stabilization of stochastic quantum dynamics via open and ... · dynamics via open and closed loop...

Claudio Altafini, Dijon 2013 Stabilization of SME – p. 1/24

Stabilization of Stochastic QuantumDynamics via Open and Closed Loop Control

QUAINT Workshop, Dijon, April 2013

Claudio Altafini (SISSA)

Kazunori Nishio (Tokyo Institute of Technology)

Francesco Ticozzi (Univ. of Padova)

● Introduction

● SME

● MME

● State decomposition

● Invariance & attractivity for MME

● Invariance & attractivity of SME

● Feedback-assisted stabilization

● Examples

● Conclusion


Introduction

■ Stochastic Master Equation (SME) for quantum filtering■ Averaging over noise: Markovian Master Equation (MME)■ Invariance & attractivity of subspaces for MME

⇐⇒ Global Asymptotic Stability (GAS)■ Environment-assisted stabilization:

◆ block-structure of the dissipation/measurement operators◆ open-loop control

■ Invariance & attractivity of subspaces for the SME⇐⇒ Global Asymptotic Stability in probability

■ Same environment-assisted stabilization properties for MMEand SME

■ Feedback-assisted stabilization for SME■ Examples

● Introduction

● SME

● MME





● Examples

● Conclusion


Quantum filtering

■ quantum filtering: Stochastic Master Equation (SME) à la Itô:◆ SME

dρt =

(

F(H, ρt) +∑

k

D(Lk, ρt) +D(M,ρt)

)

dt+ G(M,ρt)dWt

◆ ρt ∈ D(H) = set of density operators in n-dimensionalHilbert space H

◆ continuous weak measurement: output equation

dYt =√η1

2tr(ρt(M +M†))dt+ dWt

■ M = measurement operator ∈ B(H)■ η ∈ [0, 1] = efficiency of the measurement■ dWt = “innovation process”

◆ example: homodyne detection

● Introduction

● SME

● MME





● Examples

● Conclusion


Stochastic Master Equation (SME)

dρt =

(

F(H, ρt) +∑

k


)

dt+ G(M,ρt)dWt

■ HamiltonianF(H, ρ) := −i[H, ρ]

■ Lindbladian dissipation

D(Lk, ρ) := LkρL†k − 1

2(L†

kLkρ+ ρL†kLk)

■ measurement◆ drift

D(M,ρ) := MρM† − 1

2(M†Mρ+ ρM†M)

◆ diffusion

G(M,ρ) :=√η(Mρ+ ρM† − tr((M +M†)ρ)ρ)

● Introduction

● SME

● MME





● Examples

● Conclusion


Stochastic Master Equation (SME)

■ infinitesimal generator

A[·] = 1

2tr

(

(F(H, ρt) +∑

k

D(Lk, ρt) +D(M,ρt))∂ [·]∂ρ

+∂ [·]∂ρ

(F(H, ρt) +∑

k

D(Lk, ρt) +D(M,ρt))

+1

2G2(M,ρt))

∂2 [·]∂ρ2

+1

2

∂2 [·]∂ρ2

G2(M,ρt))

)

■ solution:

ρt = T Wt (ρ0), ρ0 ∈ D(H)

= ρ0 +

∫ t

0

(

F(H, ρs) +

r∑

k=1

D(Lk, ρ) +D(M,ρs)

)

ds+

∫ t

0

G(M,ρs)dWs

◆ ρt ∃ uniquely◆ ρt adapted to the filtration Et associated to {Wt, t ∈ R

+}◆ ρt D(H)-invariant by construction

● Introduction

● SME

● MME





● Examples

● Conclusion


Markovian Master Equation (MME)

■ averaging the SME over the noise trajectories■ =⇒Markovian Master Equation

ρ̇(t) = L(ρ(t)) = F(H, ρt) +∑

k


■ Quantum dynamical semigroup T (ρ) is a TPCP(Trace-Preserving Completely Positive) map

ρ(t) = Tt(ρ0), ρ0 ∈ D(H)

= exp

∫ t

0

(

F(H, ρs) +

r∑

k=1

D(Lk, ρ) +D(M,ρs)

)

ds

■ Assumption: Hamiltonian H can be chosen arbitrarily

● Introduction

● SME

● MME





● Examples

● Conclusion


State space decomposition

F. Ticozzi, L. Viola. Quantum Markovian subsystems: Invariance, attractivity

and control, IEEE Tr. Autom. Contr., 2008

■ State decomposition

H = HS ⊕HR

◆ HS = “target” subspace, dim(HS) = s◆ HR = “remainer” subspace, dim(HR) = n− s

◆ =⇒block structure for densities in D(H)

ρ =

(

ρS ρP

ρQ ρR

)

◆ =⇒block structure for operators on H

H =

(

HS HP

H†P HR

)

, L =

(

LS LP

LQ LR

)

, M =

(

MS MP

MQ MR

)

● Introduction

● SME

● MME





● Examples

● Conclusion


Invariance & attractivity of HS for the MME

H = HS ⊕HR

■ density initialized in HS

IS(H) ={

ρ ∈ D(H) | ρ =

[

ρS 0

0 0

]

, ρS ∈ D(HS)}

Definition: HS is an invariant subspace for the system underthe TPCP maps {Tt(·)}t≥0 if IS(H) is an invariant subset ofD(H), i.e. if

ρ ∈ IS(H) =⇒ Tt(ρ) ∈ IS(H) ∀ t ≥ 0

Definition: HS supports an attractive subsystem with respectto a family of TPCP maps {Tt}t≥0 if ∀ρ ∈ D(H) the followingcondition is asymptotically obeyed

limt→∞

∥∥∥∥∥Tt(ρ)−

(

ρS(t) 0

0 0

)∥∥∥∥∥= 0.

■ HS invariant and attractive ⇐⇒ Globally Asymptotically Stable (GAS)

● Introduction

● SME

● MME





● Examples

● Conclusion


Invariance for the MME

Theorem HS supports an invariant subspace iff

Lk =

(

LS,k LP,k

0 LR,k

)

∀ k, M =

(

MS MP

0 MR

)

iHP − 1

2

(∑

k

L†S,kLP,k +M†

SMP

)

= 0.

■ idea of the proof:

d

dtρ =

(

LS(ρ) LP (ρ)

LQ(ρ) LR(ρ)

)

∀ ρS ∈ IS(H) =⇒(

LS(ρS) 0

0 0

)

■ necessary and sufficient conditions on the structure of theblocks of H, Lk and M◆ MQ = 0 and LQ,k = 0 ∀k◆ Hamiltonian HP is used to compensate for LP,k 6= 0

and/or MP 6= 0 =⇒open-loop “matching”

● Introduction

● SME

● MME





● Examples

● Conclusion


Attractivity for the MME

Theorem Assume HS supports an invariant subsystem. ThenIS(H) can be made attractive iff IR(H) is not invariant.

■ idea: LP,k 6= 0 and/or MP 6= 0 =⇒HR can never be madeinvariant by Hamiltonian compensation

■ if LP,k = LQ,k = 0 and MP = MQ = 0 then IS(H) and IR(H)both invariant (when HP = 0) or non-invariant (HP 6= 0)

■ if LP,k = LQ,k 6= 0 and/or MP = MQ 6= 0 then neither IS(H)nor IR(H) can be invariant

● Introduction

● SME

● MME





● Examples

● Conclusion


Attractivity for the MME

Theorem Assume HS supports an invariant subsystem. ThenIS(H) can be made attractive iff IR(H) is not invariant.

■ idea: LP,k 6= 0 and/or MP 6= 0 =⇒HR can never be madeinvariant by Hamiltonian compensation

■ if LP,k = LQ,k = 0 and MP = MQ = 0 then IS(H) and IR(H)both invariant (when HP = 0) or non-invariant (HP 6= 0)

■ if LP,k = LQ,k 6= 0 and/or MP = MQ 6= 0 then neither IS(H)nor IR(H) can be invariant

■ Summary: rendering HS attractive for the MME is “to theexpenses of HR”, and can be accomplished by means of1. non-hermitian Lk and/or non-hermitian M

=⇒(block) “ladder-like” operators2. open-loop control

◆ HS invariant and attractive ⇐⇒ Globally Asymptotically Stable (GAS)

=⇒environment-assisted stabilization

● Introduction

● SME

● MME





● Examples

● Conclusion


Extension to SME

■ Problem formulation: global asymptotic stability for the SME

■ IS(H) globally asymptotically stable in probability⇐⇒ IS(H) is invariant and attractive in probability a.s.

Problem Consider the SME

dρt =

(

F(H, ρt) +∑

k


)

dt+ G(M,ρt)dWt

and a target subspace HS such that H = HS ⊕HR.Do there exist

◆ dissipation operators Lk and/or measurement operator M◆ Hamiltonian H

such that the target set IS(H) is invariant under T Wt (·) and at-

tractive in probability a.s.?

● Introduction

● SME

● MME





● Examples

● Conclusion


Condition for invariance

dρ =

(

LS(ρ) LP (ρ)

LQ(ρ) LR(ρ)

)

dt+

(

GS(M,ρ) GP (M,ρ)

GQ(M,ρ) GR(M,ρ)

)

dWt

■ Approach: in order for IS(H) to be invariant for the SME, ithas to be invariant for both its diffusion and drift parts

∀ ρS ∈ IS(H) =⇒(

LS(ρS) 0

0 0

)

,

(

GS(M,ρS) 0

0 0

)

=⇒block structure of Lk and M matrices

■ for ρS ∈ IS(H)

◆ diffusion

G(M,ρS) =

(

(MS − Tr(MSρS))ρS + ρS(M†S − Tr(M†

SρS)) ρSM†Q

MQρS 0

)

=⇒IS(H) is invariant to G(M,ρS) if MQ = 0

● Introduction

● SME

● MME





● Examples

● Conclusion


Condition for invariance

■ drift

F(H, ρS) +∑

k

D(Lk, ρS) +D(M,ρS) =

−iHSρS + iρSHS + d1(Lk,M, ρS) iρSHP +∑

k

d2(Lk, ρS)−1

2ρSM

†SMP

∗

∑

k

LQ,kρSL†Q,k

where we define:

d1(Lk,M, ρS) = MSρSM†S −

1

2(M†

SMSρS + ρSM†SMS) +

∑

k

{

LS,kρSL†S,k

−1

2(L†

S,kLS,kρS + ρSL†S,kLS,k + L

†Q,kLQ,kρS + ρSL

†Q,kLQ,k)

}

,

d2(Lk, ρS) = LS,kρSL†Q,k −

1

2ρS(L

†S,kLP,k + L

†Q,kLR,k).

■ =⇒same conditions on HP , MP and LP,k as for the MME

● Introduction

● SME

● MME





● Examples

● Conclusion


Invariance for the SME

■ putting together

Proposition IS(H) is invariant for the SME iff

Lk =

(

LS,k LP,k

0 LR,k

)

∀k,

M =

(

MS MP

0 MR

)

,

iHP − 1

2

(∑

k

L†S,kLP,k +M†

SMP

)

= 0,

CorollaryIS(H) invariant for the SME ⇐⇒ IS(H) invariant for the MME

■ More rigorous proof: support theorem

● Introduction

● SME

● MME





● Examples

● Conclusion


Attractivity for the SME

■ attractivity of SME ⇐⇒ attractivity of MME

Proposition Assume IS(H) is attractive for the MME for some H,Lk and M . Then with these H, Lk and M , IS(H) attractive inprobability also for the SME

● Introduction

● SME

● MME





● Examples

● Conclusion


Attractivity for the SME

■ attractivity of SME ⇐⇒ attractivity of MME

Proposition Assume IS(H) is attractive for the MME for some H,Lk and M . Then with these H, Lk and M , IS(H) attractive inprobability also for the SME

■ Proof:1. stability of IS(H) in probability

◆ candidate Lyapunov function

V (ρ) = tr(ΠRρ)

where ΠR = projection on HR

V (ρ) = 0 for ρ ∈ IS(H)

V (ρ) > 0 for ρ /∈ IS(H)

AV (ρ) = −tr

((∑

k

L†P,kLP,k +M†

PMP

)

ρR

)

6 0 ∀ρ ∈ D(H)

by ciclicity of the trace

● Introduction

● SME

● MME





● Examples

● Conclusion


Attractivity for the SME (cont.)

2. attractivity (by contradiction)■ Suppose ∃ ρ0 ∈ D(H) \ IS(H) for which IS(H) is not

attractive

=⇒ P(

limt→∞

d(T Wt (ρ0), IS(H)) = 0

)

= 1− p, p > 0

=⇒ P(

limt→∞

V (T Wt (ρ0)) = 0

)

= 1− p

=⇒ for t → ∞ set {V (T Wt (ρ0)) > 0} has measure > 0

■ in expectation then ∃ ξ(ρ0) > 0 for which

lim supt→∞

E[V (T W

t (ρ0))]= lim sup

t→∞

∫

{V (T W

t(ρ0))>0}

V (T Wt (ρ0))dP

≥ ξ(ρ0) lim supt→∞

∫

{V (T W

t(ρ0))>0}

dP

= ξ(ρ0) lim supt→∞

{1− P

(V (T W

t (ρ0)) = 0)}

= ξ(ρ0)p

=⇒ ∃k > 0 s.t. lim supt→∞

d(E[T Wt (ρ0)], IS(H)) ≥ ξ(ρ0)p

k> 0.

● Introduction

● SME

● MME





● Examples

● Conclusion


Invariance & attractivity of SME

■ Summary: Invariance and attractivity of MME⇐⇒ invariance and attractivity of the SME⇐⇒ GAS of MME and SME

■ both can be accomplished by means of1. non-hermitian Lk and/or non-hermitian M

=⇒(block) “ladder-like” operators2. open-loop control

Theorem IS(H) is GAS in probability for the SME ⇐⇒ Lk, Mand H have the structure

Lk =

(

LS,k LP,k

0 LR,k

)

∀k, M =

(

MS MP

0 MR

)

,

iHP − 1

2

(∑

k

L†S,kLP,k +M†

SMP

)

= 0,

with at least one LP,k 6= 0 and/or MP 6= 0

=⇒environment-assisted stabilization

● Introduction

● SME

● MME





● Examples

● Conclusion


Block diagonal case

■ how about the case

Lk =

(

LS,k 0

0 LR,k

)

∀k, M =

(

MS 0

0 MR

)

■ open-loop, time-invariant control: both HS and HR areinvariant =⇒neither is attractive

Proposition Given SME with Lk and M as above, no time-invariant Hexists rendering IS(H) GAS in probability

■ Proof: if HP = 0 then IR(H) is invariant;if instead HP 6= 0 then IS(H) cannot be invariant

● Introduction

● SME

● MME





● Examples

● Conclusion


Feedback-assisted stabilization

■ Assumptions: dim(HS) = 1 (pure state stabilization)dim(H) > 3

■ further split of HR: H = HS ⊕HC ⊕HZ︸︷︷︸

HR

H = Hc+u(ρ)Hf =

0 0 0

0 Hc,C Hc,W

0 H†c,W Hc,Z

+u(ρ)

0 Hf,U 0

H†f,U 0 0

0 0 0

Lk =

Lk,S 0 0

0 Lk,C Lk,W

0 Lk,Y Lk,Z

M =

MS 0 0

0 MC MW

0 MY MZ

■ idea:◆ design Hc,W so as to keep "reshuffling" inside HR = HC ⊕HZ

◆ use Hf,U to "drain" probability out of HR

● Introduction

● SME

● MME





● Examples

● Conclusion


Feedback-assisted stabilization

■ use "patchy" feedback law similar toM. Mirrahimi, R. van Handel. Stabilizing feedback controls for quantum

systems, SIAM J. Control Optim., 46, 445-467, 2007

Theorem Given SME with Lk and M as above and H = Hc + u(ρt)Hf ,where the feedback control law u(ρt) s.t. for ρd ∈ HS

1. If tr(ρtρd) ≥ γ u(ρt) = −tr(i[Hf , ρt]ρd)

2. If tr(ρtρd) ≤ γ/2, u(ρt) = 1;3. If ρt ∈ B = {ρ : γ/2 < tr(ρtρd) < γ}, then

■ u(ρt) = −tr(i[Hf , ρt]ρd) if ρt last entered B through the boundarytr(ρdρ) = γ,

■ ut = 1 otherwise.Then ∃ γ > 0 such that u(ρt) renders the SME GAS in probability

■ differences with M. Mirrahimi, R. van Handel:◆ valid for more general class of Lindbladians◆ uses feedback in a "minimal" way (to enable state

transitions otherwise impossible)◆ uses the environment as much as possible

● Introduction

● SME

● MME





● Examples

● Conclusion


Example 1: environment only

ρd =

1 0 0

0 0 0

0 0 0

M =

0 1 0

0 0 0

0 0 0

L1 =

0 0 0

0 1 3

0 2 1

■ HS is GAS without any open/closed loop Hamiltonian

Hc = 0 Hf = 0

energy population sample trajectories

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

ρ11

ρ22

ρ33

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time

1 −

tr (

ρ d ρ)

● Introduction

● SME

● MME





● Examples

● Conclusion


Example 2: environment + open loop

ρd =

1 0 0

0 0 0

0 0 0

M =

0 1 0

0 0 0

0 0 0

L1 =

0 0 0

0 1 0

0 0 2

■ HS is rendered GAS by the following open loop Hamiltonian

Hc =

0 0 0

0 0 1

0 1 0

Hf = 0


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

ρ11

ρ22

ρ33

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

1 −

tr (

ρ d ρ)

● Introduction

● SME

● MME





● Examples

● Conclusion


Example 3: open loop + closed loop

ρd =

1 0 0

0 0 0

0 0 0

M =

1 0 0

0 2 0

0 0 3

L1 = 0

■ HS is rendered GAS by the open loop and feedbackHamiltonians

Hc =

0 0 0

0 0 1

0 1 0

Hf =

0 −i 0

i 0 0

0 0 0


0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

time

ρ11

ρ22

ρ33

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

1 −

tr (

ρ d ρ)

● Introduction

● SME

● MME





● Examples

● Conclusion


Conclusion

■ Environment (i.e., dissipation) and open-loop control caneasily lead to conditions for GAS of a subsystem in bothMME and SME

■ Philosophy: make the most use of environment and the leastof feedback

■ Crucial ingredients:◆ non-hermitian part (”ladder operator”) in the dissipation

and/or measurement operator◆ open-loop control design by ”matching”

■ GAS of MME ⇐⇒ GAS of SME ⇐⇒ invariance & attractivity

■ When environment and open loop are not enough: feedbackcan be useful to get GAS

F. Ticozzi, K. Nishio, C. Altafini. Stabilization of Stochastic Quantum Dynamics

via Open and Closed Loop Control, IEEE Tr. Autom. Contr., Jan. 2013

stabilization of stochastic quantum dynamics via open and ... · dynamics via open and closed loop...

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