate Models: from Parametric Statistics to Infinite Dimensional Stochastic Analysis René Carmona Bendheim Center for Finance ORFE & PACM, Princeton University email: rcarmna@princeton.edu
More informationExponential martingales: uniform integrability results and applications to point processes
Exponential martingales: uniform integrability results and applications to point processes Alexander Sokol Department of Mathematical Sciences, University of Copenhagen 26 September, 2012 1 / 39 Agenda
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationConvergence of Particle Filtering Method for Nonlinear Estimation of Vortex Dynamics
Convergence of Particle Filtering Method for Nonlinear Estimation of Vortex Dynamics Meng Xu Department of Mathematics University of Wyoming February 20, 2010 Outline 1 Nonlinear Filtering Stochastic Vortex
More informationarxiv:math-ph/ v2 10 Nov 2005
STABILIZING FEEDBACK CONTROLS FOR QUANTUM SYSTEMS MAZYAR MIRRAHIMI AND RAMON VAN HANDEL arxiv:math-ph/5166v2 1 Nov 25 Abstract. No quantum measurement can give full information on the state of a quantum
More informationItô s formula. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Itô s formula Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Itô s formula Probability Theory
More informationGillespie s Algorithm and its Approximations. Des Higham Department of Mathematics and Statistics University of Strathclyde
Gillespie s Algorithm and its Approximations Des Higham Department of Mathematics and Statistics University of Strathclyde djh@maths.strath.ac.uk The Three Lectures 1 Gillespie s algorithm and its relation
More informationStochastic Calculus. Kevin Sinclair. August 2, 2016
Stochastic Calculus Kevin Sinclair August, 16 1 Background Suppose we have a Brownian motion W. This is a process, and the value of W at a particular time T (which we write W T ) is a normally distributed
More informationLecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University
Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral
More informationNon white sample covariance matrices.
Non white sample covariance matrices. S. Péché, Université Grenoble 1, joint work with O. Ledoit, Uni. Zurich 17-21/05/2010, Université Marne la Vallée Workshop Probability and Geometry in High Dimensions
More informationMean-field SDE driven by a fractional BM. A related stochastic control problem
Mean-field SDE driven by a fractional BM. A related stochastic control problem Rainer Buckdahn, Université de Bretagne Occidentale, Brest Durham Symposium on Stochastic Analysis, July 1th to July 2th,
More informationJyrki Piilo. Lecture II Non-Markovian Quantum Jumps. Turku Centre for Quantum Physics Non-Markovian Processes and Complex Systems Group
UNIVERSITY OF TURKU, FINLAND Lecture II Non-Markovian Quantum Jumps Jyrki Piilo Turku Centre for Quantum Physics Non-Markovian Processes and Complex Systems Group Contents Lecture 1 1. General framework:
More informationRough paths methods 4: Application to fbm
Rough paths methods 4: Application to fbm Samy Tindel Purdue University University of Aarhus 2016 Samy T. (Purdue) Rough Paths 4 Aarhus 2016 1 / 67 Outline 1 Main result 2 Construction of the Levy area:
More informationJyrki Piilo NON-MARKOVIAN OPEN QUANTUM SYSTEMS. Turku Centre for Quantum Physics Non-Markovian Processes and Complex Systems Group
UNIVERSITY OF TURKU, FINLAND NON-MARKOVIAN OPEN QUANTUM SYSTEMS Jyrki Piilo Turku Centre for Quantum Physics Non-Markovian Processes and Complex Systems Group Turku Centre for Quantum Physics, Finland
More informationBV functions in a Gelfand triple and the stochastic reflection problem on a convex set
BV functions in a Gelfand triple and the stochastic reflection problem on a convex set Xiangchan Zhu Joint work with Prof. Michael Röckner and Rongchan Zhu Xiangchan Zhu ( Joint work with Prof. Michael
More informationBackward Stochastic Differential Equations with Infinite Time Horizon
Backward Stochastic Differential Equations with Infinite Time Horizon Holger Metzler PhD advisor: Prof. G. Tessitore Università di Milano-Bicocca Spring School Stochastic Control in Finance Roscoff, March
More informationJump-type Levy Processes
Jump-type Levy Processes Ernst Eberlein Handbook of Financial Time Series Outline Table of contents Probabilistic Structure of Levy Processes Levy process Levy-Ito decomposition Jump part Probabilistic
More informationarxiv:quant-ph/ v1 6 Nov 2006
A Straightforward Introduction to Continuous Quantum Measurement Kurt Jacobs 1, and Daniel A. Steck 3 1 Department of Physics, University of Massachusetts at Boston, Boston, MA 14 Quantum Sciences and
More informationLecture 4: Introduction to stochastic processes and stochastic calculus
Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London
More informationGARCH processes continuous counterparts (Part 2)
GARCH processes continuous counterparts (Part 2) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/
More informationGaussian processes for inference in stochastic differential equations
Gaussian processes for inference in stochastic differential equations Manfred Opper, AI group, TU Berlin November 6, 2017 Manfred Opper, AI group, TU Berlin (TU Berlin) inference in SDE November 6, 2017
More informationStochastic Integration and Stochastic Differential Equations: a gentle introduction
Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process
More informationSome Tools From Stochastic Analysis
W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click
More informationBranching Processes II: Convergence of critical branching to Feller s CSB
Chapter 4 Branching Processes II: Convergence of critical branching to Feller s CSB Figure 4.1: Feller 4.1 Birth and Death Processes 4.1.1 Linear birth and death processes Branching processes can be studied
More informationWiener Measure and Brownian Motion
Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u
More informationOn the stochastic nonlinear Schrödinger equation
On the stochastic nonlinear Schrödinger equation Annie Millet collaboration with Z. Brzezniak SAMM, Paris 1 and PMA Workshop Women in Applied Mathematics, Heraklion - May 3 211 Outline 1 The NL Shrödinger
More informationViscosity Solutions of Path-dependent Integro-Differential Equations
Viscosity Solutions of Path-dependent Integro-Differential Equations Christian Keller University of Southern California Conference on Stochastic Asymptotics & Applications Joint with 6th Western Conference
More informationA REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH SPACE
Theory of Stochastic Processes Vol. 21 (37), no. 2, 2016, pp. 84 90 G. V. RIABOV A REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH
More informationGoodness of fit test for ergodic diffusion processes
Ann Inst Stat Math (29) 6:99 928 DOI.7/s463-7-62- Goodness of fit test for ergodic diffusion processes Ilia Negri Yoichi Nishiyama Received: 22 December 26 / Revised: July 27 / Published online: 2 January
More informationHandbook of Stochastic Methods
Springer Series in Synergetics 13 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences von Crispin W Gardiner Neuausgabe Handbook of Stochastic Methods Gardiner schnell und portofrei
More informationLogFeller et Ray Knight
LogFeller et Ray Knight Etienne Pardoux joint work with V. Le and A. Wakolbinger Etienne Pardoux (Marseille) MANEGE, 18/1/1 1 / 16 Feller s branching diffusion with logistic growth We consider the diffusion
More informationInformation and Credit Risk
Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information
More informationHomogenization for chaotic dynamical systems
Homogenization for chaotic dynamical systems David Kelly Ian Melbourne Department of Mathematics / Renci UNC Chapel Hill Mathematics Institute University of Warwick November 3, 2013 Duke/UNC Probability
More informationAffine Processes. Econometric specifications. Eduardo Rossi. University of Pavia. March 17, 2009
Affine Processes Econometric specifications Eduardo Rossi University of Pavia March 17, 2009 Eduardo Rossi (University of Pavia) Affine Processes March 17, 2009 1 / 40 Outline 1 Affine Processes 2 Affine
More informationStochastic Gradient Descent in Continuous Time
Stochastic Gradient Descent in Continuous Time Justin Sirignano University of Illinois at Urbana Champaign with Konstantinos Spiliopoulos (Boston University) 1 / 27 We consider a diffusion X t X = R m
More informationLecture 12. F o s, (1.1) F t := s>t
Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let
More informationApplications of controlled paths
Applications of controlled paths Massimiliano Gubinelli CEREMADE Université Paris Dauphine OxPDE conference. Oxford. September 1th 212 ( 1 / 16 ) Outline I will exhibith various applications of the idea
More informationLecture 12: Detailed balance and Eigenfunction methods
Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More information