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Introduction to quantum mechanics Statistical part Main result

Wigner function estimation in QHT with noisy data

Joint work with Lounici, K. and Peyr´ e, G.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Contents

I/ Physical part II/ Statistical part

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

I./ Introduction to quantum optics

Generally, in quantum mechanics, the result of a physical measurement is random...

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Quantum system Its measurable properties, or ”observables” (ex: spin, energy, position, ...) : X Result of the measurement is random : X = x Measurement From n measurement, one wants to reconstruct the quantum state of the quantum system!

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Quantum STATE Its measurable properties, or ”observables” (ex: spin, energy, position, ...) : X Result of the measurement is random : X = x Measurement The quantum state of a system encodes the probabilities of its measurable ”observables”. That is, the probability of obtaining each of the possible

• utcomes when measuring an observable.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Quantum state in general

The most common representation of a quantum state is an operator (density

• perator) ρ on a complex Hilbert space H (called the space of states) s.t.:

1

Self adjoint: ρ∗ = ρ,

2

Positif: ψ, ρψ ≥ 0, for all ψ ∈ H,

3

Trace 1: Tr(ρ) = 1. A quantum state ρ encodes the probabilities of the measurable properties (observables) of the considered quantum system.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Quantum state in our setting

• The quantum system : a monochromatic light in a cavity.
• The space of state : the space of square integrable complex valued

functions on the real line H = L2(R) =

• f : R → C,
• |f (x)|2dx < ∞
• A particular orthonormal basis : the Fock basis 1:

ψn(x) = 1 √π2nn! Hn(x)e−x2/2.

• In the Fock basis, a state is described by an infinite density matrix

ρ = [ρj,k]j,k∈N called a density matrix.

1Hn(x)= nth Hermite polynomial. Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Observable in general

To each measurable property (position,energy...) corresponds an observable. An observable X is described by a self adjoint operator on the space of states H and X =

dimH

• a

xaPa, where

• the eigenvalues {xa}a of the observable X are real,
• Pa is the projection onto the one dimensional space generated by the

eigenvector of X corresponding to the eigenvalue xa.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Measurement in general

When performing a measurement of the observable X of a quantum state ρ, the result is a random variable X with values in the set of the eigenvalues {xa}a

• f the observable X s.t.:
• the probability distribution is

Pρ(X = xa) = Tr(Paρ),

• the expectation function

Eρ(X) = Tr(Xρ).

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Observable in our setting

• A monochromatic light in a cavity described by a quantum harmonic
• scillator. Usually, the observables we deal with are Q and P (resp. the

electric and magnetic fields).

• According to Heisenberg’s uncertainty principle, Q and P are

non-commuting observables, they may not be simultaneously measurable.

• However, for a phase φ ∈ [0, π] we can measure the quadrature
• bservables

Xφ := Q cos φ + P sin φ.

• Each of these quadratures could be measured on a laser beam by a

technique put in practice for the first time by Smithey and called Quantum Homodyne Tomography (QHT).

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Measurement: QHT

• Mix the laser prepared in state ρ with an

additional laser of high intensity |z| >> 1 (local oscillator (LO)).

• The phase Φ of the LO is choosen s.t.

Φ ∼ U[0, π] .

• Split the mixing in 2 beams and each

beam is measured by a photodetector which gives reps. integrated currents I1 and I2 proportional to the number of photons.

• Result of the measurement of Xφ is a r.v.

X|Φ = φ of probability density pρ(·|φ).

I1 I2 z = |z|eiφ

I1−I2 √2η|z| ∼ pη ρ(x|φ)

vacuum2 vacuum1 beam splitter signal detector

• scilator

local detector

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Result of ideal QHT measurements and Wigner function

Result of the measurement by QHT of Xφ is a r.v. X|Φ = φ of probability density pρ(·|φ) s.t.: F1[pρ(·|φ](t) = Tr(ρeitXφ) := Wρ(t cos φ, t sin φ), where Wρ is the Wigner function, an equivalent representation for a quantum state ρ Wρ : R2 → R and

• Wρ(q, p)dqdp = 1

Wρ plays the role of a ”quasi-probability density” of (Q, P). Its Radon transform is always a probability density and is s.t. : pρ(x|φ) := ℜ[Wρ](x, φ) = ∞

−∞

Wρ(x cos φ + t sin φ, x sin φ − t cos φ)dt. Therefore the result of an ideal measurement : (X, Φ) ∼ pρ(x, φ) = 1 π ℜ[Wρ](x, φ)1 l[0,π](φ)

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Mathematical formalism

Quantum state of ↔ • ρ infinite density matrix a monochromatic light

• Wρ Wigner function

Observable ↔ Xφ := Q cos φ + P sin φ Φ ∼ U[0, π] Result of an ideal ↔ r.v. (X, Φ) of prob. density measurement by QHT pρ(x, φ) = 1

π ℜ[Wρ](x, φ)1

l[0,π](φ)

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

II/ Statistical part

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Severely ill-posed inverse problem

In the noisy setting, we collect by QHT, n i.i.d. r.v. (Zℓ, Φℓ)ℓ=1,...,n i.i.d. such that Zℓ = Xℓ +

• (1 − η)/(2η) ξℓ
• Detection process is inefficient, photons fail to be detected :

η ∈]0, 1] (η ≈ 0.9 in practice)

• An independent gaussian noise interferes additively with the

ideal data 2. From now: γ := (1 − η)/(4η) ∈ [0, 1/4[ and the ideal setting : γ = 0.

2Note that the gaussian nature of the noise is imposed by the gaussian

nature of the vacuum state which interferes additively.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Severely ill-posed inverse problem

In the noisy setting, we collect by QHT, n i.i.d. r.v. (Zℓ, Φℓ)ℓ=1,...,n i.i.d. such that Zℓ = Xℓ + √2γ ξℓ of probability density: pγ

ρ(z, φ) =

1 πℜ[Wρ](·, φ) ∗ Nγ

• (z)1

l[0,π](φ), Nγ(·) = N(0, 2γ) and γ := (1 − η)/(4η) ∈ [0, 1/4[ and the ideal setting : γ = 0.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Severely ill-posed inverse problem

In the noisy setting, we collect by QHT, n i.i.d. r.v. (Zℓ, Φℓ)ℓ=1,...,n i.i.d. such that Zℓ = Xℓ + √2γ ξℓ of probability density: pγ

ρ(z, φ) =

1 πℜ[Wρ](·, φ) ∗ Nγ

• (z)1

l[0,π](φ), Nγ(·) = N(0, 2γ) and γ := (1 − η)/(4η) ∈ [0, 1/4[ and the ideal setting : γ = 0. Goal: Reconstruct the quantum state from (Zℓ, Φℓ)ℓ=1,...,n.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Vacuum, 1-photon and 3-coherent states

5 10 15 20 5 10 15 20 25 0.2 0.4 0.6 0.8 1 5 10 15 20 5 10 15 20 25 0.2 0.4 0.6 0.8 1 5 10 15 20 5 10 15 20 25 0.05 0.1 0.15 0.2

1 π e−q2−p2. 1 π (2q2 + 2p2 + 1)e−q2−p2. 1 π e−(q−a)2−p2.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

A realistic class of quantum states3

For C ≥ 1, B > 0 and 0 < r ≤ 2, R(C, B, r) := {ρ quantum state : |ρj,k| ≤ C exp(−B(j + k)r/2)}.

In [Aubry, Butucea and M.(2009)], these decrease condition on ρj,k has been translated on the corresponding Wigner function : Wρ.

3The quantum states which can be created at this moment in laboratory

and belong to the class R(C, B, r) with r = 2.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

A realistic class of quantum states4

[Aubry, Butucea and M.(2009)] For β > 0, L > 0 and 0 < r ≤ 2, A(β, r, L) :=

• Wρ,
• |

Wρ(u, v)|2e2β(u,v)r dudv (2π)2L

• ,

where Wρ denotes its Fourier transform.

4The quantum states which can be created at this moment in laboratory

and belong to the class R(C, B, r) with r = 2.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

A statistical point of view

Given a norm || · || and for θn an estimator (estimation procedure)

• f an unknown quantity θ, we define:
• The maximal risk of

θn on a given class Cα : sup

θ∈Cα

E||θ − θn||

• And we call adaptive procedure

θad,n if it does not depend on the unknown regularity parameter α.

• The rate of convergence ϕn → 0 as n → ∞ is s.t.

sup

θ∈Cα

E||θ − θn|| ≤ cϕn.

• Moreover the rate of convergence is said to be minimax if

inf

T sup θ∈Cα

E||θ − T|| ≥ Cϕn, where the infimum is taken over all possible estimators.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Outline

1

Estimation of ρ

• [Artiles, Gill and Gut

¸˘ a(2005)] : η = 1,

• [Aubry, Butucea and M.(2009)]: η ∈]0, 1] (γ ∈ R+)⇒ UB for

Frobenius norm,

• Adaptation [Kahn (2008)] : η = 1 (γ = 0) by models selection,
• Adaptation [Alquier, M. and Peyr´

e (2013)] : η ∈]1/2, 1], UB for Frobenius norm.

2

Estimation of Wρ :

• [Gut

¸˘ a and Artiles(2007)]: η = 1, minimax rate, ponctual risk,

• [Butucea, Gut

¸˘ a and Artiles(2007)]: η ∈]0, 1] (ponctual risk)+ Adaptation r ∈]0, 1[,

• [Aubry, Butucea and M.(2009)]: η ∈]0, 1] (UB L2-risk).
• [Lounici, M. and Peyr´

e (2015+)]: η ∈]0, 1], (UP& LB& adaptation sup-norm), (LB L2-norm)

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Estimation procedure

For γ ∈ [0, 1/4[, we define5

• W γ

h (q, p) =

1 2πn

n

• ℓ=1

K γ

h ([z, Φℓ] − Zℓ) ,

where z = (q, p) and [z, φ] = q cos φ + p sin φ. The kernel is defined in the Fourier domain by

• K γ

h (t) = |t|eγt21

l|t|≤1/h, and h = hn > 0 tends to 0 when n → ∞.

5As in [Butucea, Gut

¸˘ a and Artiles(2007), Aubry, Butucea and M.(2009)]

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Theoritical results

Theorem 1: upper bounds Assume that Wρ ∈ (β, r, L) for some r ∈ (0, 2] and β, L > 0. Consider the procedure with h∗ = h∗(r) such that   

γ (h∗)2 + β (h∗)r = 1 2 log(n)

if 0 < r < 2, h∗ =

• 2(β+γ)

log n

1/2 if r = 2. Then we have E W γ

h∗ − Wρ∞ ≤ cϕn(r),

where ϕn(r) =

• (h∗)(r−2)/2e−β(h∗)−r

if 0 < r < 2, n−

β 2(β+γ)

if r = 2.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Theoritical results

Theorem 2 : Lower bounds For any β, L > 0 and r = 2 there exists a constant C := C(β, L, γ) > 0 such that for n large enough inf

Wn supWρ∈A(β,2,L)

E Wn − Wρ2

p

≥ Cϕ2

n =

• Cn−

β 2(β+γ) log−3/2(n)

if p = ∞, Cn−

β β+γ

if p = 2. where the infimum is taken over all possible estimators Wn based

• n the i.i.d. sample {(Zi, Φi)}n

i=1.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Idea of the proof of the upper bounds (Theo 1)

E[ W γ

h − Wρ∞ ≤ E

W γ

h − E[

W γ

h ]∞ + E[

W γ

h ] − Wρ∞

Where the stochastic term (ST) is treated as follow : It has been proved that Hh = {δ−1

h K η h (· − t), t ∈ R},

h > 0 is uniformly bounded by U :=

h 2γπ and the following entropy

bound6 sup

Q

N(ǫ, Hh, L2(Q)) ≤ (A/ǫ)v. Therefore using the same tool as in [Gin´ e and Nickl(2009)] as Hh is VC, we get a bound for the ST. By taking the derivative (ST+BT), we get the result.

6where the supremum extends over all probability measures Q on R. Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Idea of the proof of the lower bounds (Theo 2)

Let M = ⌊√log n⌋ and δ := log−1(n). Construct of a family of M Wigner functions s.t. ∀w ∈ R2: Wm,h(w) = W0(w) + Vm,h(w), 1 ≤ m ≤ M, for h = h(n) → 0 as n → ∞. In the noisy setting, we set

pγ

m,h(z, φ) = [pm,h(·, φ) ∗ Nγ] (z)

and pγ

0 (z, φ) = [p0(·, φ) ∗ Nγ] (z).

(C1) If ∀m = 1 · · · M, Wm,h ∈ A(β, L). (C2) If ∀1 ≤ k = m ≤ M, we have for ||Wk,h − Wm,h||2

2 ≥ 4ϕ2 n,

(C3) If ∀1 ≤ m ≤ M, nX 2(pγ

m,h, pγ 0) ≤ M 4 .

Then Theorem 2.6 in [Tsybakov(2009)] gives the lower bound.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Construction of the set of Wigner functions Wm,h for the L2-norm

• W0 ∈ A(β, L) is the same as in [Butucea, Gut

¸˘ a and Artiles(2007)].

• Introduce M infinitely differentiable functions s.t. ∀m and

h =

• log n

2(β+γ)

−1/2 : ∗ Vm,h : R2 → R is an odd real-valued function. ∗ Set t =

• w 2

1 + w 2 2 , then Vm,h is s.t.

• Vm,h(w) := F2[Vm,h](w) := iaC0h−1eβh−2e−2β|t|2gm(|t|2−h−2)g(w2),

where a > 0 is a numerical constant chosen sufficiently small. where ∗ ∀m, gm : R → [0, 1], s.t. Supp(gm) = (mδ, (m + 1)δ) . ∗ And ∀t ∈ [(m + 1/3)δ, (m + 2/3)δ] , gm(t) = 1. ∗ An odd function g : R → [−1, 1], s.t. for some fixed ǫ > 0, g(x) = 1 for any x ≥ ǫ.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Remarks

• Vm,h(p, q)dpdq =
• R[Vm,h](x, φ)dx = 0,

∀φ ∈ [0, π]

Vm,h ∈ S(R2)7 as it is infinitely differentiable with compact support.

• The Fourier transform being a continuous mapping of S(R2) onto itself

⇒ Vm,h ∈ S(R2).

• As

Vm,h(w) is an odd function with purely imaginary values ⇒ Vm,h is an odd real-valued function. Define pm,h(x, φ) = 1

πR[Wm,h](x, φ)1

l(0,π(φ), and ρ(m,h)

j,k

= π

• pm,h(x, φ)fj,k(x)e(j−k)φdxdφ.

The matrix ρ(m,h) is a density matrix. (Lemma)

7Schwartz class of fast decreasing functions on R2. Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Condition (C1)

• Vm,h(w) := F2[Vm,h](w) := iaC0h−1eβh−2e−2β|t|2gm(|t|2 − h−2)g(w2),

By the triangle inequality and as g is bounded by 1

• Wm,he·22 ≤

W0e·22 + Vm,he·22.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Condition (C2)

• By Plancherel Theorem, the change w = (t cos φ, t sin φ), and since

Supp(gk) ∩ Supp(gm) = ∅ ⇒ Wk,h − Wm,h2

2 is equal to

a2C 2 4π2 h−2e2βh−2 π

• |t|e−4βt2g 2(t sin φ)
• g 2

k (t2 − h−2) + g 2 m(t2 − h−2)

• dtdφ.
• Define

Am :=

• w ∈ R2 : (m + 1/3)δ ≤ w2 ≤ (m + 2/3)δ
• and

fixed8 µ ∈]0, π/4[ ⇒ π

• can be lower bounded by

π−µ

µ

• Ak∪

Am

. ⇒ ∀(t, φ) ∈ ( Ak ∪ Am)×]µ, π − µ[ : g 2(t sin(φ)) = 1 for n large enough ⇒ On Am : g 2

m(t2 − h−2) = 1.

8∃c > 0 s.t. sin(φ) > c on ]µ, π − µ[ Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Condition (C3)

Lemma There exists numerical constants c′ > 0 and c′′ > 0 such that pγ

0 (z) ≥ c′z−2,

∀|z| ≥ 1 +

• 2γ,

(1) and pγ

0 (z) ≥ c′′,

∀|z| ≤ 1 +

• 2γ.

(2) nX 2(pγ

m,h, pγ 0 ) ≤

n c′′ π 1+√2γ

−(1+√2γ)

• pγ

m,h(z, φ) − pγ 0 (z, φ)

2 dzdφ + n c′ π

• R\(1+√2γ,1+√2γ)

z2 pγ

m,h(z, φ) − pγ 0 (z, φ)

2 dzdφ =: n c′′ I1 + n c′ I2. (3)

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Condition (C3)

Main ingredients for : n c′′ I1 = n c′′ π 1+√2γ

−(1+√2γ)

• pγ

m,h(z, φ) − pγ 0 (z, φ)

2 dzdφ ≤ a2 C √log n,

• Plancherel Theorem.
• F1[pγ

m,h(·, φ)](t) =

Wm,h(t cos φ, t sin φ) Nγ(t) =

• Vm,h(t cos φ, t sin φ) +

W0(t cos φ, t sin φ)

• e−γt2,
• F1[pγ

0 (·, φ)](t) =

W0(t cos φ, t sin φ)e−γt2,

• g bounded by 1, Supp(gm) = (mδ, (m + 1)δ)

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Condition (C3)

Main ingredients for : n c′ I2 = n c′ π

• R\(1+√2γ,1+√2γ)

z2 pγ

m,h(z, φ) − pγ 0 (z, φ)

2 dzdφ ≤ a2 C

• log n,
• Same tools used for I1.
• In addition the spectral representation of the differential operator

I2 ≤ π

• z2

pγ

m,h(z, φ) − pγ 0 (z, φ)

2 dzdφ = π

• ∂

∂t

• F1[pγ

m,h(·, φ)] − F1[pγ 0 (·, φ)]

• (t)
• 2

dtdφ

• Since gm and g belong to the Schwartz class, there exists a numerical

constant cS > 0 such that max{gm∞, g ′

m∞, g∞, g ′∞} ≤ cS.

⇒ Taking the numerical constant a > 0 small enough, we deduce from the previous display that nX 2(pγ

k,h, pγ 0 ) ≤ M

4 , since M = ⌊

• log n⌋.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Simulation

We call from now for sake of brievety

• The true : the true Wigner function,
• The oracle: The result of the procedure estimation for the

best (oracle) bandwidth h

• The Lepski : Our adaptive estimation procedure for κ = 0.1

and x = log(M) We consider η = 0.9 and n = 105.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Simulation Single photon state

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Simulation Schrodinger’s cat state

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Thank you ...

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Alquier, P., Meziani, K. and Peyr´ e,G., Adaptive Estimation of the Density Matrix in Quantum Homodyne Tomography with Noisy Data. Inverse Problems, 29, 7, 075017,2013. Artiles, L. and Gill, R. and Gut ¸˘ a, M. An invitation to quantum tomography

• J. Royal Statist. Soc. B (Methodological), 67,109–134, 2005.

Aubry, J.-M. and Butucea, C. and Meziani, K., State estimation in quantum homodyne tomography with noisy data. Inverse Problems, 25, 1,2009. Butucea, C. and Gut ¸˘ a, M. and Artiles, L.. Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data.

• Ann. Statist.,2,35,465–494, 2007.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT

Introduction to quantum mechanics Statistical part Main result

Gut ¸˘ a, M. and Artiles, L. Minimax estimation of the Wigner in quantum homodyne tomography with ideal detectors.

• Math. Methods Statist., 16, 1,1–15, 2007.

Gin´ e, E. and Nickl, R., Uniform limit Theorems for wavelet density estimators.

• Ann. Probab., 37, 4,1605–1646, 2009.

Lounici, K., Meziani, K. and Peyr´ e,G., Minimax and Adaptive Estimation of the Wigner function in Quantum Homodyne Tomography with Noisy Data, Coming soon, 2015. Tsybakov, A.B. Introduction to Nonparametric Estimation. Springer Series in Statistics, New York, 2009.

Lounici, Meziani and Peyr´ e Wigner function estimation in QHT