Asymptotic equivalence of pure quantum state estimation and Gaussian - - PowerPoint PPT Presentation

Asymptotic equivalence of pure quantum state estimation and Gaussian white noise

• M. Nussbaum

Cornell University joint work with C. Butucea and M. Guta

High-dimensional Problems and Quantum Physics

Marne-la-Vallée, June 2015

( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 1 / 16

Information contained in additional observations

Le Cam, 1974: consider product experiments: En =

• Pn

ϑ , ϑ 2 Θ

• ,

Θ Rk, regularity (LAN)

I compare En and En+m with regard to the information they contain about ϑ (m

additional observations)

I shown: if ∆ (, ) is the de…ciency distance between experiments then

∆ (En, En+m) = O rm n

• .

I Recall Le Cam’s de…ciency distance: I De…nition. (Le Cam ∆-distance). Suppose that P = fPϑ, ϑ 2 Θg and

Q = fQϑ, ϑ 2 Θg are two experiments indexed by the same parameter set Θ, but with possibly di¤erent sample space. The de…ciency of P with respect to Q is δ(P, Q) = inf

K sup ϑ2Θ

kQϑ KPϑkTV (inf over all Markov kernels)

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Characterization in terms risk functions:

I δ(P, Q) ε i¤: for every decision problem with loss function L(d, ϑ) 1 we

have:

I for every decision rule in Q there is a decision rule in P such that risk is

increased by ε (at most), uniformly over ϑ 2 Θ.

I the symmetric version (i.e. ∆-distance) is

∆(P, Q) = max(δ(P, Q), δ(Q, P)

I Sequences Pn, Qn are asymptotically equivalent if ∆(Pn, Qn) ! 0.

Quantum analog ∆q of de…ciency distance: Guta, 06

I experiments: families of quantum states P = fρϑ, ϑ 2 Θg; space Θ arbitrary I replace total variation distance kkTV by trace norm distance

kρ σk = Tr jρ σj

I replace Markov kernel by quantum analog (tp-cp maps) I for two quantum experiments P and Q, on di¤erent spaces but same Θ,

∆q (P, Q) is de…ned

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Equivalent and asymptotically equivalent experiments

Experiments P, Q are called equivalent if ∆ (P, Q) = 0. We then write P $ Q.

I if P, Q live on the same sample space (Ω, F) then a one-to-one data

transformation generates equivalence.

I if they live on di¤erent sample spaces (Ω, F) and (Ω0, F0) then a su¢cient

statistic T : (Ω, F) ! (Ω0, F0) generates equivalence (if Qϑ = L (T jPϑ) , ϑ 2 Θ).

I primary example: Gaussian shift Xi N (ϑ, 1), i = 1, . . . , n; equivalent

experiment: sample mean ¯ Xn N

• ϑ, n1

, ϑ 2 R

I P is called more informative than Q if δ (P, Q) = 0 (one sided de…ciency of

P wrt Q). We write P Q.

I P Q typically happens if Qϑ = L (T jPϑ) and the map T is not su¢cient

(loses information)

Asymptotic versions: sequences Pn, Qn are called asy. equivalent if ∆ (Pn, Qn) ! 0. We write Pn ' Qn.

I Pn is called asy. more informative than Qn if δ (Pn, Qn) ! 0. We write

P % Q

Quantum versions: straighforward by using q-Markov kernels. Connection to q-su¢ciency has been worked out (Jenµ cova, Petz, 07).

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Additional observations in nonparametric case (classical)

starting point: white noise experiment dYt = f 1/2(t)dt + n1/2dWt, t 2 [0, 1], f 2 F (α, M)

I background: N (96), f probability density for n i.i.d. observations, F (α, M)

function class of smoothness α

I n replaces observation number of i.i.d. case; I below we will replace f 1/2 by f (for shortness) I model is more ‡exible than i.i.d. model; allows easy formulation of equivalent

(exactly) experiments: ∆ (, ) = 0

I multiply observations by n1/2: equivalent experiment (up to f 1/2 6= f )

dYt = n1/2f (t) dt + dWt, t 2 [0, 1], f 2 . . ..

I Call this experiment Gn, compare with Gn+m (m "additional observations") I …rst consider local versions: for a …xed function f0 de…ne neighborhood

Σn (f0) = ff : kf f0k γng where kk is L2-norm

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I subtract the "known" function f0 from the data, multiply by n1/2: equivalent

local experiment dYt = n1/2 (f (t) f0(t)) dt + dWt, t 2 [0, 1], f 2 Σn (f0) \ F (α, M)

I do the same in Gn+m; compare the above with

dYt = (n + m)1/2 (f (t) f0(t)) dt + dWt, t 2 [0, 1], f 2 . . .

I squared Hellinger distance between the two measures (drift functions g1, g2,

say ) bounded by kg1 g2k2 =

• (n + m)1/2 n1/22

kf f0k2 =

• (1 + m/n)1/2 1

2 n kf f0k2

I a Taylor expansion, assuming m/n = o(1) gives

C m n 2 n kf f0k2 C m2 n γ2

n.

I smoothness assumption f0 2 F (α, M) means that f0 can be estimated with

rate γn = nα/(2α+1) (in terms of kk) So assuming γn of that rate will allow us to "get into the neighborhoods" Σn (f0) later.

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I we need m2n1γ2

n = o (1) for Hellinger distance to be o (1). That gives a

condition m n1/2γ1

n

= n1/2nα/(2α+1) = n1s(α), s(α) = 1 4α + 2 for asymptotic equivalence of the two local experiments, with f 2 Σn (f0), dYt = n1/2f (t)dt + dWt, dYt = (n + m)1/2 f (t)dt + dWt

I extreme cases: a) α = ∞ (…nite dimension): s(α) = 0, hence m n su¢ces

(Le Cam)

I b) α > 0 , condition is m n1/2+ε

phenomenon: the larger the parameter space, the smaller the allowed m for

• asy. equivalence (= the higher the informational content of additional
• bservations)

I (holds for local equivalence, but for global too: see below) ( ) Quantum Asymptotic Equivalence Marne-la-Vallée, June 2015 7 / 16

Globalization: get rid of f 2 Σn (f0), keep only smoothness f 2 F (α, M).

I idea: sample splitting; start with global experiment Gn with observations

dYt = f (t)dt + n1/2dWt, f 2 F (α, M)

I this is equivalent (by su¢ciency argument) to indep. observations

• Y (1), Y (2)

where dY (i)

t

= f (t)dt + (n/2)1/2 dW (i)

t

, i = 1, 2.

I use local equivalence under f 2 Σn (f0) of Y (2) and an estimator ˆ

f0 of f0 based on Y (1), to establish global equivalence to

• Y (1), ˜

Y (2) where d ˜ Y (2)

t

= f (t)dt + ((n + m) /2)1/2 dW (2)

t

.

I Repeat, with roles of Y (1) and ˜

Y (2) reversed, to obtain global equivalence to

• ˜

Y (1), ˜ Y (2) where d ˜ Y (1)

t

= f (t)dt + ((n + m) /2)1/2 dW (1)

t

.

I take average of ˜

Y (1), ˜ Y (2) (su¢ciency argument) to obtain d ˜ Yt = f (t)dt + (n + m)1/2 dWt, f 2 F (α, M) .

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Result: the Gaussian experiments Gn and Gn+m are asy. equivalent, globally under f 2 F (α, M) if m n1s(α), s(α) = 1 (4α + 2) (informational content of "additional observations"). similar method used before for i.i.d. experiment (N 96):

I recall experiment En: i.i.d. observations, root density f 2 F (α, M) (=density

f 2, on unit interval etc.)

I we have ∆ (En, Gn) ! 0 if Gn is the Gaussian white noise experiment with drift

function f

I method applied for proof: …rst localization around f0, rate γn n1/4; I consider local experiments En(f0), Gn(f0), approximation

∆ (En (f0) , Gn (f0)) ! 0

I globalization by sample splitting: En $ En/2 En/2, (symbol $ :exact

equivalence)

I then use estimate of f0 based on En/2 to replace 2nd En/2 by Gn/2 I obtain En/2 Gn/2 then (reverse) obtain Gn/2 Gn/2 $ Gn I fact ∆ (En, Gn) ! 0 implies ∆ (En, En+m) ! 0 (carry over results about

additional observations from Gaussian to i.i.d. case)

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Quantum experiments

pure quantum state: unit vector φ 2 H (complex) Hilbert space

I random variables are created by observables A: self-adjoint operators. I A discrete spectrum: A = ∑k

j=1 aj Πj spectral decomposition (all real

eigenvalues di¤erent, possibly 0, ∑k

j=1 Πj = I). Then

• Πj

k

j=1 is a

measurement, and XA is de…ned by P

• XA = aj

= pj = φΠj φ.

I expectations: EXA = ∑k

j=1 aj φΠj φ = φAφ = trAφφ.

I (A continuous spectrum: XA continuous r.v., e.g. Gaussian)

quantum analog of i.i.d. observations: tensor product state φn

I consider nonparametric quantum i.i.d. experiment. I assume smoothness class for φ: Hilbert space H- in…nite dimensional,

complex, exists ONB

I de…ne F (α, M) =

• φ : ∑∞

j=1 j2α

• φj
• 2

M

• : Sobolev-type smoothness class

I thus the i.i.d. experiment is En =

• φn, φ 2 F (α, M)
• ( )

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q-Gaussian experiments

try a minimalist, non-technical description of quantum Gaussians:

I jφig denotes a Gaussian pure state: element of a certain Hilbert space HF

• Fock space associated to H,

I a translation by φ of the Gaussian ground state j0ig in HF I also called coherent states in quantum optics I case a) assume H =C F φ is just a complex number. Fock space HF can be taken L2(R), j0ig can be

taken p ϕ (x) (ϕ standard Gaussian density) and jφig = Uφ j0ig - a rotated vector, Uφ is a unitary depending on φ.

F quantum analog of one-dimensional Gaussian shift N (φ, 1) for real φ. (In the

q-model, φ is complex)

I case b) assume H = `2 (square summable complex sequences) F φ is a sequence. Fock space HF is a more involved construction, ground state

j0ig is de…ned, then also jφig = Uφ j0ig with Uφ a unitary depending on φ.

F quantum analog of white noise in sequence space:

yj = φj + ξj, j 2 N (assuming φj real; in q-model φj complex)

• r of continuous white noise model

dYt = ˇ φ (t) dt + dWt, t 2 [0, 1] (assuming ˇ φ (t) real-valued function);

details cf. quantum stochastic calculus (Parthasarathy, 92)

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Asymptotic equivalence in the quantum case

consider q-Gaussian shift

• n1/2φ

E

g where n is "sample size"

I aim: for Gn =

• n1/2φ

E

g , φ 2 F (α, M)

• show En ' Gn ! 0 for α > some

α0 (global asy. equivalence)

I assume local equivalence has been shown: for a "center" φ0 2 H and

γn = o

• n1/4

:

I de…ne shrinking neighborhood: Σn (φ0) = ff : kφ φ0k γng I consider local experiment En (φ0) =

• φn, φ 2 F (α, M) \ Σn (φ0)
• ,

similarly Gn (φ0)

I assume En (φ0) ' Gn (φ0) (extended q-LAN for pure states) I question: how to globalize ? I method of preliminary estimation of φ0 from n/2 observations not possible.

Observer e¤ect in quantum mechanics: measurement changes the state.

I consider observable A = ∑d

j=1 aj Πj where Πj = uju j (uj eigenvectors)

I the r.v. XA is observed; suppose it takes value XA = aj (happens with

probability pj = φΠj φ).

I afterwards the system is in new state uj ("collapse of the wave function") I for our problem: using n/2 data for ˆ

φ0 means big information loss - equivalence to Gn not possible

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basic idea: estimate center φ0 from a fraction of the sample m n (traditional "preliminary estimation"), these data are lost, check whether the loss of m observations is informationally negligible.

I consider splitting. En $ Em Enm (exact equivalence), local exper. En (φ0)

is shrinking with rate γn = o

• n1/4

I we have En (φ0) ' Gn (φ0)(local asy. equivalence to Gaussian) I from Em obtain estimator ˆ

φ0 of φ0 2 F (α, M) with standard rate : E

• ˆ

φ0 φ0

• mα/(2α+1) (nonparametric estimation)

I to "get into the neighborhoods" we need mα/(2α+1) γn = o

• n1/4

I this requires m n1/2+1/4α ( hence restriction α > 1/2, since m n) I let E

m be the experiment obtained from Em by quantum measurement

("corrupted")

I E

m less informative than Em : use symbol Em E m

I use an estimator ˆ

φ0 based on E

m to globalize local equivalence

I similar to N 96: for each φ0 there are (two) Markov kernels realizing

Enm (φ0) ' Gnm (φ0)

I use ˆ

φ0 to put together Markov kernels realizing E

m Enm ' E m Gnm

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I we now have En $ Em Enm E

m Enm ' E m Gnm

I next we want to replace Gnm by Gn. Apply same idea: …rst local result, then

estimate center φ0 from E

m (same estimator ˆ

φ0).

I for local result on the Gaussians: Gnm (φ0) ' Gn (φ0) we need m n1/2γ1

n

I indeed: relevant estimates for classical Gaussian white noise hold analogously

in q-Gaussian case

I (e.g. trace norm distance for q-Gaussian states jφig , jψig is bounded by

kφ ψk in H)

I and since γn = o

• n1/4

it su¢ces to have m Cn3/4.

I compare the conditions on m: m n1/2+1/4α and m Cn3/4. This is

possible if α > 1.

I globalization now gives: E

m Gnm ' E m Gn.

I Next, we drop the "corrupted" experiment E

m to obtain E m Gn Gn

I Combining all this and and using symbol % for "asymptotically more

informative" we obtain En % Gn. This means the one sided quantum de…ciency of En wrt Gn tends to zero: δq (En, Gn) ! 0.

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It is possible to reverse this reasoning and start with the Gaussian Gn, to

• btain Gn % En.

I but: we don’t want to deal with "additional observations" on the i.i.d. side

En; deal with it only on the Gaussian side.

I modify reasoning: Gn $ Gm Gnm G

m Gnm (get estimator ˆ

φ0 from Gm, which is then corrupted to G

m)

I G

m Gnm ' G m Gn: put in m additional observations in Gaussian model,

using Gnm (φ0) ' Gn (φ0) and ˆ φ0

I G

m Gn ' G m En using Gn (φ0) ' En (φ0) and ˆ

φ0

I G

m En En dropping G m

I put together: Gn % En.

We thus show En % Gn and Gn % En which means Gn ' En (asy. equivalence by "bracketing") condition was φ 2 F (α, M) for α > 1 caution: assumed standard estimation rate E

• ˆ

φ0 φ0

• nα/(2α+1) in

both Gn and En for given α. If this estimation rate is slower, we need α > α0 for some α0 > 1

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Square root density as signal in white noise

common framework for quantum and classical probability: density matrices

I classical …nite probab. distribution p1, . . . , pd I write as diagonal matrix ρ =

@ p1 . . . pd 1 A

I general quantum case: complex, self adjoint, nonnegat. de…n. matrix of

Tr [ρ] = 1 (density matrix)

I special quantum case: rank(ρ) = 1, hence ρ = φφ, φ unit vector (pure

state)

I asymptotic equivalence to Gaussian model, classical case, …nite dimension (Le

Cam, 86): yj = p1/2

j

+ n1/2ξj, j = 1, . . . , d

I classical case, nonparametric (N 96)

dYt = f 1/2(t)dt + n1/2dWt, t 2 [0, 1]

I now: quantum case, pure state ,nonparametric:

• n1/2φ

E

g , quantum Gaussian shift

common feature: square root of density (. . .matrix) is location in Gaussian shift model

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