Convergence of Quantum Statistical Experiments

Transcription

1 Convergence of Quantum Statistical Experiments M!d!lin Gu"! University of Nottingham Jonas Kahn (Paris-Sud) Richard Gill (Leiden) Anna Jen#ová (Bratislava)

2 Statistical experiments and statistical decision problems Statistical experiment Measure space: (Ω, Σ, P) Statistical experiment over Θ (dominated): E = {P θ P : θ Θ} Statistical decision problem Decision space: (D, D, ν) Loss function: W : D Θ R + Decision procedure: T : L 1 (Ω, Σ, P) L 1 (D, D, ν) (randomization, e.g. Markov kernel, randomized statistic) Risk of the decision procedure T : R(T, θ) = T P θ W θ := W (y, θ)t ( dp θ dp ) ν(dy) Standard examples: Estimation: D = Θ, W (ˆθ, θ) = (ˆθ θ) 2 Hypothesis testing: D = {0, 1}, W (h, θ) = 1 (0) if θ (/ ) Θ h where Θ = Θ 0 Θ 1

3 Statistical experiments and statistical decision problems Statistical experiment Measure space: (Ω, Σ, P) Statistical experiment over Θ (dominated): E = {P θ P : θ Θ} Statistical decision problem Decision space: (D, D, ν) P θ T TP θ Loss function: W : D Θ R + Decision procedure: T : L 1 (Ω, Σ, P) L 1 (D, D, ν) (randomization, e.g. Markov kernel, randomized statistic) Risk of the decision procedure T : R(T, θ) = T P θ W θ := W (y, θ)t ( dp θ dp ) ν(dy) Standard examples: Estimation: D = Θ, W (ˆθ, θ) = (ˆθ θ) 2 Hypothesis testing: D = {0, 1}, W (h, θ) = 1 (0) if θ (/ ) Θ h where Θ = Θ 0 Θ 1

4 Comparison of experiments Experiments which are close to each other should have similar optimal risks Weak convergence of experiments: E n = {P n θ : θ Θ} E = {P θ : θ Θ} if { } dp n { } θ dpθ dp n : θ Θ : θ Θ θ 0 dp θ0 Convergence in distribution of finite marginals Strong convergence of experiments: E n E iff max(δ(e n, E), δ(e, E n )) 0, δ(e n, E) = inf T sup θ T P n θ P θ tv Le Cam deficiency Theorem: Weak and strong convergence are equivalent for finite sets Θ. Local asymptotic normality: let Y 1,..., Y n i.i.d. with P θ0 +u/ n be a smooth family with u R. Then {( Pθ0 +u/ n) n : u R } { N ( u, I(θ0 ) 1 ) )}

5 Quantum systems Quantum system: light in a microcavity, atom, nanomechanical systems... Observables = quantities which can be measured form a non-commutative algebra A (for ex. M(C d )) State = preparation: determines the prob. distribution of results of any measurements Duality between A and A: S A = A +,1 = {ρ M(C d ) : ρ 0, Tr(ρ) = 1}, : A A C ρ, A := E ρ (A) = Tr(ρ A) Classical case: A = L (Ω, Σ, P), A = L 1 (Ω, Σ, P), P : L 1 (Ω, Σ, P) L (Ω, Σ, P) C p, X P := p(ω) X(ω) P(dω)

6 Cl. Statistical Experiments Measure space: (Ω, Σ, P) Q. Statistical Experiments No little ω s Space of random variables: L (Ω, Σ, P) Algebra of Observables: A = M(C d ) Space of probability densities: L 1 (Ω, Σ, P) Q P = q := dq dp L1 (Ω, Σ, P) Space of States (density matrices): S = {ρ M(C d ) : ρ 0, Tr(ρ) = 1} Duality between the L 1 and L spaces:, P : L 1 (Ω, Σ, P) L (Ω, Σ, P) C p, X P := p(ω) X(ω) P(dω) Duality between A and A:, : A A C ρ, A = E ρ (A) := Tr(ρ A) Statistical experiment over Θ (dominated): E = {P θ P : θ Θ} Quantum statistical experiment over Θ: Q = {φ θ : θ Θ} φ θ : A C φ θ (A) = Tr(ρ θ A)

7 Quantum and Classical Randomizations Classical-Classical: T : L 1 (Ω, Σ, P) L 1 (Ω, Σ, P ) linear, positive, normalized P θ T TP θ Quantum-Classical (measurement): M : A L 1 (Ω, Σ, P) linear, positive, normalized ρ θ M P θ M Quantum-Quantum (channel): M : A ρ θ C A C(ρ θ ) linear, c. positive, normalized Classical-Quantum (preparation): P : L 1 (Ω, Σ, P) A P θ P PP θ linear, positive, normalized

8 Some quantum statistical problems Estimation of an unknown state (preparation) of a quantum system through measurements ρ θ ρ θ M X P θ M,n ˆθ n ρ θ Estimation of a quantum channel (transformation) C θ ρ n C θ M n X P θ M,n ˆθn C θ Cloning (copying) an unknown quantum state ρ θ C ρ θ ρ θ

9 Connes cocycles as likelihood ratio process Likelihood ratio process: Experiment E = (Ω, Σ, {P θ : θ Θ}) Λ θ0 : θ dp θ dp θ0 (ω) is sufficient statistic for E: if Σ 0 := σ(λ θ0 ) then E 0 := E Σ0 E. Quantum likelihood ratio is not uniquely defined: Experiment Q := (A, {ρ θ : θ Θ}) ρ θ (ρ θ0 ) 1 (ρ θ0 ) 1 ρ θ Quantum likelihood ratio process : Connes cocycles [Dρ θ, Dρ θ0 ] t := (ρ θ ) it (ρ θ0 ) it (analogue of ( dpθ dp θ0 ) it ) Theorem: [Petz] Let A 0 A be the algebra generated by {[Dρ θ, Dρ θ0 ] t : θ Θ, t R}. Then Q 0 := Q A0 Q.

10 Convergence of quantum statistical experiments Weak convergence of quantum experiments: Q n := (A n, {ρ n θ : θ Θ}) Q := (A, {ρ θ : θ Θ}) if for all k N, θ i Θ, t i R ( lim Tr n ρ n θ 0 k i=1 [Dρ n θ i, Dρ n θ 0 ] ti ) = Tr ( ρ θ0 k i=1 [Dρ θi, Dρ θ0 ] ti ), Strong convergence of quantum experiments: Q n Q if max(δ(q n, Q), δ(q, Q n )) 0 δ(q n, Q) = inf C sup θ C(ρ n θ ) ρ θ 1 Theorem: If Q n are based on discrete sums of matrix algebras, then weak and strong convergence are equivalent for finite sets Θ.

11 Local asymptotic normality for quantum states z Qubit state ρ = 1 + r σ 2 := ( 1 + rz r x ir y r x + ir y 1 r z ) 0, for r 1. r x Consider a local neighborhod (small ball) of the state ( ) µ 0 ρ 0 :=, 0 1 µ y z Parametrize the local neighborhood ρ = ρ u/ n := U ( u/ n ) ( µ + u z / ) n 0 U ( u/ n ), 0 1 µ u z where u = (u x, u y, u z ) R 3 is the local parameter, U(u) := exp(i(u x σ x + u y σ y )). y x The i.i.d. experiment : n identically prepared quatum systems { (ρu/ ) } Q n = n n : u R 3

12 The two convergence Theorems Locally, the joint state of n qubits converges to a quantum Gaussian shift model ρ n u/ n N u φ u Theorem: Let ρ (n) u := ( ρ u/ n ) n. Then as n { ρ (n) u : u R 3} { N u φ u : u R 3}. Theorem: Let ρ (n) u := ( ) ρ n. u/ n Then there exist quantum randomizations (q. channels) T n, S n such that for any η < 1/4 and ɛ > 0: lim n sup u <n η T n (ρ (n) u ) φ u 1 = O(n 1/4+η+ɛ ), and lim n sup u <n η u S n (φ u ) = O(n 1/4+η+ɛ ). 1 ρ (n)

13 References:! M. Gu"! and J. Kahn Local asymptotic normality for qubit states, Physical Review A., 73, (2006)! M. Gu"! and A. Jen#ová Local asymptotic normality in quantum statistics quant-ph/ ! M. Gu"!, B. Janssens and J. Kahn Optimal estimation of qubit states with continuous time measurements quant-ph/ ! M. Gu"!: Quantum statistical experiments and quantum decision problems in preparation