Local Asymptotic Normality in Quantum Statistics M d lin Gu - PowerPoint PPT Presentation

Local Asymptotic Normality in Quantum Statistics M � d � lin Gu �� School of Mathematical Sciences University of Nottingham Richard Gill (Leiden) Jonas Kahn (Paris XI) Bas Janssens (Utrecht) Anna Jencova (Bratislava) Luc Bouten (Caltech)

Outline: • Quantum state estimation and optimality • Local Asymptotic Normality in classical statistics • Local Asymptotic Normality for qubits • Local Asymptotic Normality for d-dimensional state

Quantum state estimation Problem: given n identically prepared systems in the state ρ θ with θ ∈ Θ , perform a measurement M ( n ) and construct an estimator ˆ θ n of θ from the result X ( n ) . ρ θ ⊗ ρ θ ⊗ · · · ⊗ ρ θ �− → X ( n ) �− → ˆ θ n

Quantum state estimation Problem: given n identically prepared systems in the state ρ θ with θ ∈ Θ , perform a measurement M ( n ) and construct an estimator ˆ θ n of θ from the result X ( n ) . ρ θ ⊗ ρ θ ⊗ · · · ⊗ ρ θ �− → X ( n ) �− → ˆ θ n ρ θ ∼ ✲ X 1 ∼ M 1 ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ρ θ ∼ ✲ X 2 ✲ ˆ θ n ∼ M 2 ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ρ θ ∼ ✲ X n ∼ M n ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼

Quantum state estimation Problem: given n identically prepared systems in the state ρ θ with θ ∈ Θ , perform a measurement M ( n ) and construct an estimator ˆ θ n of θ from the result X ( n ) . ρ θ ⊗ ρ θ ⊗ · · · ⊗ ρ θ �− → X ( n ) �− → ˆ θ n ρ θ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ✲ ˆ X ( n ) ρ θ M ( n ) θ n ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ✲ ρ θ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼

Quantum state estimation Problem: given n identically prepared systems in the state ρ θ with θ ∈ Θ , perform a measurement M ( n ) and construct an estimator ˆ θ n of θ from the result X ( n ) . ρ θ ⊗ ρ θ ⊗ · · · ⊗ ρ θ �− → X ( n ) �− → ˆ θ n ρ θ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ✲ ˆ X ( n ) ρ θ M ( n ) θ n ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ✲ ρ θ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ Risk and Optimality: The quality of the estimation strategy ( M, ˆ θ n ) is given by the risk θ n ) = E � ρ θ − ρ ˆ ˆ R ( θ , ˆ R ( θ , ˆ θ n � 2 θ n ) = 1 − E F ( ρ θ , ρ θ n ) or 1

Bayesian vs frequentist optimality Bayesian: prior π ( d θ ) Frequentist � R θ 0 (ˆ R ( θ , ˆ R π (ˆ R ( θ , ˆ θ n ) := sup θ n ) θ n ) := θ n ) π ( d θ ) θ ∈ B ( θ 0 ,n − 1 / 2 ) M n R θ 0 (ˆ M n R π (ˆ := inf θ n ) R θ 0 ,n := inf θ n ) R π ,n n →∞ nR θ 0 ,n = C H ( θ 0 ) := lim R θ 0 := lim R π n →∞ nR π ,n � R π = R θ π ( d θ )

A rough classification of state estimation problems Separate measurements Joint measurements Practically feasible More difficult to implement Optimal for pure states Optimal for mixed states Parametric Optimal for one parameter R n ≈ C sep /n R n ≈ C joint /n Q. Homodyne Tomography, Conjecture/Program: Direct detection of Wigner fct... Non � parametric L.A.N. = convergence to model non-parametric rates for of displaced quasifree states of estimation of state as a whole infinite dimensional CCR alg. R n = O ( (log n ) k /n, n − α , ... )

Asymptotically things become easier... Idea of using (local) asymptotic normality in optimal estimation: • as n → ∞ the n particle model gets ‘close’ to a Gaussian shift model Φ θ • the latter has fixed, known variance and unknown mean (related to) θ , • the mean can be estimated optimally by simple measurements (heterodyne) • the measurement can be ‘pulled back’ to the n systems 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 H ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼

Motivation / earlier work • Classical L.A.N. theory of Le Cam asymptotic equivalence of statistical models optimal estimation rates • Central Limit behaviour for quantum systems Coherent spin states Gaussian description of atoms-light interaction (Mabuchi, Polzik experiments) • Work by Hayashi and Matsumoto on asymptotics of state estimation M. Hayashi, Quantum estimation and the quantum central limit theorem (in Japanese), Bull. Math. Soc. Japan 55 (2003) English translation: quant-ph/0608198 M. Hayashi, K. Matsumoto, Asymptotic performance of optimal state estimation in quantum two level system arXiv:quant-ph/0411073

Local Asymptotic Normality for coin toss Repeated coin toss: X 1 , . . . , X n i.i.d. with P [ X i = 1] = θ , P [ X i = 0] = 1 − θ Su ffi cient statistic: ˆ � n θ n := 1 i =1 X i unbiased estimator since E ( X ) = θ n Central Limit Theorem: √ n (ˆ D θ n − θ ) → N (0 , θ (1 − θ )) − √

Local Asymptotic Normality for coin toss 0.1 0.05 0 0 20 40 60 80 k Binomial n=100 p=0.6 Normal m=60 v=25

Local Asymptotic Normality for coin toss 0.1 0.05 0 0 20 40 60 80 k Binomial n=100 p=0.5 Normal m=50 v=25

Local Asymptotic Normality for coin toss 0.1 0.05 0 0 20 40 60 80 k Binomial n=100 p=0.4 Normal m=40 v=25

Local Asymptotic Normality for coin toss 0.1 0.05 0 0 20 40 60 80 k Binomial n=100 p=0.3 Normal m=30 v=25

Local Asymptotic Normality for coin toss Repeated coin toss: X 1 , . . . , X n i.i.d. with P [ X i = 1] = θ , P [ X i = 0] = 1 − θ Su ffi cient statistic: ˆ � n θ n := 1 i =1 X i unbiased estimator since E ( X ) = θ n Central Limit Theorem: √ n (ˆ D θ n − θ ) → N (0 , θ (1 − θ )) − Local parameter: let θ = θ 0 + u/ √ n for a fixed known θ 0 , then Gaussian shift u n := √ n (ˆ ˆ θ n − θ 0 ) ≈ N ( u, θ 0 (1 − θ ) 0 ) model

Local Asymptotic Normality for coin toss Repeated coin toss: X 1 , . . . , X n i.i.d. with P [ X i = 1] = θ , P [ X i = 0] = 1 − θ Su ffi cient statistic: ˆ � n θ n := 1 i =1 X i unbiased estimator since E ( X ) = θ n Central Limit Theorem: √ n (ˆ D θ n − θ ) → N (0 , θ (1 − θ )) − Local parameter: let θ = θ 0 + u/ √ n for a fixed known θ 0 , then Gaussian shift u n := √ n (ˆ ˆ θ n − θ 0 ) ≈ N ( u, θ 0 (1 − θ ) 0 ) model Why can we restrict to a local neighbourhood ? You can construct a θ 0 from the data and the true θ will be in a ‘1 / √ n -neighbourhood’ with high probability

Local Asymptotic Normality: general case Let ( Y 1 , . . . , Y n ) be i.i.d. with P θ 0 + u/ √ n a ‘smooth’ family with u ∈ R k . Then P θ 0 + u/ √ n � n �� : u ∈ R k � θ 0 ) : u ∈ R k � N ( u, I − 1 � ❀

Local Asymptotic Normality: general case Let ( Y 1 , . . . , Y n ) be i.i.d. with P θ 0 + u/ √ n a ‘smooth’ family with u ∈ R k . Then P θ 0 + u/ √ n � n �� : u ∈ R k � θ 0 ) : u ∈ R k � N ( u, I − 1 � ❀ Strong convergence: there exist randomizations (Markov kernels) T n , S n such that P θ 0 + u/ √ n � n � � � − N ( u, I − 1 n →∞ sup lim θ 0 ) tv = 0 � T n � � � � u � <a and P θ 0 + u/ √ n � n � � � − S n N ( u, I − 1 n →∞ sup lim θ 0 ) tv = 0 � � � � � u � <a

Local Asymptotic Normality: general case Let ( Y 1 , . . . , Y n ) be i.i.d. with P θ 0 + u/ √ n a ‘smooth’ family with u ∈ R k . Then P θ 0 + u/ √ n � n �� : u ∈ R k � θ 0 ) : u ∈ R k � N ( u, I − 1 � ❀ Strong convergence: there exist randomizations (Markov kernels) T n , S n such that P θ 0 + u/ √ n � n � � � − N ( u, I − 1 n →∞ sup lim θ 0 ) tv = 0 � T n � � � � u � <a and P θ 0 + u/ √ n � n � � � − S n N ( u, I − 1 n →∞ sup lim θ 0 ) tv = 0 � � � � � u � <a Importance: • Shows that for large n the statistical model is ‘locally easy’: Gaussian shift model • Asymptotically, we only need to solve the statistical problem for the Gaussian shift model

L. A. N. for finite dimensional quantum systems � ⊗ n be the joint state of n i.i.d. systems with ρ θ ∈ M ( C d ) ‘smooth’. Then � Let ρ θ 0 + u/ √ n � ⊗ n : u ∈ R d 2 − 1 � �� � θ 0 ) : u ∈ R d 2 − 1 � Φ ( u, H − 1 ρ θ 0 + u/ √ n ❀

L. A. N. for finite dimensional quantum systems � ⊗ n be the joint state of n i.i.d. systems with ρ θ ∈ M ( C d ) ‘smooth’. Then � Let ρ θ 0 + u/ √ n � ⊗ n : u ∈ R d 2 − 1 � �� � θ 0 ) : u ∈ R d 2 − 1 � Φ ( u, H − 1 ρ θ 0 + u/ √ n ❀ Strong convergence: there exist quantum channels T n , S n such that � � � ⊗ n − Φ ( u, H − 1 � ρ θ 0 + u/ √ n n →∞ sup lim θ 0 ) 1 = 0 � T n � � � � u � <a and � � � ⊗ n − S n Φ ( u, H − 1 � ρ θ 0 + u/ √ n n →∞ sup lim θ 0 ) 1 = 0 � � � � � u � <a