Local Asymptotic Normality in Quantum Statistics

Transcription

1 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)

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

3 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 ˆθ n of θ from the result X (n). ρ θ ρ θ ρ θ X (n) ˆθ n

4 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 ˆθ n of θ from the result X (n). ρ θ ρ θ ρ θ X (n) ˆθ n ρ θ M 1 X 1 ρ θ M 2 X 2 ρ θ M n X n ˆθ n

5 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 ˆθ n of θ from the result X (n). ρ θ ρ θ ρ θ X (n) ˆθ n ρ θ ρ θ M (n) X (n) ˆθ n ρ θ

6 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 ˆθ n of θ from the result X (n). ρ θ ρ θ ρ θ X (n) ˆθ n ρ θ ρ θ M (n) X (n) ˆθ n ρ θ Risk and Optimality: The quality of the estimation strategy (M, ˆθ n ) is given by the risk R(θ, ˆθ n ) = E ρ θ ρˆθ n 2 1 or R(θ, ˆθ n ) = 1 E F (ρ θ, ρˆθ n )

7 Bayesian vs frequentist optimality Bayesian: prior π(dθ) Frequentist R π (ˆθ n ) := R(θ, ˆθ n )π(dθ) R θ0 (ˆθ n ) := sup R(θ, ˆθ n ) θ B(θ 0,n 1/2 ) R π,n := inf M n R π (ˆθ n ) R π := lim n nr π,n R θ0,n := inf M n R θ0 (ˆθ n ) R θ0 := lim n nr θ 0,n = C H (θ 0 ) R π = R θ π(dθ)

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

9 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 ρ θ M n X n P(M n, ρ θ ) ˆθ n ρ θ n Φ θ H Y P(H, Φ θ ) ˆθ

10 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/ M. Hayashi, K. Matsumoto, Asymptotic performance of optimal state estimation in quantum two level system arxiv:quant-ph/

11 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 Sufficient statistic: ˆθ n := 1 n n i=1 X i unbiased estimator since E(X) = θ Central Limit Theorem: n(ˆθ n θ) D N(0, θ(1 θ))

12 Local Asymptotic Normality for coin toss k Binomial n=100 p=0.6 Normal m=60 v=25

13 Local Asymptotic Normality for coin toss k Binomial n=100 p=0.5 Normal m=50 v=25

14 Local Asymptotic Normality for coin toss k Binomial n=100 p=0.4 Normal m=40 v=25

15 Local Asymptotic Normality for coin toss k Binomial n=100 p=0.3 Normal m=30 v=25

16 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 Sufficient statistic: ˆθ n := 1 n n i=1 X i unbiased estimator since E(X) = θ Central Limit Theorem: n(ˆθ n θ) D N(0, θ(1 θ)) Local parameter: let θ = θ 0 + u/ n for a fixed known θ 0, then û n := n(ˆθ n θ 0 ) N(u, θ 0 (1 θ) 0 ) Gaussian shift model

17 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 Sufficient statistic: ˆθ n := 1 n n i=1 X i unbiased estimator since E(X) = θ Central Limit Theorem: n(ˆθ n θ) D N(0, θ(1 θ)) Local parameter: let θ = θ 0 + u/ n for a fixed known θ 0, then û n := n(ˆθ n θ 0 ) N(u, θ 0 (1 θ) 0 ) Gaussian shift 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

18 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 } { N(u, I 1 θ 0 ) : u R k}

19 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 } { N(u, I 1 θ 0 ) : u R k} Strong convergence: there exist randomizations (Markov kernels) T n, S n such that ( T n P θ 0+u/ ) n n N(u, I 1 = 0 tv lim n sup u <a θ 0 ) and lim sup n u <a ( P θ 0+u/ ) n n Sn N(u, I 1 θ 0 ) = 0 tv

20 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 } { N(u, I 1 θ 0 ) : u R k} Strong convergence: there exist randomizations (Markov kernels) T n, S n such that ( T n P θ 0+u/ ) n n N(u, I 1 = 0 tv lim n sup u <a θ 0 ) and lim sup n u <a ( P θ 0+u/ ) n n Sn N(u, I 1 θ 0 ) = 0 tv 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

21 L. A. N. for finite dimensional quantum systems Let ( ρ θ0 +u/ n n) be the joint state of n i.i.d. systems with ρθ M(C d ) smooth. Then { ) { } (ρθ0+u/ n n : u R d 1} 2 Φ(u, H 1 θ 0 ) : u R d2 1

22 L. A. N. for finite dimensional quantum systems Let ( ρ θ0 +u/ n n) be the joint state of n i.i.d. systems with ρθ M(C d ) smooth. Then { ) { } (ρθ0+u/ n n : u R d 1} 2 Φ(u, H 1 θ 0 ) : u R d2 1 Strong convergence: there exist quantum channels T n, S n such that ( ) T n ρθ0 +u/ n n Φ(u, H 1 = 0 1 lim n sup u <a θ 0 ) and lim n sup u <a ( ) ρ θ0 +u/ n n Sn Φ(u, H 1 θ 0 ) = 0 1

23 L. A. N. for finite dimensional quantum systems Let ( ρ θ0 +u/ n n) be the joint state of n i.i.d. systems with ρθ M(C d ) smooth. Then { ) { } (ρθ0+u/ n n : u R d 1} 2 Φ(u, H 1 θ 0 ) : u R d2 1 Strong convergence: there exist quantum channels T n, S n such that ( ) T n ρθ0 +u/ n n Φ(u, H 1 = 0 1 lim n sup u <a θ 0 ) and lim n sup u <a ( ) ρ θ0 +u/ n n Sn Φ(u, H 1 θ 0 ) = 0 1 Importance: Shows that for large n the statistical model is locally easy : Gaussian shift model Provides a two step adaptive measurement strategy which is asymptotically optimal

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

25 L. A. N. for qubit states (d=2) An arbitrary qubit (spin) state: ρ r := 1 ( ) 1 + rz r x ir y = 1 2 r x + ir y 1 r z 2 (1 + r xσ x + r y σ y + r z σ z ), r 1 z Non-commuting spin components: σ x σ y σ y σ x = 2iσ z Probability distributions : P([σ a = +1]) = (1 + r a )/2 P([σ a = 1]) = (1 r a )/2 y r x ( )

26 L. A. N. for qubit states (d=2) An arbitrary qubit (spin) state: ρ r := 1 ( ) 1 + rz r x ir y = 1 2 r x + ir y 1 r z 2 (1 + r xσ x + r y σ y + r z σ z ), r 1 z Non-commuting spin components: σ x σ y σ y σ x = 2iσ z Probability distributions : P([σ a = +1]) = (1 + r a )/2 P([σ a = 1]) = (1 r a )/2 y r x A local neighborhood of ρ 0 := ( µ µ ) is parametrised by u = (u x, u y, u z ) R 3 z ( ) ( ρ u µ + u z u/ n := U n 0 n 0 1 µ u z n ) ( ) u U, n x where U(u) SU(2) is the unitary U(u) := exp(i(u x σ x + u y σ y )) y

27 LAN for qubits: the big ball picture Quantum coin toss : ρ 0 = µ + (1 µ) = P([σ x = ±1]) = P([σ y = ±1]) = 1/2, P([σ z = 1]) = µ n identically prepared systems: ρ 0 ρ 0 Central Limit Theorem... Collective spin L x,y,z := n i=1 σ(i) x,y,z 1 n L x D N(0, 1), z 1 n L y D N(0, 1), n (2µ 1)n...with a quantum twist [ 1 n L x, 1 n L y ] = 2i 1 n L z 2i(2µ 1)1 x y

28 Quantum particle (harmonic oscillator) 1 L x Q 2n(2µ 1) 1 L y P 2n(2µ 1) Gaussian states = [Q, P ] = i1 Heisenberg commutation relation z n Thermal equilibrium state: < Q 2 >=< P 2 >= 1 2(2µ 1) (2µ 1)n φ 0 := (1 p) k=0 p k k k, p = 1 µ µ < 1 x Quantum Gaussian shift: spin rotations become displacements y ( φ u := D(u) φ 0 D(u), D(u) := exp i ) 2(2µ 1)(u x Q + u y P ) Classical Gaussian shift: diagonal parameter behaves like in coin toss N u := N(u z, µ(1 µ))

29 Local Asymptotic Normality for qubit states Classical Gaussian shift gives info about the eigenvalues of ρ N u := N(u z, µ(1 µ)) Quantum Gaussian shift gives info about the eigenvectors of ρ ( Q N ) 2(2µ 1)u y, 1/2(2µ 1) φ u := D(u) φ 0 D(u), ( 2(2µ ) P N 1)ux, 1/2(2µ 1) Theorem: Let ρ (n) u := ( ) ρ n. u/ n Then there exist q. channels (randomizations) T n, S n such that for any η < 1/4 lim sup n u <n η T n ( ρ (n) u ) N u φ u 1 = 0, and lim sup n u <n η ρ (n) u S n (N u φ u ) = 0. 1

30 Localisation + Local Asymptotic Normality = optimal estimation localise the state within the small local neighborhood of ρ 0 while waisting ñ n qubits use local asymptotic normality on the bigger ball to design the second stage measurement N u φ u n 1/2+η Observe N u Do heterodyne on φ u n 1/2+ɛ

31 Implementation with continuous time measurements Couple the qubits with a Bosonic field, let the state leak into the field and do heterodyne (quantum part) followed by a L z measurement (classical part) P vacuum n Qubits vacuum Heterodyne detector Q Unitary evolution on ( C 2) n F(L 2 (R)) given by the QSDE: du n (t) = (a n da t a nda t 1 2 a na n dt)u n (t), a n := 1 n(2µ 1) n i=1 σ (i) +

32 Idea of the proof: typical SU(2) representations ρ n u/ n N u φ u Two commuting group actions: SU(2) rotations and permutations ( C 2 ) n = ρ n = n/2 j=0,1/2 n/2 j=0,1/2 H j K j p n (j)ρ n,j 1 n j Classical part: measuring j ( which block ) does not disturb the state Distribution p u n(j) converges to N u as in coin toss (L = L z Bin(µ + u z / n, n)) τ n (j) := (j (µ 1/2)n)/ n N u = N(u z, µ(1 µ))

33 Idea of the proof: typical SU(2) representations Quantum part: conditional on j we remain with a typical block state ρ u n,j Structure of π j : Spin operators L ± = L x ± il y act as ladder operators on basis vectors H j = Lin{ m, j : m = j,..., j} Creation and annihilation operators a, a act similarly on the number basis { k : k 0} Embed irrep H j into the harmonic oscillator l 2 (N) by isometry V j : m, j j m By Quantum Central Limit Theorem, collective observables J ± become Gaussian and m=-j V j ρ u j,nv j φ u n 2j + 1 J + V j a m=j-1 m=j J 1 0 a

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

35 Local asymptotic normality for d-dimensional states Local neighbourhood around ρ 0 := Diag(µ 1,..., µ d ) with µ 1 > µ 2 > > µ d > 0 ρ θ = µ 1 + u 1 ζ1,2... ζ 1,d. ζ 1,2 µ 2 + u ζd 1,d ζ 1,d... ζ d 1,d µ d, d 1 i=1 u i θ = ( u, ζ ) R d 1 C d(d 1)/2 In first order, ρ θ/ n := U µ 1 + u 1 / n ( ) ζ 0 µ 2 + u 2 / n.... n µ d d 1 i=1 u i/ U n ( ζ n ), where U( ζ) := exp 1 j<k d ζj,k E k,j ζ j,k E j,k SU(d) µ j µ k

36 L. A. N. Theorem Diagonal parameters give rise to a classical (d 1)-dimensional Gaussian N u := N( u, V µ ) Off-diagonal parameters decouple from the diagonal ones and from each other Φ ξ := j<k φ ξ j,k where φ ξ j,k is a displaced thermal equilibrium state with β = ln µ j /µ k Total classical-quantum limit model: Φ θ := N u Φ ξ

37 L. A. N. Theorem Diagonal parameters give rise to a classical (d 1)-dimensional Gaussian N u := N( u, V µ ) Off-diagonal parameters decouple from the diagonal ones and from each other Φ ξ := j<k φ ξ j,k where φ ξ j,k is a displaced thermal equilibrium state with β = ln µ j /µ k Total classical-quantum limit model: Φ θ := N u Φ ξ Theorem. Let ρ (n) θ := ρ n θ/ n. Then there exist q. channels T n, S n such that lim n lim n sup θ Θ n,β,γ Φ θ T n (ρ θ,n ) 1 = 0, sup θ Θ n,β,γ Sn (Φ θ ) ρ θ,n 1 = 0, where Θ n,β,γ := {θ = ( ξ, u) : ξ n β, u n γ }, β < 1/9, γ < 1/4.

38 Local asymptotic normality for d-dimensional states Two commuting group actions: SU(d) rotations and permutations ( C d ) n = λ H λ K λ ρ n = λ p n (λ)ρ n,λ 1 n λ SU(d) irreps λ are labelled by Young diagrams with d rows and n boxes λ 1 nµ 1 Classical part: measure which Young diagram λ λ d nµ d Distribution p n,θ (λ) converges to multivariate Gaussian shift same as the multinomial model Mult {(λ i nµ i )/ n : i = 1,... d 1} N( u, I 1 ( µ 1 + u 1 n,... µ d i ) n u i ; n! µ ) Quantum part: conditional on λ we remain with a typical block state ρ θ λ,n

39 Incursion into SU(d) irreps Structure of H λ : Write tensors into λ-tableaux e a := e a(1) e a(n) t a, e.g. e 2 e 1 e Young symmetriser Y λ is minimal projection in Alg(S(n)) Y λ = Q λ P λ := sgn(τ)τ τ C(λ) σ R(λ) σ Non-orthogonal basis of H λ indexed by semi-standard Young tableaux, e.g f a := Y λ e a Number basis : t a m = {m i,j = j s in row i : i < j} m, λ := f a / f a Lemma: Not far from vacuum m = 0, basis is almost ON If l, m = O(n η ), for some 0 < η < 2/9 then m, λ l, λ = O(n c(η) ), c(η) > 0

40 Incursion into SU(d) irreps Structure of π λ : Ladder operators L i,j = π λ (E i,j ), L i,j = π λ(e j,i ) for 1 i < j d don t act as ladder... L 2,3 : However they do so on typical vectors m = O(n η ) n L 2,3 : O(n η ) O(n) After normalisation first term drops and we get an a i,j creation operator L 2,3/ n : {m 1,2, m 1,3, m 2,3 }, λ = m 2,3 + 1 {m 1,2, m 1,3, m 2,3 +1}, λ!! Asymptotically, L i,j / n acts only on row i and they all commute with each other... We have convergence to a tensor product of harmonic oscillators (a i,j, a i,j ) in the vacuum

41 Further work: Local asymptotic normality for i.i.d. infinite dimensional quantum states general parametric families of states of light optimal estimation rate for states of light Local asymptotic normality for quantum Markov chains (processes) optimal rates for interaction parameters in realistic dynamical models (next talk) Central Limit Theorem for quantum Markov chains Testing with non-discrete hypotheses eg ρ = ρ 0 vs ρ ρ 0 Weak and strong convergence of quantum statistical experiments Q n = {ρ θ n : θ Θ} Q = {ρ θ : θ Θ} Quantum statistical decision theory quantum experiment Q = {ρ θ M(C d ) : θ Θ} non-commuting loss functions 0 W θ M(C k ) decision C : ρ θ C(ρ θ ) M(C k ) with risk R(C, θ) = Tr(C(ρ θ )W θ ) applications in quantum memory, quantum cloning

42 References! M. Guta, J. Kahn Local asymptotic normality for qubit states P.R.A 73, 05218, (2006)! M. Guta, A. Jencova Local asymptotic normality in quantum statistics Commun. Math. Phys. 276, , (2007)! M. Guta, B Janssens and J. Kahn Optimal estimation of qubit states with continuous time measurements Commun. Math. Phys. 277, , (2008)! M. Guta, J. Kahn: Local asymptotic normality for finite dimensional quantum systems arxiv: (Commun. Math. Phys, to appear)