Local Asymptotic Normality in Quantum Statisticsmasahito/workshop/presentation/Guta.pdf · Local Asymptotic Normality f or coin toss Rep eated coin toss: X 1, . . .. , X n i.i.d

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

!! ! !! ! · · ·! !! "#$ X(n) "#$ "n

Problem: given n identically prepared systems in the state !! with " ! !, performa measurement M (n) and construct an estimator "n of " from the result X(n).


Mesures individuelles adaptatives

!" !!!!!!!!!! M1! X1

!" !!!!!!!!!!!!!!M2! X2

!" !!!!!!!!!!!!!!!!!!Mn! Xn

! "n

Mesure optimale depend de "

!! ! !! ! · · ·! !! "#$ X(n) "#$ "n



!! ! !! ! · · ·! !! "#$ X(n) "#$ "n

!! !!!!!!!!!!

!! !!!!!!!!!!

!! !!!!!!!!!!

M(n) ! X(n) ! "n


Risk and Optimality: The quality of the estimation strategy (M, !n) is given by the risk

R(!, !n) = E!"! " "!n!2

1 or R(!, !n) = 1 " E F ("!, "!n)


!! ! !! ! · · ·! !! "#$ X(n) "#$ "n

!! !!!!!!!!!!

!! !!!!!!!!!!

!! !!!!!!!!!!

M(n) ! X(n) ! "n


Bayesian: prior !(d")

R!("n) :=!

R(", "n)!(d")

R!,n := infMn

R!("n)

R! := limn!"

nR!,n

Frequentist

R!0(!n) := sup!!B(!0,n!1/2)

R(!, !n)

R!0,n := infMn

R!0(!n)

R!0 := limn"#

nR!0,n = CH(!0)

Bayesian vs frequentist optimality

R! =!

R" !(d")

Joint measurements

Non!parametric

Practically feasible

Optimal for pure states

Optimal for one parameter

More difficult to implement

Optimal for mixed states

Q. Homodyne Tomography,

Direct detection of Wigner fct...

non-parametric rates for

estimation of state as a whole

Conjecture/Program:

L.A.N. = convergence to model

of displaced quasifree states of

infinite dimensional CCR alg.

Separate measurements

Parametric

Rn ! Csep/n Rn ! Cjoint/n

A rough classification of state estimation problems

Rn = O( (log n)k/n, n!!, ...)

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 !!

• the latter has fixed, known variance and unknown mean (related to) !,

• the mean can be estimated optimally by simple measurements (heterodyne)

• the measurement can be ‘pulled back’ to the n systems

!! !!!!!!!!!! H ! Y ! P(H,!!) ! !

!! !!!!!!!!!!

!! !!!!!!!!!!

!! !!!!!!!!!!

Mn ! Xn ! P(Mn, !!) ! "n

n ! "

Motivation / earlier work

• Classical L.A.N. theory of Le Cam

asymptotic equivalence of statistical modelsoptimal estimation rates

• Central Limit behaviour for quantum systems

Coherent spin statesGaussian 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 systemarXiv:quant-ph/0411073

Local Asymptotic Normality for coin toss

Repeated coin toss: X1, . . . ,Xn i.i.d. with P[Xi = 1] = !, P[Xi = 0] = 1! !

Su!cient statistic: !n := 1n

!ni=1 Xi unbiased estimator since E(X) = !

Central Limit Theorem:"

n(!n ! !) D!# N(0, !(1! !))

Local parameter: let ! = !0 + u/"

n for a fixed known !0, then

un :="

n(!n ! !0) $ N(u, !0(1! !)0)

Why can we restrict to a local neighbourhood ?

You can construct a !0 from the data and the true ! will be in a

‘1/"

n-neighbourhood’ with high probability


0.2

k

0.15

80

0.1

0.05

600

40200

Binomial n=100 p=0.6

Normal m=60 v=25


0.2

k

0.15

80

0.1

0.05

600

40200


Normal m=50 v=25


0.2

k

0.15

80

0.1

0.05

600

40200


Normal m=40 v=25


0.2

k

0.15

80

0.1

0.05

600

40200


Normal m=30 v=25






n(!n ! !) D!# N(0, !(1! !))



un :="

n(!n ! !0) $ N(u, !0(1! !)0)



‘1/"


Gaussian shiftmodel






n(!n ! !) D!# N(0, !(1! !))



un :="

n(!n ! !0) $ N(u, !0(1! !)0)



‘1/"


Gaussian shiftmodel

Local Asymptotic Normality: general case

Let (Y1, . . . , Yn) be i.i.d. with P!0+u/!

n a ‘smooth’ family with u ! Rk. Then!"

P!0+u/!

n#n

: u ! Rk$

!%N(u, I"1

!0) : u ! Rk

&


Strong convergence: there exist randomizations (Markov kernels) Tn, Sn such that

limn!"

sup#u#<a

!!!Tn

"P!0+u/

$n#n!N(u, I%1

!0)!!!

tv= 0

and

limn!"

sup#u#<a

!!!"P!0+u/

$n#n! SnN(u, I%1

!0)!!!

tv= 0



P!0+u/!

n#n

: u ! Rk$

!%N(u, I"1

!0) : u ! Rk

&


Strong convergence: there exist randomizations (Markov kernels) Tn, Sn such that

limn!"

sup#u#<a

!!!Tn

"P!0+u/

$n#n!N(u, I%1

!0)!!!

tv= 0

and

limn!"

sup#u#<a

!!!"P!0+u/

$n#n! SnN(u, I%1

!0)!!!

tv= 0

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



P!0+u/!

n#n

: u ! Rk$

!%N(u, I"1

!0) : u ! Rk

&

L. A. N. for finite dimensional quantum systems

!"!!0+u/

!n

#"n : u ! Rd2#1$

!!

!(u, H#1!0

) : u ! Rd2#1$

Let!!!0+u/

!n

""n be the joint state of n i.i.d. systems with !! ! M(Cd) ‘smooth’. Then


!"!!0+u/

!n

#"n : u ! Rd2#1$

!!

!(u, H#1!0

) : u ! Rd2#1$

Let!!!0+u/

!n


Strong convergence: there exist quantum channels Tn, Sn such that

limn!"

sup#u#<a

!!!Tn

"!!0+u/

$n

#%n ! !(u, H&1!0

)!!!

1= 0

and

limn!"

sup#u#<a

!!!"!!0+u/

$n

#%n ! Sn!(u, H&1!0

)!!!

1= 0


!"!!0+u/

!n

#"n : u ! Rd2#1$

!!

!(u, H#1!0

) : u ! Rd2#1$

Let!!!0+u/

!n


Strong convergence: there exist quantum channels Tn, Sn such that

limn!"

sup#u#<a

!!!Tn

"!!0+u/

$n

#%n ! !(u, H&1!0

)!!!

1= 0

and

limn!"

sup#u#<a

!!!"!!0+u/

$n

#%n ! Sn!(u, H&1!0

)!!!

1= 0

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

Outline:





z

y

x

!r

L. A. N. for qubit states (d=2)

Probability distributions :

!"

#

P([!a = +1]) = (1 + ra)/2

P([!a = !1]) = (1! ra)/2

An arbitrary qubit (spin) state:

!!r :=12

!1 + rz rx ! iry

rx + iry 1! rz

"=

12

(1 + rx"x + ry"y + rz"z) , "#r" # 1

Non-commuting spin components: "x"y ! "y"x = 2i"z

A local neighborhood of !0 :=#

µ 00 1! µ

$is parametrised by u = (ux, uy, uz) $ R3

!u/"

n := U

!u%n

" %µ + uz"

n0

0 1! µ! uz"n

&U

!u%n

"#,

where U(u) $ SU(2) is the unitary U(u) := exp(i(ux"x + uy"y))

z

y

x

!r

z

y

x

L. A. N. for qubit states (d=2)

Probability distributions :

!"

#

P([!a = +1]) = (1 + ra)/2

P([!a = !1]) = (1! ra)/2

An arbitrary qubit (spin) state:

!!r :=12

!1 + rz rx ! iry

rx + iry 1! rz

"=

12

(1 + rx"x + ry"y + rz"z) , "#r" # 1

Non-commuting spin components: "x"y ! "y"x = 2i"z

A local neighborhood of !0 :=#

µ 00 1! µ

$is parametrised by u = (ux, uy, uz) $ R3

!u/"

n := U

!u%n

" %µ + uz"

n0

0 1! µ! uz"n

&U

!u%n

"#,

where U(u) $ SU(2) is the unitary U(u) := exp(i(ux"x + uy"y))

•

z

x

!

n

y

(2µ ! 1)n

‘Quantum coin toss’: !0 = µ |!" #!| + (1$ µ) |%" #%| =&

P(["x = ±1]) = P(["y = ±1]) = 1/2, P(["z = 1]) = µ

n identically prepared systems: !0 ' · · ·' !0

Central Limit Theorem... Collective spin Lx,y,z :=!n

i=1 "(i)x,y,z

1(n

LxD$) N(0, 1),

1(n

LyD$) N(0, 1),

...with a quantum twist"

1(n

Lx ,1(n

Ly

#= 2i

1n

Lz * 2i(2µ$ 1)1

LAN for qubits: the big ball picture

Gaussian states

•

z

x

!

n

y

(2µ ! 1)n

Quantum particle (harmonic oscillator)

1!2n(2µ!1)

Lx "# Q1!

2n(2µ!1)Ly "# P

!"

# =$ [Q,P ] = i1 Heisenberg commutation relation

Thermal equilibrium state: < Q2 >=< P 2 >= 12(2µ!1)

!0 := (1" p)"$

k=0

pk |k% &k| , p =1" µ

µ< 1

Quantum Gaussian shift: spin rotations become displacements

!u := D(u) !0 D(u)#, D(u) := exp%i&

2(2µ" 1)(uxQ + uyP )'

Classical Gaussian shift: diagonal parameter behaves like in coin toss

Nu := N(uz, µ(1" µ))

Local Asymptotic Normality for qubit states

Theorem: Let !(n)u :=

!!u/

!n

""n. Then there exist q. channels (randomiza-tions) Tn, Sn such that for any " < 1/4

limn#$

sup%u%<n!

###Tn

$!(n)u

%!Nu " #u

###1

= 0,

andlim

n#$sup

%u%<n!

###!(n)u ! Sn (Nu " #u)

###1

= 0.

Classical Gaussian shift gives info about the eigenvalues of !

Nu := N(uz, µ(1! µ))

Quantum Gaussian shift gives info about the eigenvectors of !

"u := D(u) "0 D(u)!,

!"#

"$

Q " N%!

&2(2µ! 1)uy, 1/2(2µ! 1)

'

P " N%&

2(2µ! 1)ux, 1/2(2µ! 1)'

n!1/2+!

n!1/2+!

Two stage adaptive measurement

• localise the state within the small local neighborhood of !0 while waisting n ! n qubits

• use local asymptotic normality on the bigger ball to design the second stage measurement

Nu" "u

“Observe” Nu Do heterodyne on !u

Localisation + Local Asymptotic Normality= optimal estimation

Implementation with continuous time measurements

Practical implementation?

Heterodynedetector

vacuum

P

Qn Qubits

vacuum

After interaction, measure the total spin of the qubits

Couple the qubits with a Bosonic field, let the state leak into the field and doheterodyne (quantum part) followed by a Lz measurement (classical part)

Unitary evolution on!

C2"!n

! F(L2(R)) given by the QSDE:

dUn(t) = (andA"t " a"

ndAt "1

2a"

nandt)Un(t), an :=1

#

n(2µ " 1)

n$

i=1

!(i)+

Idea of the proof: typical SU(2) representations

!!nu/

"n

! Nu! "u

”“

Two commuting group actions: SU(2) rotations and permutations

!C2

"!n =n/2#

j=0,1/2

Hj !Kj

!!n =n/2#

j=0,1/2

pn(j)!n,j !1nj

Classical part: measuring j (‘which block’) does not disturb the state

Distribution pun(j) converges to Nu as in coin toss (L != Lz ! Bin(µ + uz/

"n, n))

!n(j) := (j # (µ# 1/2)n)/"

n ! Nu = N(uz, µ(1# µ))

m=j

m=j-1

m=-j

J+

J!

a

a!

Vj

|0!

|1!

|2j + 1!

Idea of the proof: typical SU(2) representations

Quantum part: conditional on j we remain with a typical block state !un,j

Structure of "j :

Spin operators L± = Lx ± iLy act as ladder operators on basis vectors

Hj = Lin{|m, j! : m = "j, . . . , j}

Creation and annihilation operators a!, a act similarly on the number basis {|k! : k # 0}

Embed irrep Hj into the harmonic oscillator #2(N) by isometry

Vj : |m, j! $% |j "m!

By Quantum Central Limit Theorem, collective observables J± become Gaussian and

Vj!uj,nV !j "%

n"#$u

Outline:





Local asymptotic normality for d-dimensional states

Local neighbourhood around !0 := Diag(µ1, . . . , µd) with µ1 > µ2 > · · · > µd > 0

!! =

!

""""#

µ1 + u1 "!1,2 . . . "!1,d

"1,2 µ2 + u2. . .

......

. . . . . . "!d"1,d

"1,d . . . "d"1,d µd !$d"1

i=1 ui

%

&&&&', # = (#u,

#" ) " Rd"1 # Cd(d"1)/2

In first order,

!!/$

n := U

($"$n

)!

""""#

µ1 + u1/$

n 0 . . . 0

0 µ2 + u2/$

n. . .

......

. . . . . . 00 . . . 0 µd !

$d"1i=1 ui/

$n

%

&&&&'U!

($"$n

),

where

U($") := exp

!

#*

1%j<k%d

"!j,kEk,j ! "j,kEj,k

µj ! µk

%

' " SU(d)

L. A. N. Theorem

• Diagonal parameters give rise to a classical (d! 1)-dimensional Gaussian

N!u := N(!u, Vµ)

• O!-diagonal parameters decouple from the diagonal ones and from each other

"!" :=!

j<k

""j,k

where ""j,k is a displaced thermal equilibrium state with # = lnµj/µk

• Total classical-quantum limit model: "# := N!u " "!"

Theorem. Let !(n)! := !!n

!/"

n. Then there exist q. channels Tn, Sn such that

limn#$

sup!%!n,!,"

!!!! ! Tn(!!,n)!!

1= 0,

limn#$

sup!%!n,!,"

!!Sn(!!)! !!,n!!

1= 0,

where "n,",# := {" = (#$, #u) : "#$" # n" , "#u" # n#}, % < 1/9, & < 1/4.

L. A. N. Theorem

• Diagonal parameters give rise to a classical (d! 1)-dimensional Gaussian

N!u := N(!u, Vµ)

• O!-diagonal parameters decouple from the diagonal ones and from each other

"!" :=!

j<k

""j,k

where ""j,k is a displaced thermal equilibrium state with # = lnµj/µk

• Total classical-quantum limit model: "# := N!u " "!"

Local asymptotic normality for d-dimensional states

!1 ! nµ1

!d ! nµd

SU(d) irreps ! are labelled by Young diagrams with d rows and n boxes

Quantum part: conditional on ! we remain with a typical block state "!",n

Classical part: measure ‘which Young diagram !’

Distribution pn,!(!) converges to multivariate Gaussian shift

{(!i ! nµi)/"

n : i = 1, . . . d! 1} ! N("u, I!1µ )

same as the multinomial model Mult!µ1 + u1"

n, . . . µd !

"i

ui"n;n

#!

Two commuting group actions: SU(d) rotations and permutations

!Cd

"!n=

#

!

H! !K!

!!n =#

!

pn(")!n,! !1n!

Incursion into SU(d) irreps

Structure of H!:

• Write tensors into !-tableaux

ea := ea(1) ! · · ·! ea(n) "#$ ta, e.g. e2 ! e1 ! e1 "$ 2 11

• Young symmetriser Y! is minimal projection in Alg(S(n))

Y! = Q! · P! :=!

"!C(!)

sgn(")" ·!

#!R(!)

#

• Non-orthogonal basis of H! indexed by semi-standard Young tableaux, e.g.1 2 22 33

fa := Y!ea

• ‘Number basis’: ta %$ m = {mi,j = $j"s in row i : i < j}

|m, !& := fa/'fa'

Lemma: Not far from ’vacuum’ m = 0, basis is almost ONIf |l, |m| = O(n$), for some 0 < % < 2/9 then

|(m, !|l, !&| = O(n#c($)), c(%) > 0

Incursion into SU(d) irreps

Structure of !!:

• ‘Ladder operators’ Li,j = !!(Ei,j), L!i,j = !!(Ej,i) for 1 ! i < j ! d don’t act as ladder...

L!2,3 :1 1 22 23

"#1 1 32 23

+ 21 1 22 33

• However they do so on ‘typical vectors’ |m| = O(n")$ n

L!2,3 :1 1 1 1 1 1 1 2 2 32 2 2 2 3 33 3 3

"# O(n")1 1 1 1 1 1 1 2 3 32 2 2 2 3 33 3 3

+ O(n)1 1 1 1 1 1 1 2 2 32 2 2 3 3 33 3 3

After normalisation first term drops and we get an a!i,j creation operator

L!2,3/%

n : |{m1,2, m1,3, m2,3}, "&"='"#

!m2,3 + 1 |{m1,2, m1,3, m2,3+1}, "& !!

• Asymptotically, Li,j/%

n acts only on row i and they all commute with each other...

We have convergence to a tensor product of harmonic oscillators (ai,j , a!i,j) in the vacuum

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 ! = !0 vs ! != !0

Weak and strong convergence of quantum statistical experiments

Qn = {!!n : " " !} ! Q = {!! : " " !}

Quantum statistical decision theory• quantum experiment Q = {!! "M(Cd) : " " !}

• non-commuting “loss functions” 0 #W! "M(Ck)

• “decision” C : !! $% C(!!) "M(Ck) with risk R(C, ") = Tr(C(!!)W!)

• applications in quantum memory, quantum cloning

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, 341-379, (2007)

! M. Guta, B Janssens and J. Kahn Optimal estimation of qubit states with continuous time measurements Commun. Math. Phys. 277, 127-160, (2008)

! M. Guta, J. Kahn: Local asymptotic normality for finite dimensional quantum systems arXiv:0804.3876 (Commun. Math. Phys, to appear)

Documents

Local Asymptotic Normality in Quantum Statisticsmasahito/workshop/presentation/Guta.pdf · Local Asymptotic Normality f or coin toss Rep eated coin toss: X 1, . . .. , X n i.i.d