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 ρ θ X ∼ P M ˆ M θ θ Estimation problem: estimate θ by performing a measurement M on system in state ρ θ 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 bounds 1 : if ˆ θ is unbiased h i (ˆ θ − θ ) T · (ˆ ≥ I M ( θ ) − 1 ≥ F ( θ ) − 1 E θ − θ ) Classical Quantum Fisher info Fisher info 1 A. Holevo. Probabilistic and Statistical Aspects of Quantum Theory 1982; S. L. Braunstein and C. M. Caves, P.R.L. 1994

Pure states models one parameter pure state rotation model: | ψ θ � := e − iθG | ψ � , � ψ | G | ψ � = 0 Quantum Fisher information: 2 � dψ θ � = 4Var ψ ( G ) = 4 � ψ � � G 2 � � ψ � F ( θ ) = 4 � � dθ � � Quantum Gaussian shift model: CV system [ Q , P ] = i 1 P � � �� � � F/ 2 θ coherent state with mean F/ 2 θ, 0 ◮ � � Q �� � ◮ quantum Fisher information = 4Var F/ 2 P = F ◮ QFI achievable by measuring Q 2D quantum Gaussian shift model incompatibility of P and Q ⇒ F not achievable

Convergence to Gaussian model for i.i.d. ensembles P � E � ψ ⊗ n � θ 0 + v/ p n Q p F/ 2 u p F/ 2 v � E � ψ ⊗ n � θ 0 + u/ p n Quantum data: ensemble of n identically prepared systems e iθG | ψ � � ⊗ n , | ψ θ � ⊗ n := � � ψ | G | ψ � = 0 Local asymptotic normality (Holstein-Primakov): In an “uncertainty neighbourhood" of size n − 1 / 2 around θ 0 , the overlaps of joint states are approximately equal to those of a Gaussian model with QFI = F �� � � ψ | e i ( u − v ) G/ √ n � � ψ � n − → e ( u − v )2 F/ 8 = � � � = � � ψ ⊗ n � ψ ⊗ n F/ 2 u � F/ 2 v θ 0+ u/ √ n θ 0+ v/ √ n � General LAN for mixed states & multi-dimensional models 2 2 J. Kahn and MG, Commun. Math. Phys. 2009

Gaussian limit for qubits n identically prepared qubits � � ivσ x − uσ y � � � ψ := exp √ n | ↑ � � u v √ n , √ n Collective observables L x,y,z := � n i =1 σ ( i ) x,y,z Quantum Central Limit Theorem ( u = 0 , v = 0 ) z D L x → N � 0 , 1 � − √ 2 2 n √ n n D L y → N � 0 , 1 � − √ 2 2 n • x l.l.n. � � L y L x = 2 i 2 n , 2 n L z − − − → i 1 √ √ 2 n y

Gaussian limit for qubits n identically prepared qubits � � ivσ x − uσ y � � � ψ := exp √ n | ↑ � � u v √ n , √ n Collective observables L x,y,z := � n i =1 σ ( i ) x,y,z Quantum Central Limit Theorem ( u � = 0 , v � = 0 ) z → N � √ D L x 2 u, 1 � − √ 2 2 n √ n n → N � √ D L y 2 v, 1 � − √ 2 2 n • x l.l.n. � � L y L x = 2 i 2 n , 2 n L z − − − → i 1 √ √ 2 n y

Gaussian limit for qubits n identically prepared qubits � � ivσ x − uσ y � � � ψ := exp √ n | ↑ � � u v √ n , √ n Collective observables L x,y,z := � n i =1 σ ( i ) x,y,z Local asymptotic normality = q Gaussian on tangent space z L x 2 n − → Q √ √ n n L y 2 n − → P √ • x √ √ � � � ψ − → | 2 u, 2 v � � u v √ n , √ n y

Optimal estimation using local asymptotic normality ρ θ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ - ˆ ρ θ - X n ∼ P ( M n , ρ θ ) M n θ n ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ρ θ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ n → ∞ - ˆ Φ θ - Y ∼ P ( H , Φ θ ) θ [L. Le Cam] H ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ Sequence of I.I.D. quantum statistical models Q n = { ρ ⊗ n : θ ∈ Θ } θ Q n converges (locally) to simpler Gaussian shift model Q Optimal measurement for limit Q can be pulled back to Q n

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 Input Output t ij j i Y n +2 Y n +1 Y n X n Z n − 1 Z n − 2 Z n − 3 Evolution: scattering interaction between input and system: ( X n , Y n ) �→ ( X n +1 , Z n ) = F ( X n , Y n ) ◮ Input: i.i.d. random variables Y 1 , Y 2 , . . . ◮ System: Markov process X 1 , X 2 , . . . induced by the interaction ◮ Output: correlated random variables Z 1 , Z 2 , . . . (hidden Markov chain) Statistical Problem: estimate dynamics law F by observing the output 3 3 T. 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 U θ | χ i | χ i | χ i 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 U θ System identification problem 4 : 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 filter 5 , maximum likelihood, compressed sensing 6 , q.f.tomo 7 4 M.G., J. Kiukas, Commun. Math. Phys. 2015 5 H. Mabuchi Quant. Semiclass. Optics 1996; J. Gambetta and H. M. Wiseman Phys. Rev. A 2001 6 M. Cramer et al, Nat. Commun. 2010 7 Steffens et al, N. J. Phys. 2014

Quantum Markov dynamics | χ i | χ i | χ i U C k ⊗ C k ⊗ C k ⊗ C k ⊗ ⊗ ⊗ C k C k C k ⊗ C D Input Output System Dynamics determined by isometry V : C D → C D ⊗ C k � V | ψ � := U | ψ ⊗ χ � = K i | ψ � ⊗ | i � i System-output state after n steps is of matrix product form 8 | Ψ( n ) � = U ( n ) | χ � ⊗ | ψ � ⊗ n = � K i n . . . K i 1 | χ � ⊗ | i n � ⊗ · · · ⊗ | i 1 � i 1 ,...,i n Reduced system evolution given by transition operator T : M ( C D ) → M ( C D ) Tr out ( | Ψ( n ) �� Ψ( n ) | ) = T n ( ρ in ) , ρ ( n ) = ρ in = | χ �� χ | k � K i ρK † T ( ρ ) = i i =1 8 M. Fannes, B. Nachtergale and R. Werner, 1992; D. Perez-Garcia, F. Verstraete, M. Wolf and I. Cirac, 2007

Quantum Perron-Frobenius Theorem 9 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 ( ρ ss ) = ρ ss ◮ the eigenvalues on the unit circle form a group If T is primitive (irreducible and aperiodic) then r(S) ◮ | λ | < 1 for all remaining eigenvalues ◮ convergence to stationary state n →∞ T n ( σ ) = ρ ss lim Key observation: if T ǫ is a small perturbation of primitive T ⇒ dominant eigenvalue λ ǫ varies smoothly and determines the asymptotics 9 D. 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 A 3 A 2 A 1 | χ � | χ � | χ � U C k ⊗ C k ⊗ C k ⊗ C k ⊗ C k ⊗ C k ⊗ C k ⊗ C D Input Output System Observable A = � i a i | i �� i | measured on each unit − → outcomes A 1 , A 2 , . . . � n Statistic: time (empirical) average S n ( A ) = 1 i =1 A i n Moment generating function φ ( s ) := E � e snS n ( A ) � = Tr( T n s ( ρ in )) Deformed transition operator T s : M ( C D ) → M ( C D ) (CP, non-TP) � e sa i K i ρK ∗ T s : ρ �→ i i

Central Limit Theorem for the measurement process Theorem (Central Limit) Let T be primitive. Then 1) time averages converge to stationary means S n ( A ) → E ss ( A ) 2) fluctuations are normal F n ( A ) := √ n ( S n ( A ) − E ss ( A )) L − n →∞ N (0 , V ( A )) − − − → with variance � E ss ( A 2 ) + 2 E ss ( A ⊗ (Id − T ) − 1 ( B )) , B := � χ | U ∗ ( 1 ⊗ A ) U | χ � V ( A ) = d 2 log λ s , λ s = dominant eigenvalue of T s ds 2 Remarks 1) similar CLT holds for time averages of multiple-outcomes functions f ( A 1 , . . . , A r ) 10 2) similar CLT holds for the total counts and integrated homodyne current in continuous-time 11 10 M. van Horssen and M.G., J. Math. Phys. 2015 11 C. Catana, L. Bouten, M.G., arXiv:1407.5131