System identification for quantum Markov processes Mdlin Gu School - - PowerPoint PPT Presentation

System identification for quantum Markov processes

Mădălin Guţă School of Mathematical Sciences University of Nottingham

High dimensional problems and Quantum Physics Paris 2015

Outline

Quantum parameter estimation

◮ classical and quantum Fisher information ◮ local asymptotic normality for i.i.d. models

Quantum Markov chains

◮ ergodic dynamics, output state ◮ CLT and LAN for time averages of output measurements ◮ quantum Fisher information and quantum LAN ◮ identifiability classes

Quantum enhanced metrology

◮ atom maser ◮ dynamical phase transitions and Heisenberg scaling ◮ metastability

Quantum parameter estimation

M

ρθ

X ∼ PM

θ

ˆ θ

Estimation problem: estimate θ by performing a measurement M on system in state ρθ What is quantum about this ?

◮ fixed measurement: "classical stats" problem with special probabilistic structure ◮ "optimal" measurement: need to understand structure of quantum statistical model

Classical and quantum Cramér-Rao bounds1: if ˆ θ is unbiased

E h (ˆ θ − θ)T · (ˆ θ − θ) i ≥ IM(θ)−1 ≥ F(θ)−1

Classical Fisher info Quantum Fisher info

• 1A. Holevo. Probabilistic and Statistical Aspects of Quantum Theory 1982; S. L. Braunstein and C. M. Caves, P.R.L. 1994

Pure states models

• ne parameter pure state rotation model: |ψθ := e−iθG|ψ,

ψ|G|ψ = 0 Quantum Fisher information: F(θ) = 4

• dψθ

dθ

• 2

= 4Varψ(G) = 4 ψ

G2 ψ

Quantum Gaussian shift model: CV system [Q, P] = i1

◮

• F/2 θ
• coherent state with mean
• F/2 θ, 0
• ◮ quantum Fisher information = 4Var
• F/2P
• = F

◮ QFI achievable by measuring Q

Q P

2D quantum Gaussian shift model incompatibility of P and Q ⇒ F not achievable

Convergence to Gaussian model for i.i.d. ensembles

p F/2u p F/2v Q P

• ψ⊗n

θ0+u/pn

E

• ψ⊗n

θ0+v/pn

E

Quantum data: ensemble of n identically prepared systems |ψθ⊗n := eiθG|ψ⊗n , ψ|G|ψ = 0 Local asymptotic normality (Holstein-Primakov): In an “uncertainty neighbourhood" of size n−1/2 around θ0, the overlaps of joint states are approximately equal to those of a Gaussian model with QFI = F

• ψ⊗n

θ0+u/√n

• ψ⊗n

θ0+v/√n

• =

ψ|ei(u−v)G/√n

ψn −

→ e(u−v)2F/8 =

• F/2 u
• F/2 v
• General LAN for mixed states & multi-dimensional models2
• 2J. Kahn and MG, Commun. Math. Phys. 2009

Gaussian limit for qubits

n identically prepared qubits

• ψ

u √n , v √n

• := exp
• ivσx − uσy

√n

• | ↑

Collective observables Lx,y,z := n

i=1 σ(i) x,y,z

Quantum Central Limit Theorem (u = 0, v = 0)

Lx √ 2n D

− → N 0, 1

2

• Ly

√ 2n D

− → N 0, 1

2

• Lx

√ 2n, Ly √ 2n

• = 2i

2nLz l.l.n.

− − − → i1

• z

x y √n n

Gaussian limit for qubits

n identically prepared qubits

• ψ

u √n , v √n

• := exp
• ivσx − uσy

√n

• | ↑

Collective observables Lx,y,z := n

i=1 σ(i) x,y,z

Quantum Central Limit Theorem (u = 0, v = 0)

Lx √ 2n D

− → N √ 2u, 1

2

• Ly

√ 2n D

− → N √ 2v, 1

2

• Lx

√ 2n, Ly √ 2n

• = 2i

2nLz l.l.n.

− − − → i1

• z

x y √n n

Gaussian limit for qubits

n identically prepared qubits

• ψ

u √n , v √n

• := exp
• ivσx − uσy

√n

• | ↑

Collective observables Lx,y,z := n

i=1 σ(i) x,y,z

Local asymptotic normality = q Gaussian on tangent space

Lx √ 2n −

→ Q

Ly √ 2n −

→ P

• ψ

u √n , v √n

• −

→ | √ 2u, √ 2v

• z

x y √n n

Optimal estimation using local asymptotic normality

Φθ

∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼

H

• Y ∼ P(H, Φθ)
• ˆ

θ

ρθ

∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼

ρθ

∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼

ρθ

∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼

Mn

• Xn ∼ P(Mn, ρθ)
• ˆ

θn

n → ∞

[L. Le Cam]

Sequence of I.I.D. quantum statistical models Qn = {ρ⊗n

θ

: θ ∈ Θ} Qn converges (locally) to simpler Gaussian shift model Q Optimal measurement for limit Q can be pulled back to Qn

Outline

Quantum parameter estimation

◮ classical and quantum Fisher information ◮ local asymptotic normality for i.i.d. models

Quantum Markov chains

◮ ergodic dynamics, output state ◮ CLT and LAN for time averages of output measurements ◮ quantum Fisher information and quantum LAN ◮ identifiability classes

Quantum enhanced metrology

◮ atom maser ◮ dynamical phase transitions and Heisenberg scaling ◮ metastability

Hidden Markov chain in an input-output setting

i j Xn Yn Yn+1 Yn+2 Zn−1 Zn−2 Zn−3 tij

Input Output

Evolution: scattering interaction between input and system: (Xn, Yn) → (Xn+1, Zn) = F(Xn, Yn)

◮ Input: i.i.d. random variables Y1, Y2, . . . ◮ System: Markov process X1, X2, . . . induced by the interaction ◮ Output: correlated random variables Z1, Z2, . . . (hidden Markov chain)

Statistical Problem: estimate dynamics law F by observing the output3

• 3T. Petrie, Ann. of Math. Stat. (1969).

P.J. Bickel, Y. Ritov, and T. Ryden Ann. Stat. (1998)

Discrete Markov dynamics a.k.a. channels with memory

System Input Output Uθ |χi |χi |χi

Feedback control of cavity state in the atom maser [C. Sayrin et al, Nature 2011]

Input-output dynamics: successive interactions with (memory) system via unitary Uθ System identification problem4: estimate θ by measuring the output state

◮ which parameters can be identified ? ◮ How does the output QFI scale with "time" n ? ◮ How does this related to dynamical properties, e.g. ergodicity, spectral gap...?

Methods: Bayesian/extended filter5, maximum likelihood, compressed sensing6, q.f.tomo7

4M.G., J. Kiukas, Commun. Math. Phys. 2015

• 5H. Mabuchi Quant. Semiclass. Optics 1996; J. Gambetta and H. M. Wiseman Phys. Rev. A 2001
• 6M. Cramer et al, Nat. Commun. 2010

7Steffens et al, N. J. Phys. 2014

Quantum Markov dynamics

System Output Input Ck Ck Ck CD Ck |χi |χi |χi Ck Ck Ck ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ U

Dynamics determined by isometry V : CD → CD ⊗ Ck V |ψ := U|ψ ⊗ χ =

• i

Ki|ψ ⊗ |i System-output state after n steps is of matrix product form8 |Ψ(n) = U(n)|χ ⊗ |ψ⊗n =

• i1,...,in

Kin . . . Ki1|χ ⊗ |in ⊗ · · · ⊗ |i1 Reduced system evolution given by transition operator T : M(CD) → M(CD) ρ(n) = Trout(|Ψ(n)Ψ(n)|) = T n(ρin), ρin = |χχ| T(ρ) =

k

• i=1

KiρK†

i

• 8M. Fannes, B. Nachtergale and R. Werner, 1992; D. Perez-Garcia, F. Verstraete, M. Wolf and I. Cirac, 2007

Quantum Perron-Frobenius Theorem9

Theorem (Quantum Perron-Frobenius Theorem)

If T is an irreducible CP-TP map (no invariant subspaces)

◮ spectral radius r(T) = 1 is a non-degenerate eigenvalue of T ◮ unique, strictly positive stationary state: T(ρss) = ρss ◮ the eigenvalues on the unit circle form a group

If T is primitive (irreducible and aperiodic) then

◮ |λ| < 1 for all remaining eigenvalues ◮ convergence to stationary state

lim

n→∞ T n(σ) = ρss r(S)

Key observation: if Tǫ is a small perturbation of primitive T ⇒ dominant eigenvalue λǫ varies smoothly and determines the asymptotics

• 9D. E. Evans and R. Hoegh-Krohn, J. London Math. Soc 1978; M. Sanz, et al, IEEE Trans. Inform. Th., 2010

Quantum Markov chains: sequential output measurements

System Output Input Ck Ck Ck CD Ck |χ |χ |χ Ck Ck Ck ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ U A1 A2 A3

Observable A =

i ai|ii| measured on each unit −

→ outcomes A1, A2, . . . Statistic: time (empirical) average Sn(A) = 1

n

n

i=1 Ai

Moment generating function φ(s) := E esnSn(A) = Tr(T n

s (ρin))

Deformed transition operator Ts : M(CD) → M(CD) (CP, non-TP) Ts : ρ →

• i

esaiKiρK∗

i

Central Limit Theorem for the measurement process

Theorem (Central Limit)

Let T be primitive. Then 1) time averages converge to stationary means Sn(A) → Ess(A) 2) fluctuations are normal Fn(A) := √n(Sn(A) − Ess(A))

L

− − − − →

n→∞ N(0, V (A))

with variance V (A) =

• Ess(A2) + 2Ess(A ⊗ (Id − T)−1(B)),

B := χ|U∗(1 ⊗ A)U|χ

d2 log λs ds2

, λs = dominant eigenvalue of Ts Remarks 1) similar CLT holds for time averages of multiple-outcomes functions f(A1, . . . , Ar)10 2) similar CLT holds for the total counts and integrated homodyne current in continuous-time 11

• 10M. van Horssen and M.G., J. Math. Phys. 2015
• 11C. Catana, L. Bouten, M.G., arXiv:1407.5131

Classical Fisher information of time average

System Output Input Ck Ck Ck CD Ck |χ |χ |χ Ck Ck Ck ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ U A1 A2 A3

Dynamics with isometry Vθ with unknown parameter θ = θ0 + u/√n Time average Sn = 1

n

n

i=1 Ai captures deviations form mean µθ0 = Eθ0(A)

√n(Sn − µθ0)

L

− → N

dµ

dθ u, V (A)

• Classical Fisher information = signal to noise ratio (in terms of dom. eigenv. λs,θ of Ts,θ)

IA(θ0) =

dµ

dθ

2

V (A) =

• ∂2λs,θ

∂s∂θ

• s=0,θ=θ0

2

∂2λs,θ ∂s2

• s=0

Quantum Markov dynamics

System Output Input Ck Ck Ck CD Ck |χi |χi |χi Ck Ck Ck ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ U

Dynamics determined by isometry V : CD → CD ⊗ Ck V |ψ := U|ψ ⊗ χ =

• i

Ki|ψ ⊗ |i System-output state after n steps is of matrix product form12 |Ψ(n) = U(n)|χ ⊗ |ψ⊗n =

• i1,...,in

Kin . . . Ki1|χ ⊗ |in ⊗ · · · ⊗ |i1 If U = Uθ, what is the quantum Fisher information of the output state ?

• 12M. Fannes, B. Nachtergale and R. Werner, 1992; D. Perez-Garcia, F. Verstraete, M. Wolf and I. Cirac, 2007

Linear scaling of quantum Fisher information (for primitive chains)

One step generator Gθ := i dUθ

dθ · U∗ θ

Total generator Gθ(n) as fluctuations operator d dθ |Ψθ(n) = −iGθ(n)|Ψθ(n) Gθ(n) :=

n

• i=1

U(n)

θ

. . . U(i+1)

θ

G(i)

θ U(i+1)∗ θ

. . . U(n)∗

θ

= √nFn(Gθ) QFI = Markov variance of generator Gθ(n) increases linearly with n and the constant is Fθ := lim

n→∞

1 nF n

θ = lim n→∞

4 nVar (Gθ(n)) = 4 G2

θ + 2Re[Gθ((Id − T)−1(Kθ) ⊗ 1)] ss ,

Kθ := χ|Gθ|χ Remarks

◮ linearity is related to the finite correlation time τ ∝

1 gap of the output

◮ Constant in linear scaling may blow up as (Id − T)−1 ∝

1 gap

◮ Conjecture: all fluctuation operators satisfy Central Limit Theorem

Local asymptotic normality for the system + output state

Primitive quantum Markov chain V = Vθ with θ unknown Localise parameter in a region of “uncertainty” size θ = θ0 + u/√n

p F/2u p F/2v Q P

• Ψθ0+u/

√ t(n)

E

• Ψθ0+v/

√ t(n)

E

Theorem (LAN for quantum Markov chains)

The system + output quantum model |Ψn

u := |Ψθ0+u/√n(n) converges (weakly) to the

quantum Gaussian shift model

• F/2u

lim

n→∞Ψn u|Ψn v =

• F/2u
• F/2v
• = exp(−F(u − v)2/8)

Remark

◮ General LAN for the (mixed) output state ρθ(n) = Trs(|Ψθ(n)Ψθ(n)|)13 ◮ LAN for all parameters with convergence to Gaussian shift on CCR algebra 13M.G. and J. Kiukas, Commun. Math. Phys. 2015

Quantum Fisher information as susceptibility

Overlap can be reduced to dominant eigenvalue of a deformed transition operator Ψn

u|Ψn v = Tr

• T n

u √n , v √n (ρin)

• ≈ exp(n log λ

u √n , v √n )

Expanding in

u √n, v √n and setting ∂λa,b ∂a

• a=b=0

= 0 Ψn

u|Ψn v −

→ exp

1

2 ∂ log λa,b ∂a∂b

• a=b=0

(u − v)2 so that F = −4 ∂ log λa,b ∂a∂b

• a=b=0

Similar methods have been used in 14

• 14M. Cozzini, R. Ionicioiu, and P. Zanardi, Phys. Rev. B, 2007; S. Gammelmark and K. Mølmer, Phys. Rev. Lett., 2014

Equivalence classes with identical (stationary outputs)17

Definition Two primitive chains with isometries V1 and V2 are called equivalent if for all n, ρout

V1 (n) = ρout V2 (n)

Theorem

Two primitive chains with isometries V1 and V2 are equivalent if and only if there exists a phase eiφ and a unitary W : CD → CD such that V2 = eiφ(W ⊗ 1)V1W ∗

• r equivalently

KV2

i

= eiφ(W ⊗ 1)KV1

i

W ∗, i = 1, . . . , k.

Remarks 1) quantum extension of the "classical" result by Petrie15 on equivalence classes of ergodic hidden Markov chains 2) similar result holds in continuos-time: LV2

i

= W LV1

i

W ∗ and HV2 = W HV1W ∗ + c1 3) similar result holds for (passive) linear systems16

• 15T. Petrie, Annals of Math. Statistics, 1969

16M.G. and N. Yamamoto, IEEE Proceedings 52nd CDC 2013, IEEE TC (2016) 17M.G. and J. Kiukas, Commun. Math. Phys. 2015; M Fannes, B Nachtergaele, and R.F. Werner. J. Funct. Anal. 1994.

Idea of the proof

Define the off-diagonal transition operator T12 : ρ →

d

• i=1

KV1

i ρKV2∗ i

Overlap of the two system-output states ΨV2ψ(n)|ΨV1ψ(n) ≈ λn

1,2

Two alternatives:

A) |λ12| = 1 = ⇒ KV2

i

= eiφWKV1

i

W ∗ = ⇒ equivalent systems B) |λ12| < 1 = ⇒ overlap decays exponentially = ⇒ non-equivalent systems

Outline

Quantum parameter estimation

◮ classical and quantum Fisher information ◮ local asymptotic normality for i.i.d. models

Quantum Markov chains

◮ ergodic dynamics

CLT and LAN for time averages of output measurements

◮ quantum Fisher information and quantum LAN ◮ identifiability classes

Quantum enhanced metrology

◮ atom maser ◮ dynamical phase transitions and Heisenberg scaling ◮ metastability

Open systems in the input-output formalism

Open system dynamics: dissipative evolution of reduced system state dρ dt = L(ρ) = −i[H, ρ] +

k

• j=1

LjρL†

j − 1

2 {ρ, L†

jLj}

Input-output formalism18: system interacts with environment in a Markov fashion System Input Output A1(t) Aout

1

(t) Aout

k (t)

Ak(t) N(t) H L1, . . . , Lk B(t) Unitary dynamics: singular coupling with incoming input fields (Q Stoch Diff Eq19) dU(t) =

• −iHdt +

k

• j=1

LjdA†

j(t) − L† jdAj(t) − 1

2 L†

jLjdt

• U(t)

18C.W. Gardiner and P Zoller, Quantum Noise, Springer 2004

• 19K. R. Parthasarathy, An introduction to quantum stochastic calculus, Springer Birkhäuser, 1992

The atom maser12

( ( ( ( ( (

ck

|k |k |k + 1

+ sk

Jaynes-Cummings interaction between a two-level atom and a cavity U : |1 ⊗ |k → cos

• φ
• k + 1
• |1 ⊗ |k + sin
• φ
• k + 1
• |0 ⊗ |k + 1

Coarse grained cavity dynamics for Poisson distributed input atoms with rate Nex dρ dt = L∗(ρ) =

4

• i=1
• LiρL∗

i − 1

2 {L∗

i Li, ρ}

• with jump operators

◮ L1 : |k → √Nex sin(φ

√ k + 1) |k + 1 (excitation absorbed from atom)

◮ L2 : |k → √Nex cos(φ

√ k + 1) |k (atom remains in excited state)

◮ L3 : |k →

• k(ν + 1) |k − 1 (photon emitted in the bath)

◮ L4 : |k →

• (k + 1)ν |k + 1 (photon absorbed from the bath)

1H.-J. Briegel, B.-G. Englert, N. Sterpi, and H. Walther, Phys. Rev. A 1994

• 2H. Walther., Rep. Phys. 1992

Stationary state and phase transitions

                  

 

Mean photon number and photon distribution in the stationary state as function of α = √Nexφ

unique stationary state

ρss(n) = ρss(0)

n

• k=1
• ν

ν + 1 + Nex ν + 1 sin2(φ √ k) k

• jumps in mean photon number around α = 1, 2π, 4π

bistable stationary distribution around α = 2π, 4π can be undestood via large deviations for the counting process1

• 1J. P. Garrahan and I. Lesanovsky, Phys. Rev. Lett. 2010

The many Fisher informations of the atom maser20, 21

red: quantum Fisher info black: observe cavity + bath blue: observe cavity

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 50 100 150 200 250 300 φ Fisher.Info

Maximum Likelihood Counting statistics

red: Fisher info total counts blue: Fisher info counting process

• 20C. Catana, M van Horssen, M.G., Phil. Trans. Royal Soc. A 2012

21C.Catana, T. Kypraios and M.G. J. Phys. A: Math. Theor. 2014

Ergodicity and dynamical phase transitions22

If L is ergodic (spectral gap ∆λ := −Reλ2 > 0) then

◮ system converges to stationary state ρ(t) = etL(ρin)

t→∞

− − − − − → ρss

◮ Counting operator N(t) has normal fluctuations (∆N(t) ∝

√ t) around mean tµ If L is near phase transition (∆λ ≈ 0) then

◮ slow convergence to stationarity, long correlation time τ = 1/∆λ ◮ intermittent trajectories, counting operator N(t) has bimodal distribution up to times τ

If L has degenerate stationary states then

◮ infinite correlation times ◮ counting operator N(t) remains bimodal all times and variance increases as t2

• 22J. Garrahan, I. Lesanovsky, P.R.L. 2010; I. Lesanovsky, M. van Horssen, M. G., J. P. Garrahan, P.R.L. 2013

Phase estimation: Heisenberg limit at the DPT

System Input Output H L eiφµat|passivei + eiφµpt|activei φ First order phase transition: system with two "stationary phases" (H = Hi ⊕ Ha) with different emission rates µi = µa Initial state: superposition √pi|χi + √pa|χa with |χa,i ∈ Hi,a GHZ-type system-output state with generator N(t) |ψφ(t) = eiφN(t)|ψ(t) ≈ √pieiφµit|ψi(t) + √paeiφµat|ψa(t) Heisenberg limit: F(t) = 4Var(N(t)) ≈ t2pipa(µa − µi)2 must measure sys+out to achieve QFI

[ K. Macieszczak, M.G. I. Lesanovsky, J. P. Garrahan arXiv:1411.3914 ]

50 100 150 200 0.00 0.02 0.04 0.06 0.08

µit µat ∝ t γi √ t γa √ t

Phase estimation: QFI time behaviour near phase transition

System Input Output H L φ

System near first order DPT: metastability = ⇒ counting trajectories exhibit intermittency Short time (t ≪ τ) distribution of generator N(t) is bimodal = ⇒ quadratic growth of QFI Long time (t ≫ τ) ergodicity and normal fluctuations win = ⇒ linear growth of QFI ∆2N(t) τ = ∆λ−1 ∝ t2 ∝ tτ t

Outlook

Stationary (primitive) quantum Markov chains can be characterised completely up to unitary "change of coordinates" by measuring the output The output state is asymptotically Gaussian with quantum Fisher information equal to the "Markov variance of the generator" Chains with multiple stationary states can exhibit Heisenberg (quadratic) scaling of QFI Ongoing / other work

◮ Multiparameter LAN = convergence to CCR model ◮ Enhanced metrology & dynamical phase transitions ◮ Metastability ◮ General quantum Markov CLT ◮ use of coherent feedback in system identification ◮ design better input states ◮ Linear systems: frequency domain analysis with time dependent/ stationary input