Invariance & attractivity of SME - Semantic Scholar
Stabilization of Stochastic Quantum Dynamics 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)
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 1/24
Introduction
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization
■ ■ ■
● Examples ● Conclusion
■
■
■
■ ■
Claudio Altafini, Dijon 2013
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 MME and SME Feedback-assisted stabilization for SME Examples
Stabilization of SME – p. 2/24
Quantum filtering
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
quantum filtering: Stochastic Master Equation (SME) à la Itô: ◆ SME ! X dρt = F(H, ρt ) + D(Lk , ρt ) + D(M, ρt ) dt + G(M, ρt )dWt k
◆ ◆
ρt ∈ D(H) = set of density operators in n-dimensional Hilbert space H continuous weak measurement: output equation dYt =
√ 1 η tr(ρt (M + M † ))dt + dWt 2
M = measurement operator ∈ B(H) ■ η ∈ [0, 1] = efficiency of the measurement ■ dW = “innovation process” t example: homodyne detection ■
◆
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 3/24
Stochastic Master Equation (SME) dρt =
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization
■
F(H, ρt ) +
k
D(Lk , ρt ) + D(M, ρt ) dt + G(M, ρt )dWt
Hamiltonian F(H, ρ) := −i[H, ρ]
● Examples ● Conclusion
■
X
!
Lindbladian dissipation
1 D(Lk , ρ) := Lk ρL†k − (L†k Lk ρ + ρL†k Lk ) 2 ■
measurement ◆ drift 1 D(M, ρ) := M ρM − (M † M ρ + ρM † M ) 2 †
◆
diffusion G(M, ρ) :=
Claudio Altafini, Dijon 2013
√
η(M ρ + ρM † − tr((M + M † )ρ)ρ) Stabilization of SME – p. 4/24
Stochastic Master Equation (SME) ■
infinitesimal generator X ∂ [·] 1 D(Lk , ρt ) + D(M, ρt )) A[·] = tr (F(H, ρt ) + 2 ∂ρ
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
k
X ∂ [·] (F(H, ρt ) + + D(Lk , ρt ) + D(M, ρt )) ∂ρ k 2 2 1 ∂ [·] 1 ∂ [·] 2 + G 2 (M, ρt )) 2 + G (M, ρt )) 2 2 ∂ρ 2 ∂ρ
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
solution: ρt = TtW (ρ0 ), ρ0 ∈ D(H) ! Z t Z t r X = ρ0 + D(Lk , ρ) + D(M, ρs ) ds + F(H, ρs ) + G(M, ρs )dWs 0
◆ ◆ ◆ Claudio Altafini, Dijon 2013
k=1
0
ρt ∃ uniquely ρt adapted to the filtration Et associated to {Wt , t ∈ R+ } ρt D(H)-invariant by construction Stabilization of SME – p. 5/24
Markovian Master Equation (MME)
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■ ■
● Invariance & attractivity of SME ● Feedback-assisted stabilization
averaging the SME over the noise trajectories =⇒Markovian Master Equation ρ(t) ˙ = L(ρ(t)) = F(H, ρt ) +
● Examples ● Conclusion
■
X k
D(Lk , ρt ) + D(M, ρt )
Quantum dynamical semigroup T (ρ) is a TPCP (Trace-Preserving Completely Positive) map ρ(t) = Tt (ρ0 ), ρ0 ∈ D(H) ! Z t r X = exp D(Lk , ρ) + D(M, ρs ) ds F(H, ρs ) + 0
■
Claudio Altafini, Dijon 2013
k=1
Assumption: Hamiltonian H can be chosen arbitrarily
Stabilization of SME – p. 6/24
State space decomposition
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization
F. Ticozzi, L. Viola. Quantum Markovian subsystems: Invariance, attractivity and control, IEEE Tr. Autom. Contr., 2008 ■
State decomposition H = HS ⊕ HR
● Examples ● Conclusion
◆ ◆ ◆
HS = “target” subspace, dim(HS ) = s HR = “remainer” subspace, dim(HR ) = n − s =⇒block structure for densities in D(H) ! ρS ρP ρ= ρQ ρR
=⇒block structure for operators on H ! ! LS LP H S HP , L= , H= † H P HR LQ LR
◆
Claudio Altafini, Dijon 2013
M=
MS MQ
MP MR
Stabilization of SME – p. 7/24
!
Invariance & attractivity of HS for the MME ● Introduction
■
● SME ● MME
H = HS ⊕ HR
density initialized in HS n
"
ρS IS (H) = ρ ∈ D(H) | ρ = 0
● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
#
o 0 , ρS ∈ D(HS ) 0
HS is an invariant subspace for the system under the TPCP maps {Tt (·)}t≥0 if IS (H) is an invariant subset of D(H), i.e. if ρ ∈ IS (H) =⇒ Tt (ρ) ∈ IS (H) ∀ t ≥ 0 Definition:
HS supports an attractive subsystem with respect to a family of TPCP maps {Tt }t≥0 if ∀ρ ∈ D(H) the following condition is asymptotically obeyed
!
ρS (t) 0
lim Tt (ρ) −
= 0. t→∞ 0 0 Definition:
■
Claudio Altafini, Dijon 2013
HS invariant and attractive ⇐⇒ Globally Asymptotically Stable (GAS) Stabilization of SME – p. 8/24
Invariance for the MME HS supports an invariant subspace iff ! MS LS,k LP,k ∀ k, M= Lk = 0 LR,k 0 1 X † iHP − LS,k LP,k + MS† MP = 0. 2
Theorem ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
MP MR
!
k
■
idea of the proof: d ρ= dt
■
Claudio Altafini, Dijon 2013
LS (ρ) LQ (ρ)
LP (ρ) LR (ρ)
!
∀ ρS ∈ IS (H) =⇒
LS (ρS ) 0
0 0
!
necessary and sufficient conditions on the structure of the blocks 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” Stabilization of SME – p. 9/24
Attractivity for the MME Assume HS supports an invariant subsystem. Then IS (H) can be made attractive iff IR (H) is not invariant. Theorem
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■ ■
Claudio Altafini, Dijon 2013
idea: LP,k 6= 0 and/or MP 6= 0 =⇒HR can never be made invariant 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
Stabilization of SME – p. 10/24
Attractivity for the MME Assume HS supports an invariant subsystem. Then IS (H) can be made attractive iff IR (H) is not invariant. Theorem
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■ ■
■
idea: LP,k 6= 0 and/or MP 6= 0 =⇒HR can never be made invariant 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 rendering HS attractive for the MME is “to the expenses of HR ”, and can be accomplished by means of 1. non-hermitian Lk and/or non-hermitian M =⇒(block) “ladder-like” operators 2. open-loop control Summary:
◆
HS invariant and attractive ⇐⇒ Globally Asymptotically Stable (GAS)
=⇒environment-assisted stabilization Claudio Altafini, Dijon 2013
Stabilization of SME – p. 10/24
Extension to SME
● Introduction
■
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.
● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
Problem
dρt =
Consider the SME F(H, ρt ) +
X k
!
D(Lk , ρt ) + D(M, ρt ) 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 TtW (·) and attractive in probability a.s.?
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 11/24
Condition for invariance
● Introduction
dρ =
● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
LP (ρ) LR (ρ)
LS (ρ) LQ (ρ)
!
dt +
GS (M, ρ) GQ (M, ρ)
GP (M, ρ) GR (M, ρ)
!
dWt
Approach: in order for IS (H) to be invariant for the SME, it has to be invariant for both its diffusion and drift parts ! GS (M, ρS ) LS (ρS ) 0 , ∀ ρS ∈ IS (H) =⇒ 0 0 0
0 0
!
=⇒block structure of Lk and M matrices ■
for ρS ∈ IS (H) ◆ diffusion G(M, ρS ) =
ρS (MS†
(MS − Tr(MS ρS ))ρS + MQ ρS
−
Tr(MS† ρS ))
† ρ S MQ
0
=⇒IS (H) is invariant to G(M, ρS ) if MQ = 0 Claudio Altafini, Dijon 2013
Stabilization of SME – p. 12/24
!
Condition for invariance ■
drift
● Introduction ● SME ● MME
X
F(H, ρS ) +
● State decomposition ● Invariance & attractivity for MME
D(Lk , ρS ) + D(M, ρS ) =
k
● Invariance & attractivity of SME ● Feedback-assisted stabilization
● Examples ● Conclusion
−iHS ρS + iρS HS + d1 (Lk , M, ρS )
∗
1 † iρS HP + d2 (Lk , ρS ) − ρS MS MP 2 k X † LQ,k ρS LQ,k X
k
where we define: Xn 1 † † LS,k ρS L†S,k d1 (Lk , M, ρS ) = − (MS MS ρS + ρS MS MS ) + 2 k 1 − (L†S,k LS,k ρS + ρS L†S,k LS,k + L†Q,k LQ,k ρS + ρS L†Q,k LQ,k ) , 2 1 d2 (Lk , ρS ) = LS,k ρS L†Q,k − ρS (L†S,k LP,k + L†Q,k LR,k ). 2 MS ρS MS†
■
Claudio Altafini, Dijon 2013
=⇒same conditions on HP , MP and LP,k as for the MME Stabilization of SME – p. 13/24
Invariance for the SME ■
putting together
● Introduction ● SME ● MME
Proposition
● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
IS (H) is invariant for the SME iff ! LS,k LP,k Lk = ∀k, 0 LR,k ! MS MP , M= 0 MR 1 iHP − 2
X
L†S,k LP,k + MS† MP
k
!
= 0,
Corollary
IS (H) invariant for the SME ■
Claudio Altafini, Dijon 2013
⇐⇒
IS (H) invariant for the MME
More rigorous proof: support theorem
Stabilization of SME – p. 14/24
Attractivity for the SME ■ ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
attractivity of SME ⇐⇒ attractivity of MME Assume IS (H) is attractive for the MME for some H, Lk and M . Then with these H, Lk and M , IS (H) attractive in probability also for the SME
Proposition
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 15/24
Attractivity for the SME ■ ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
attractivity of SME ⇐⇒ attractivity of MME Assume IS (H) is attractive for the MME for some H, Lk and M . Then with these H, Lk and M , IS (H) attractive in probability also for the SME
Proposition
■ Proof:
1. stability of IS (H) in probability ◆ candidate Lyapunov function V (ρ) = tr(ΠR ρ) where ΠR = projection on HR V (ρ) = 0 for ρ ∈ IS (H)
AV (ρ) = −tr
V (ρ) > 0 for ρ ∈ / IS (H) ! ! X † LP,k LP,k + MP† MP ρR 6 0 k
∀ρ ∈ D(H)
by ciclicity of the trace Claudio Altafini, Dijon 2013
Stabilization of SME – p. 15/24
Attractivity for the SME (cont.) 2. ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
attractivity (by contradiction) ■ Suppose ∃ ρ0 ∈ D(H) \ IS (H) for which IS (H) is not attractive =⇒ P lim d(TtW (ρ0 ), IS (H)) = 0 = 1 − p, p > 0 t→∞ =⇒ P lim V (TtW (ρ0 )) = 0 = 1 − p t→∞
=⇒
■
for t → ∞ set {V (TtW (ρ0 )) > 0} has measure > 0
in expectation then ∃ ξ(ρ0 ) > 0 for which Z lim supE V (TtW (ρ0 )) = lim sup t→∞
t→∞
≥ ξ(ρ0 ) lim sup t→∞
Z
dP {V
(TtW (ρ0 ))>0}
= ξ(ρ0 ) lim sup 1 − P V t→∞
=⇒ ∃k > 0 s.t. Claudio Altafini, Dijon 2013
{V (TtW (ρ0 ))>0}
V (TtW (ρ0 ))dP
(TtW (ρ0 ))
=0
lim sup d(E[TtW (ρ0 )], IS (H)) t→∞
= ξ(ρ0 )p
ξ(ρ0 )p ≥ > 0. k Stabilization of SME – p. 16/24
Invariance & attractivity of SME ■ ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
Invariance and attractivity of MME ⇐⇒ invariance and attractivity of the SME ⇐⇒ GAS of MME and SME both can be accomplished by means of 1. non-hermitian Lk and/or non-hermitian M =⇒(block) “ladder-like” operators 2. open-loop control Summary:
IS (H) is GAS in probability for the SME ⇐⇒ Lk , M and H have the structure ! ! LS,k LP,k MS MP Lk = ∀k, M = , 0 LR,k 0 MR ! 1 X † iHP − LS,k LP,k + MS† MP = 0, 2
Theorem
k
with at least one LP,k 6= 0 and/or MP 6= 0 =⇒environment-assisted stabilization Claudio Altafini, Dijon 2013
Stabilization of SME – p. 17/24
Block diagonal case
● Introduction
■
● SME ● MME
how about the case
● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization
Lk =
● Examples ● Conclusion
■
LS,k 0
0 LR,k
!
∀k,
M=
MS 0
0 MR
!
open-loop, time-invariant control: both HS and HR are invariant =⇒neither is attractive
Given SME with Lk and M as above, no time-invariant H exists rendering IS (H) GAS in probability
Proposition
if HP = 0 then IR (H) is invariant; if instead HP 6= 0 then IS (H) cannot be invariant
■ Proof:
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 18/24
Feedback-assisted stabilization ■ Assumptions: ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■
● Invariance & attractivity of SME ● Feedback-assisted stabilization
dim(HS ) = 1 (pure state stabilization) dim(H) > 3
further split of HR : H = HS ⊕ HC ⊕ HZ | {z } HR
● Examples ● Conclusion
0 H = Hc +u(ρ)Hf = 0 0
■
Claudio Altafini, Dijon 2013
Lk,S Lk = 0 0
0 Lk,C Lk,Y
0 Hc,C † Hc,W
0
Lk,W Lk,Z
0
0
† Hc,W +u(ρ) Hf,U Hc,Z 0
MS M = 0 0
0 MC MY
Hf,U 0 0
0 MW MZ
idea: ◆ design Hc,W so as to keep "reshuffling" inside HR = HC ⊕ HZ ◆ use Hf,U to "drain" probability out of HR Stabilization of SME – p. 19/24
0 0 0
Feedback-assisted stabilization ■
M. Mirrahimi, R. van Handel. Stabilizing feedback controls for quantum systems, SIAM J. Control Optim., 46, 445-467, 2007
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
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(ρt ρd ) ≥ γ u(ρt ) = −tr(i[Hf , ρt ]ρd ) 2. If tr(ρt ρd ) ≤ γ/2, u(ρt ) = 1; 3. If ρt ∈ B = {ρ : γ/2 < tr(ρt ρd ) < γ}, then ■ u(ρt ) = −tr(i[Hf , ρt ]ρd ) if ρt last entered B through the boundary tr(ρd ρ) = γ, ■ ut = 1 otherwise. Then ∃ γ > 0 such that u(ρt ) renders the SME GAS in probability Theorem
■
Claudio Altafini, Dijon 2013
use "patchy" feedback law similar to
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 Stabilization of SME – p. 20/24
Example 1: environment only
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
1 ρd = 0 0
0 0 0 0 0 0
0 M = 0 0
1 0 0
0 0 0
0 L1 = 0 0
0 0 1 3 2 1
HS is GAS without any open/closed loop Hamiltonian Hc = 0
Hf = 0
energy population
sample trajectories 0.8
1
0.9
0.7
0.8
0.6
1 − tr (ρd ρ)
0.7
0.6 ρ
11
ρ
0.5
22
ρ33 0.4
0.5
0.4
0.3
0.3
0.2 0.2
0.1
0.1
0
0
2
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time
Claudio Altafini, Dijon 2013
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20
0
0
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Stabilization of SME – p. 21/24
Example 2: environment + open loop
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
1 ρd = 0 0
0 0 0 0 0 0
0 M = 0 0
1 0 0
0 0 0
0 L1 = 0 0
0 0 1 0 0 2
HS is rendered GAS by the following open loop Hamiltonian 0 0 0 H c = 0 0 1 Hf = 0 0 1 0 energy population
sample trajectories 1
0.9
0.9
0.8
0.8
0.7
0.7
1 − tr (ρ ρ)
1
0.6 11
ρ
0.5
0.6
d
ρ
22
ρ
33
0.4
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0.3
0.3
0.2
0.2
0.1
0.1
0
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0
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Stabilization of SME – p. 22/24
Example 3: open loop + closed loop
1 ρd = 0 0
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
0 0 0
0 0 0
1 0 M = 0 2 0 0
0 0 3
L1 = 0
HS is rendered GAS by the open loop and feedback Hamiltonians 0 0 0 0 −i 0 H c = 0 0 1 H f = i 0 0 0 1 0 0 0 0 energy population
sample trajectories 1
1.2
0.9 1
0.8
0.7
1 − tr (ρ ρ)
0.8
ρ
22
0.6
0.6
d
ρ11 ρ
33
0.5
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0.4
0.3
0.2 0.2
0.1
0
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Claudio Altafini, Dijon 2013
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Stabilization of SME – p. 23/24
Conclusion ■
Environment (i.e., dissipation) and open-loop control can easily lead to conditions for GAS of a subsystem in both MME and SME
■
Philosophy: make the most use of environment and the least of 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: feedback can be useful to get GAS
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
F. Ticozzi, K. Nishio, C. Altafini. Stabilization of Stochastic Quantum Dynamics via Open and Closed Loop Control, IEEE Tr. Autom. Contr., Jan. 2013 Claudio Altafini, Dijon 2013
Stabilization of SME – p. 24/24
Claudio Altafini (SISSA) Kazunori Nishio (Tokyo Institute of Technology) Francesco Ticozzi (Univ. of Padova)
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 1/24
Introduction
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization
■ ■ ■
● Examples ● Conclusion
■
■
■
■ ■
Claudio Altafini, Dijon 2013
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 MME and SME Feedback-assisted stabilization for SME Examples
Stabilization of SME – p. 2/24
Quantum filtering
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
quantum filtering: Stochastic Master Equation (SME) à la Itô: ◆ SME ! X dρt = F(H, ρt ) + D(Lk , ρt ) + D(M, ρt ) dt + G(M, ρt )dWt k
◆ ◆
ρt ∈ D(H) = set of density operators in n-dimensional Hilbert space H continuous weak measurement: output equation dYt =
√ 1 η tr(ρt (M + M † ))dt + dWt 2
M = measurement operator ∈ B(H) ■ η ∈ [0, 1] = efficiency of the measurement ■ dW = “innovation process” t example: homodyne detection ■
◆
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 3/24
Stochastic Master Equation (SME) dρt =
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization
■
F(H, ρt ) +
k
D(Lk , ρt ) + D(M, ρt ) dt + G(M, ρt )dWt
Hamiltonian F(H, ρ) := −i[H, ρ]
● Examples ● Conclusion
■
X
!
Lindbladian dissipation
1 D(Lk , ρ) := Lk ρL†k − (L†k Lk ρ + ρL†k Lk ) 2 ■
measurement ◆ drift 1 D(M, ρ) := M ρM − (M † M ρ + ρM † M ) 2 †
◆
diffusion G(M, ρ) :=
Claudio Altafini, Dijon 2013
√
η(M ρ + ρM † − tr((M + M † )ρ)ρ) Stabilization of SME – p. 4/24
Stochastic Master Equation (SME) ■
infinitesimal generator X ∂ [·] 1 D(Lk , ρt ) + D(M, ρt )) A[·] = tr (F(H, ρt ) + 2 ∂ρ
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
k
X ∂ [·] (F(H, ρt ) + + D(Lk , ρt ) + D(M, ρt )) ∂ρ k 2 2 1 ∂ [·] 1 ∂ [·] 2 + G 2 (M, ρt )) 2 + G (M, ρt )) 2 2 ∂ρ 2 ∂ρ
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
solution: ρt = TtW (ρ0 ), ρ0 ∈ D(H) ! Z t Z t r X = ρ0 + D(Lk , ρ) + D(M, ρs ) ds + F(H, ρs ) + G(M, ρs )dWs 0
◆ ◆ ◆ Claudio Altafini, Dijon 2013
k=1
0
ρt ∃ uniquely ρt adapted to the filtration Et associated to {Wt , t ∈ R+ } ρt D(H)-invariant by construction Stabilization of SME – p. 5/24
Markovian Master Equation (MME)
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■ ■
● Invariance & attractivity of SME ● Feedback-assisted stabilization
averaging the SME over the noise trajectories =⇒Markovian Master Equation ρ(t) ˙ = L(ρ(t)) = F(H, ρt ) +
● Examples ● Conclusion
■
X k
D(Lk , ρt ) + D(M, ρt )
Quantum dynamical semigroup T (ρ) is a TPCP (Trace-Preserving Completely Positive) map ρ(t) = Tt (ρ0 ), ρ0 ∈ D(H) ! Z t r X = exp D(Lk , ρ) + D(M, ρs ) ds F(H, ρs ) + 0
■
Claudio Altafini, Dijon 2013
k=1
Assumption: Hamiltonian H can be chosen arbitrarily
Stabilization of SME – p. 6/24
State space decomposition
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization
F. Ticozzi, L. Viola. Quantum Markovian subsystems: Invariance, attractivity and control, IEEE Tr. Autom. Contr., 2008 ■
State decomposition H = HS ⊕ HR
● Examples ● Conclusion
◆ ◆ ◆
HS = “target” subspace, dim(HS ) = s HR = “remainer” subspace, dim(HR ) = n − s =⇒block structure for densities in D(H) ! ρS ρP ρ= ρQ ρR
=⇒block structure for operators on H ! ! LS LP H S HP , L= , H= † H P HR LQ LR
◆
Claudio Altafini, Dijon 2013
M=
MS MQ
MP MR
Stabilization of SME – p. 7/24
!
Invariance & attractivity of HS for the MME ● Introduction
■
● SME ● MME
H = HS ⊕ HR
density initialized in HS n
"
ρS IS (H) = ρ ∈ D(H) | ρ = 0
● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
#
o 0 , ρS ∈ D(HS ) 0
HS is an invariant subspace for the system under the TPCP maps {Tt (·)}t≥0 if IS (H) is an invariant subset of D(H), i.e. if ρ ∈ IS (H) =⇒ Tt (ρ) ∈ IS (H) ∀ t ≥ 0 Definition:
HS supports an attractive subsystem with respect to a family of TPCP maps {Tt }t≥0 if ∀ρ ∈ D(H) the following condition is asymptotically obeyed
!
ρS (t) 0
lim Tt (ρ) −
= 0. t→∞ 0 0 Definition:
■
Claudio Altafini, Dijon 2013
HS invariant and attractive ⇐⇒ Globally Asymptotically Stable (GAS) Stabilization of SME – p. 8/24
Invariance for the MME HS supports an invariant subspace iff ! MS LS,k LP,k ∀ k, M= Lk = 0 LR,k 0 1 X † iHP − LS,k LP,k + MS† MP = 0. 2
Theorem ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
MP MR
!
k
■
idea of the proof: d ρ= dt
■
Claudio Altafini, Dijon 2013
LS (ρ) LQ (ρ)
LP (ρ) LR (ρ)
!
∀ ρS ∈ IS (H) =⇒
LS (ρS ) 0
0 0
!
necessary and sufficient conditions on the structure of the blocks 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” Stabilization of SME – p. 9/24
Attractivity for the MME Assume HS supports an invariant subsystem. Then IS (H) can be made attractive iff IR (H) is not invariant. Theorem
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■ ■
Claudio Altafini, Dijon 2013
idea: LP,k 6= 0 and/or MP 6= 0 =⇒HR can never be made invariant 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
Stabilization of SME – p. 10/24
Attractivity for the MME Assume HS supports an invariant subsystem. Then IS (H) can be made attractive iff IR (H) is not invariant. Theorem
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■ ■
■
idea: LP,k 6= 0 and/or MP 6= 0 =⇒HR can never be made invariant 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 rendering HS attractive for the MME is “to the expenses of HR ”, and can be accomplished by means of 1. non-hermitian Lk and/or non-hermitian M =⇒(block) “ladder-like” operators 2. open-loop control Summary:
◆
HS invariant and attractive ⇐⇒ Globally Asymptotically Stable (GAS)
=⇒environment-assisted stabilization Claudio Altafini, Dijon 2013
Stabilization of SME – p. 10/24
Extension to SME
● Introduction
■
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.
● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
Problem
dρt =
Consider the SME F(H, ρt ) +
X k
!
D(Lk , ρt ) + D(M, ρt ) 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 TtW (·) and attractive in probability a.s.?
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 11/24
Condition for invariance
● Introduction
dρ =
● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
LP (ρ) LR (ρ)
LS (ρ) LQ (ρ)
!
dt +
GS (M, ρ) GQ (M, ρ)
GP (M, ρ) GR (M, ρ)
!
dWt
Approach: in order for IS (H) to be invariant for the SME, it has to be invariant for both its diffusion and drift parts ! GS (M, ρS ) LS (ρS ) 0 , ∀ ρS ∈ IS (H) =⇒ 0 0 0
0 0
!
=⇒block structure of Lk and M matrices ■
for ρS ∈ IS (H) ◆ diffusion G(M, ρS ) =
ρS (MS†
(MS − Tr(MS ρS ))ρS + MQ ρS
−
Tr(MS† ρS ))
† ρ S MQ
0
=⇒IS (H) is invariant to G(M, ρS ) if MQ = 0 Claudio Altafini, Dijon 2013
Stabilization of SME – p. 12/24
!
Condition for invariance ■
drift
● Introduction ● SME ● MME
X
F(H, ρS ) +
● State decomposition ● Invariance & attractivity for MME
D(Lk , ρS ) + D(M, ρS ) =
k
● Invariance & attractivity of SME ● Feedback-assisted stabilization
● Examples ● Conclusion
−iHS ρS + iρS HS + d1 (Lk , M, ρS )
∗
1 † iρS HP + d2 (Lk , ρS ) − ρS MS MP 2 k X † LQ,k ρS LQ,k X
k
where we define: Xn 1 † † LS,k ρS L†S,k d1 (Lk , M, ρS ) = − (MS MS ρS + ρS MS MS ) + 2 k 1 − (L†S,k LS,k ρS + ρS L†S,k LS,k + L†Q,k LQ,k ρS + ρS L†Q,k LQ,k ) , 2 1 d2 (Lk , ρS ) = LS,k ρS L†Q,k − ρS (L†S,k LP,k + L†Q,k LR,k ). 2 MS ρS MS†
■
Claudio Altafini, Dijon 2013
=⇒same conditions on HP , MP and LP,k as for the MME Stabilization of SME – p. 13/24
Invariance for the SME ■
putting together
● Introduction ● SME ● MME
Proposition
● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
IS (H) is invariant for the SME iff ! LS,k LP,k Lk = ∀k, 0 LR,k ! MS MP , M= 0 MR 1 iHP − 2
X
L†S,k LP,k + MS† MP
k
!
= 0,
Corollary
IS (H) invariant for the SME ■
Claudio Altafini, Dijon 2013
⇐⇒
IS (H) invariant for the MME
More rigorous proof: support theorem
Stabilization of SME – p. 14/24
Attractivity for the SME ■ ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
attractivity of SME ⇐⇒ attractivity of MME Assume IS (H) is attractive for the MME for some H, Lk and M . Then with these H, Lk and M , IS (H) attractive in probability also for the SME
Proposition
● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 15/24
Attractivity for the SME ■ ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
attractivity of SME ⇐⇒ attractivity of MME Assume IS (H) is attractive for the MME for some H, Lk and M . Then with these H, Lk and M , IS (H) attractive in probability also for the SME
Proposition
■ Proof:
1. stability of IS (H) in probability ◆ candidate Lyapunov function V (ρ) = tr(ΠR ρ) where ΠR = projection on HR V (ρ) = 0 for ρ ∈ IS (H)
AV (ρ) = −tr
V (ρ) > 0 for ρ ∈ / IS (H) ! ! X † LP,k LP,k + MP† MP ρR 6 0 k
∀ρ ∈ D(H)
by ciclicity of the trace Claudio Altafini, Dijon 2013
Stabilization of SME – p. 15/24
Attractivity for the SME (cont.) 2. ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
attractivity (by contradiction) ■ Suppose ∃ ρ0 ∈ D(H) \ IS (H) for which IS (H) is not attractive =⇒ P lim d(TtW (ρ0 ), IS (H)) = 0 = 1 − p, p > 0 t→∞ =⇒ P lim V (TtW (ρ0 )) = 0 = 1 − p t→∞
=⇒
■
for t → ∞ set {V (TtW (ρ0 )) > 0} has measure > 0
in expectation then ∃ ξ(ρ0 ) > 0 for which Z lim supE V (TtW (ρ0 )) = lim sup t→∞
t→∞
≥ ξ(ρ0 ) lim sup t→∞
Z
dP {V
(TtW (ρ0 ))>0}
= ξ(ρ0 ) lim sup 1 − P V t→∞
=⇒ ∃k > 0 s.t. Claudio Altafini, Dijon 2013
{V (TtW (ρ0 ))>0}
V (TtW (ρ0 ))dP
(TtW (ρ0 ))
=0
lim sup d(E[TtW (ρ0 )], IS (H)) t→∞
= ξ(ρ0 )p
ξ(ρ0 )p ≥ > 0. k Stabilization of SME – p. 16/24
Invariance & attractivity of SME ■ ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
Invariance and attractivity of MME ⇐⇒ invariance and attractivity of the SME ⇐⇒ GAS of MME and SME both can be accomplished by means of 1. non-hermitian Lk and/or non-hermitian M =⇒(block) “ladder-like” operators 2. open-loop control Summary:
IS (H) is GAS in probability for the SME ⇐⇒ Lk , M and H have the structure ! ! LS,k LP,k MS MP Lk = ∀k, M = , 0 LR,k 0 MR ! 1 X † iHP − LS,k LP,k + MS† MP = 0, 2
Theorem
k
with at least one LP,k 6= 0 and/or MP 6= 0 =⇒environment-assisted stabilization Claudio Altafini, Dijon 2013
Stabilization of SME – p. 17/24
Block diagonal case
● Introduction
■
● SME ● MME
how about the case
● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization
Lk =
● Examples ● Conclusion
■
LS,k 0
0 LR,k
!
∀k,
M=
MS 0
0 MR
!
open-loop, time-invariant control: both HS and HR are invariant =⇒neither is attractive
Given SME with Lk and M as above, no time-invariant H exists rendering IS (H) GAS in probability
Proposition
if HP = 0 then IR (H) is invariant; if instead HP 6= 0 then IS (H) cannot be invariant
■ Proof:
Claudio Altafini, Dijon 2013
Stabilization of SME – p. 18/24
Feedback-assisted stabilization ■ Assumptions: ● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME
■
● Invariance & attractivity of SME ● Feedback-assisted stabilization
dim(HS ) = 1 (pure state stabilization) dim(H) > 3
further split of HR : H = HS ⊕ HC ⊕ HZ | {z } HR
● Examples ● Conclusion
0 H = Hc +u(ρ)Hf = 0 0
■
Claudio Altafini, Dijon 2013
Lk,S Lk = 0 0
0 Lk,C Lk,Y
0 Hc,C † Hc,W
0
Lk,W Lk,Z
0
0
† Hc,W +u(ρ) Hf,U Hc,Z 0
MS M = 0 0
0 MC MY
Hf,U 0 0
0 MW MZ
idea: ◆ design Hc,W so as to keep "reshuffling" inside HR = HC ⊕ HZ ◆ use Hf,U to "drain" probability out of HR Stabilization of SME – p. 19/24
0 0 0
Feedback-assisted stabilization ■
M. Mirrahimi, R. van Handel. Stabilizing feedback controls for quantum systems, SIAM J. Control Optim., 46, 445-467, 2007
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
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(ρt ρd ) ≥ γ u(ρt ) = −tr(i[Hf , ρt ]ρd ) 2. If tr(ρt ρd ) ≤ γ/2, u(ρt ) = 1; 3. If ρt ∈ B = {ρ : γ/2 < tr(ρt ρd ) < γ}, then ■ u(ρt ) = −tr(i[Hf , ρt ]ρd ) if ρt last entered B through the boundary tr(ρd ρ) = γ, ■ ut = 1 otherwise. Then ∃ γ > 0 such that u(ρt ) renders the SME GAS in probability Theorem
■
Claudio Altafini, Dijon 2013
use "patchy" feedback law similar to
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 Stabilization of SME – p. 20/24
Example 1: environment only
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
1 ρd = 0 0
0 0 0 0 0 0
0 M = 0 0
1 0 0
0 0 0
0 L1 = 0 0
0 0 1 3 2 1
HS is GAS without any open/closed loop Hamiltonian Hc = 0
Hf = 0
energy population
sample trajectories 0.8
1
0.9
0.7
0.8
0.6
1 − tr (ρd ρ)
0.7
0.6 ρ
11
ρ
0.5
22
ρ33 0.4
0.5
0.4
0.3
0.3
0.2 0.2
0.1
0.1
0
0
2
4
6
8
10
time
Claudio Altafini, Dijon 2013
12
14
16
18
20
0
0
2
4
6
8
10
12
14
16
18
20
time
Stabilization of SME – p. 21/24
Example 2: environment + open loop
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
1 ρd = 0 0
0 0 0 0 0 0
0 M = 0 0
1 0 0
0 0 0
0 L1 = 0 0
0 0 1 0 0 2
HS is rendered GAS by the following open loop Hamiltonian 0 0 0 H c = 0 0 1 Hf = 0 0 1 0 energy population
sample trajectories 1
0.9
0.9
0.8
0.8
0.7
0.7
1 − tr (ρ ρ)
1
0.6 11
ρ
0.5
0.6
d
ρ
22
ρ
33
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
2
4
6
8
10
time
Claudio Altafini, Dijon 2013
12
14
16
18
20
0
0
2
4
6
8
10
12
14
16
18
20
time
Stabilization of SME – p. 22/24
Example 3: open loop + closed loop
1 ρd = 0 0
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
■
0 0 0
0 0 0
1 0 M = 0 2 0 0
0 0 3
L1 = 0
HS is rendered GAS by the open loop and feedback Hamiltonians 0 0 0 0 −i 0 H c = 0 0 1 H f = i 0 0 0 1 0 0 0 0 energy population
sample trajectories 1
1.2
0.9 1
0.8
0.7
1 − tr (ρ ρ)
0.8
ρ
22
0.6
0.6
d
ρ11 ρ
33
0.5
0.4
0.4
0.3
0.2 0.2
0.1
0
0
2
4
6
8
10
time
Claudio Altafini, Dijon 2013
12
14
16
18
20
0
0
2
4
6
8
10
12
14
16
18
20
time
Stabilization of SME – p. 23/24
Conclusion ■
Environment (i.e., dissipation) and open-loop control can easily lead to conditions for GAS of a subsystem in both MME and SME
■
Philosophy: make the most use of environment and the least of 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: feedback can be useful to get GAS
● Introduction ● SME ● MME ● State decomposition ● Invariance & attractivity for MME ● Invariance & attractivity of SME ● Feedback-assisted stabilization ● Examples ● Conclusion
F. Ticozzi, K. Nishio, C. Altafini. Stabilization of Stochastic Quantum Dynamics via Open and Closed Loop Control, IEEE Tr. Autom. Contr., Jan. 2013 Claudio Altafini, Dijon 2013
Stabilization of SME – p. 24/24
