Bayesian nonparametric estimation for quantum homodyne tomography - - PowerPoint PPT Presentation

Bayesian nonparametric estimation for quantum homodyne tomography

Zacharie Naulet (CEA Saclay and Paris-Dauphine University) Joint work with: Éric Barat (CEA Saclay) Judith Rousseau (Paris-Dauphine University)

• TT. Truong (Cergy-Pontoise University)

High-Dimensional Problems and Quantum Physics | Marne-la-Vallée 9th June 2015

Zacharie Naulet BNP estimation for QHT 9th June 2015 1 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 1 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 1 / 36

QM : Short introduction

A1 With every quantum system is associated an infinite-dimensional separable Hilbert space H over C, with inner product ·, ·, the space of states.

Zacharie Naulet BNP estimation for QHT 9th June 2015 2 / 36

QM : Short introduction

A1 With every quantum system is associated an infinite-dimensional separable Hilbert space H over C, with inner product ·, ·, the space of states. A2 The set of observables A of a quantum system consists of all self-adjoint operators on H .

Zacharie Naulet BNP estimation for QHT 9th June 2015 2 / 36

QM : Short introduction

A1 With every quantum system is associated an infinite-dimensional separable Hilbert space H over C, with inner product ·, ·, the space of states. A2 The set of observables A of a quantum system consists of all self-adjoint operators on H . A3 The set of pure states S of a quantum system are projection operators P onto one-dimensional subspaces of H , with Tr P = 1. For ψ ∈ H with ψ = 1, we write Pψ : H → H the projection operator (density matrix) onto the linear span of ψ.

Zacharie Naulet BNP estimation for QHT 9th June 2015 2 / 36

QM : Short introduction

A1 With every quantum system is associated an infinite-dimensional separable Hilbert space H over C, with inner product ·, ·, the space of states. A2 The set of observables A of a quantum system consists of all self-adjoint operators on H . A3 The set of pure states S of a quantum system are projection operators P onto one-dimensional subspaces of H , with Tr P = 1. For ψ ∈ H with ψ = 1, we write Pψ : H → H the projection operator (density matrix) onto the linear span of ψ. A4 A measurement is a mapping A × S ∋ (A, P) → µA ∈ P(R).

Zacharie Naulet BNP estimation for QHT 9th June 2015 2 / 36

QM : Short introduction

A1 With every quantum system is associated an infinite-dimensional separable Hilbert space H over C, with inner product ·, ·, the space of states. A2 The set of observables A of a quantum system consists of all self-adjoint operators on H . A3 The set of pure states S of a quantum system are projection operators P onto one-dimensional subspaces of H , with Tr P = 1. For ψ ∈ H with ψ = 1, we write Pψ : H → H the projection operator (density matrix) onto the linear span of ψ. A4 A measurement is a mapping A × S ∋ (A, P) → µA ∈ P(R). µA is given by the Born-Von Neumann formula : µA(E) := Tr PA(E)Pψ = PA(E)ψ, ψ, E ∈ B(R), where PA is a positive operator-valued measure which precise definition is stated in Von Neumann’s Spectral Theorem. µA(E) is the probability that the measurement of the observable A on the system in state Pψ lies in E ∈ B(R).

Zacharie Naulet BNP estimation for QHT 9th June 2015 2 / 36

QM : Simultaneous measurements

Definition

Self-adjoint operators A and B commute if the corresponding POVM PA and PB satisfy PA(E1)PB(E2) = PB(E2)PA(E1) for all E1, E2 ∈ B(R).

Zacharie Naulet BNP estimation for QHT 9th June 2015 3 / 36

QM : Simultaneous measurements

Definition

Self-adjoint operators A and B commute if the corresponding POVM PA and PB satisfy PA(E1)PB(E2) = PB(E2)PA(E1) for all E1, E2 ∈ B(R). A5 Observables A and B can be measured simultaneously if and only if they commute.

Zacharie Naulet BNP estimation for QHT 9th June 2015 3 / 36

QM : Simultaneous measurements

Definition

Self-adjoint operators A and B commute if the corresponding POVM PA and PB satisfy PA(E1)PB(E2) = PB(E2)PA(E1) for all E1, E2 ∈ B(R). A5 Observables A and B can be measured simultaneously if and only if they commute. Otherwise, we have the celebrated Heisenberg’s uncertainty relations :

Theorem

Let A, B ∈ A and ψ ∈ D(A) ∩ D(B) with Aψ, Bψ ∈ D(A) ∩ D(B). Then, σ2

ψ(A)σ2 ψ(B) ≥ 1

4i(AB − BA)ψ, ψ2.

Zacharie Naulet BNP estimation for QHT 9th June 2015 3 / 36

QM : Simultaneous measurements

Definition

Self-adjoint operators A and B commute if the corresponding POVM PA and PB satisfy PA(E1)PB(E2) = PB(E2)PA(E1) for all E1, E2 ∈ B(R). A5 Observables A and B can be measured simultaneously if and only if they commute. Otherwise, we have the celebrated Heisenberg’s uncertainty relations :

Theorem

Let A, B ∈ A and ψ ∈ D(A) ∩ D(B) with Aψ, Bψ ∈ D(A) ∩ D(B). Then, σ2

ψ(A)σ2 ψ(B) ≥ 1

4i(AB − BA)ψ, ψ2. = ⇒ If A and B don’t commute, there is no joint distribution associated with the measurement of A, B. Example : position Q and momentum P.

Zacharie Naulet BNP estimation for QHT 9th June 2015 3 / 36

The Wigner transform

Definition

Wigner transform Let ψ ∈ L2(R) with ψ2 = 1, then the Wigner transform W : L2(R) → L2(R2) is defined as W (ψ)(q, p) := 1 π

• R

ψ

• q + y

2

• ψ
• q − y

2

• e−ipy dy,

(q, p) ∈ R2

Zacharie Naulet BNP estimation for QHT 9th June 2015 4 / 36

The Wigner transform

Definition

Wigner transform Let ψ ∈ L2(R) with ψ2 = 1, then the Wigner transform W : L2(R) → L2(R2) is defined as W (ψ)(q, p) := 1 π

• R

ψ

• q + y

2

• ψ
• q − y

2

• e−ipy dy,

(q, p) ∈ R2

Properties of the Wigner transform

1 Real valued. 2 Total energy :

• R2 W (ψ)(q, p) dqdp = 1.

3 Marginals :

• R W (ψ)(p, q) dp = |ψ(q)|2 (proba. density)

4 Marginals :

• R W (ψ)(p, q) dq = |

ψ(p)|2 (proba. density)

5 Isometry : W (ψ1), W (ψ2)L2(R2) = ψ1, ψ2L2(R)

Zacharie Naulet BNP estimation for QHT 9th June 2015 4 / 36

The Wigner transform (continued)

Properties of the Wigner transform (continued)

From the isometry property, if ψ and ϕ are orthogonal,

• R2 W (ψ)(q, p)W (ϕ)(q, p) dqdp = 0.

= ⇒ There exists locally negative Wigner transforms, and W (ψ) can’t be a joint probability density.

Zacharie Naulet BNP estimation for QHT 9th June 2015 5 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 5 / 36

Quantum Homodyne Tomography

Goal : Measuring the quantum state of light. Observables of interest are self-adjoint operators on L2(R) :

1 P : the magnetic field, Pψ(x) = −iψ′(x). 2 Q : the electric field, Qψ(x) = xψ(x). 3 (H = P2 + Q2 : the total energy of the electromagnetic field)

Zacharie Naulet BNP estimation for QHT 9th June 2015 6 / 36

Quantum Homodyne Tomography

Goal : Measuring the quantum state of light. Observables of interest are self-adjoint operators on L2(R) :

1 P : the magnetic field, Pψ(x) = −iψ′(x). 2 Q : the electric field, Qψ(x) = xψ(x). 3 (H = P2 + Q2 : the total energy of the electromagnetic field)

Obviously PQ = QP thus we can’t measure simultaneously observables P and Q. But, we can measure the quadrature observables Xφ = cos(φ)Q + sin(φ)P on n quantum systems in the same state Pψ.

1 We get iid data (X1, φ1), . . . , (Xn, φn) with distribution Pψ on R × [0, π]. 2 We aim at estimating Pψ (or equivalently ψ ∈ L2(R)) from the data.

Zacharie Naulet BNP estimation for QHT 9th June 2015 6 / 36

QHT : Experimental setup

Distribution of X|φ

We have iid data (X1, φ1), . . . , (Xn, φn) coming from the measurement of the observables Xφ on n quantum systems in the same state Pψ. The distribution of X|φ has density (wrt Lebesgue measure on R)

gψ(x|φ) = R(W (ψ))(x, φ) :=

• R

W (ψ)(x cos φ − ξ sin φ, x sin φ + ξ cos φ) dξ,

where R(W (ψ)) is the Radon transform of W (ψ).

Source : Quantum information technology, a homodyne tomogra- phy tutorial. http://www.iqst.ca/ quantech/homotomo.php.

Zacharie Naulet BNP estimation for QHT 9th June 2015 7 / 36

QHT : A more realistic model

Because of the efficiency of the detector, we should consider that instead of

• bserving X|Φ, we observe Y |Φ, η, where

Y := √ηX +

• (1 − η)/2ξ,

ξ ∼ N (0, 1) See Cristina Butucea, Mădălin Guţă, and Luis Artiles (2007). “Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data”. In: The Annals of Statistics 35.2,

• pp. 465–494 for details.

See also, Ulf Leonhardt and H Paul (1995). “Measuring the quantum state of light”. In: Progress in quantum electronics 19.2, pp. 89–130.

Zacharie Naulet BNP estimation for QHT 9th June 2015 8 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 8 / 36

Examples of quantum states (continued)

Coherent state 1-photon Fock state Squeezed state Schrödinger cat state Philippe Grangier (2011). “Make It Quantum and Continuous”. In: Science 332.6027, pp. 313–314

Zacharie Naulet BNP estimation for QHT 9th June 2015 9 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 9 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 9 / 36

Problem Statement

We consider the nonparametric regression problem of estimating the quantum state Pψ, ψ ∈ L2(R), from data Y1, . . . , Yn|φ1, . . . , φn, i = 1, . . . , n, from a Bayesian perspective. With the assumption that ψ is a random variable, the Bayes rule is Π(ψ ∈ U|Y 1, Y 2, . . . )

• posterior

∝

• U

L(ψ|Y 1, Y 2, . . . )

• likelihood

Π(dψ)

prior

. Given the posterior distribution of ψ, we can compute several approximates,

• the posterior mean of ψ, the posterior median of ψ,
• the variance, credible bands, . . .

= ⇒ But we need a prior distribution Π(dψ) on ψ

Zacharie Naulet BNP estimation for QHT 9th June 2015 10 / 36

Before going further

Q: The inverse problem is linear on Wψ (Radon transform), but quadratic on ψ (Radon-Wigner transform), so why working at the level of ψ ? (Furthermore, ψ → W (ψ) is isometric). Pros : This guarantee physical properties of the Wigner transform of the estimate,

1 Total energy 2 Marginals 3 Heisenberg uncertainty relations

Cons : This certainly makes the problem harder1 (computationally and theoretically).

1Is this really a cons ?

Zacharie Naulet BNP estimation for QHT 9th June 2015 11 / 36

Prior distributions on function spaces

Regression:

• Gaussian processes (Rasmussen, 2004)

Density estimation:

• Dirichlet processes mixtures (Escobar and West, 1995)

Idea: Use Kernel mixtures models in regression problems

• Abramovich, Sapatinas, and Silverman (2000),
• Pillai et al. (2007) and Pillai (2008),
• Wolpert, Clyde, Tu, et al. (2011),
• This talk, and Naulet and Barat (2015).

Zacharie Naulet BNP estimation for QHT 9th June 2015 12 / 36

Kernel mixtures models for C-valued regression functions

Let

• (G, ρ) be a complete separable locally compact metric space
• M(G) be the set of complex-valued measures on G
• Π∗(dQ) be a prior distribution on M(G)
• Φ : G × R → C be a kernel function

Then Π∗(dQ) induces a prior distribution on an abstract space of functions f : R → C through the mapping M(G) ∋ Q →

• G

Φ(x; ·) dQ(x). Let Π(df ) denote this prior distribution f ∼ Π(df ) ⇐ ⇒ f (·) =

• G Φ(x; ·) dQ(x)

Q ∼ Π∗(dQ).

Zacharie Naulet BNP estimation for QHT 9th June 2015 13 / 36

Kernel mixtures models for C-valued regression functions

Examples of prior distributions on M(G) (ie. random measures):

• Dirichlet processes (with adaptations)
• Lévy Random Measures (with adaptations) (Wolpert, Clyde, Tu, et al.,

2011; Pillai, 2008; Rajput and Rosinski, 1989).

• This talk : Completely Random Measures (with adaptations), (Kingman,

1967a; Kingman, 1992; Naulet and Barat, 2015). Examples of kernels:

• Bases, unions of bases, frames
• Polynomial kernels
• Coherent states, continuous wavelets, . . . , (this talk).

Zacharie Naulet BNP estimation for QHT 9th June 2015 14 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 14 / 36

Basics of Representation Theory

• G will always denote a Locally Compact Topological Hausdorff Group.
• An Irreducible, Unitary Representation (UIR) π ≡ (π, H ) of G on a

Hilbert space H is an operator on H which is

• a group morphism, ie. π(x1) ◦ π(x2) = π(x1x2), and
• H has no invariant subspace under π (except ∅ and H ), and
• π(x)(f ) = f for all f ∈ H and all x ∈ G
• π is called square integrable if at least one of the coefficients π(x)g, g,

with g = 0 is square integrable wrt to the Haar measure of G.

Zacharie Naulet BNP estimation for QHT 9th June 2015 15 / 36

Reproducing formula

If the representation π is square integrable for some g ∈ H , then the Voice transform Vg(f )(x) = π(x)g, f , is a well defined linear mapping from H to Cb(G). There is a unique positive self-adjoint operator A on H , with D(A) dense in H , such that Vg(g) ∈ L2(G) iff g ∈ D(A) and the orthogonality relation holds (Grossmann, Morlet, and Paul, 1985).

• G

Vg1(f1)(x)Vg2(f2)(x)dµ(x) = A(g2), A(g1)f1, f2, A simple restatement yields the Reproducing formula (in L2(G)): Vg(f )(y) := (Vg(f ) ∗ Vg(g))(y) =

• G

Vg(f )(x)Vg(g)(x−1y)dµ(x)

Zacharie Naulet BNP estimation for QHT 9th June 2015 16 / 36

Reproducing formula

Short version : The voice transform realize an isometry between H and a closed subspace of L2(G) with a reproducing kernel. Another restatement of the orthogonality relations yields the following representation (at least in a weak sense) for all f ∈ H : f =

• G

Vg(f )(x) π(x)(g) dµ(x).

Zacharie Naulet BNP estimation for QHT 9th June 2015 17 / 36

Representation Theory : Example 1

The reduced Heisenberg group is,

• the set R2 × T with composition law,
• (x1, ω1, θ1) · (x2, ω2, θ2) = (x1 + x2, ω1 + ω2, θ1θ2eiω1x2)

and has Unitary Irreducible Representation (UIR) on L2(R), called the Schrödinger representation, π(x, ω, θ)(f )(t) = θeiω(t− x

2 )f (t − x).

Voice transform ⇐ ⇒ Short Time Fourier Transform (up to a phase factor) Vg(f )(x, ω, θ) = θeiω x

2

+∞

−∞

f (t)g(t − x)e−iωtdt Remark : Because the reduced Heisenberg group is unimodular, the set of admissibles g ∈ L2(R) for which the coefficients π(x, ω, θ)g, g are square integrable is dense in L2(R), allowing a wide variety of analyzing windows.

Zacharie Naulet BNP estimation for QHT 9th June 2015 18 / 36

Representation Theory : Example 2

The full affine group on R is,

• the set R∗ × R with composition law,
• (a1, b1) · (a2, b2) = (a1a2, b1 + a1b2)

and has Unitary Irreducible Representation (UIR) on L2(R), called the quasi-regular representation, π(a, b)(f )(t) = |a|−1/2f (t − b a ). Square integrability condition ⇐ ⇒ Admissibility condition: +∞

−∞

| g(ξ)|2 |ξ| dξ < +∞ Voice transform ⇐ ⇒ Continuous Wavelet Transform Vg(f )(a, b) = |a|−1/2 +∞

−∞

f (t)g t − b a

• dt

Zacharie Naulet BNP estimation for QHT 9th June 2015 19 / 36

Representation theory : Examples summary

Definition

A coherent states kernel is a mapping Φ : R × R2 → C given by Φ(x; y, z) = π(y, z, 0)g(x), where (π, L2) is the Schrödinger representation of the reduced Heisenberg group and g an admissible vector. If g is a gaussian window, π(q, p, 0)g(u) = π−1/4 exp

• −1

2(u − q)2 + ip(u − q 2)

• .

Definition

A continuous wavelets kernel is a mapping Φ : R × R∗ × R → R given by Φ(x; a, b) = π(a, b)g(x), where (π, L2) is the quasi-regular representation of the full affine group and g an admissible vector.

Zacharie Naulet BNP estimation for QHT 9th June 2015 20 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 20 / 36

Completely Random Measures

Completely Random Measures (CRM) are distributions over space of measures (random measure)

Zacharie Naulet BNP estimation for QHT 9th June 2015 21 / 36

Completely Random Measures

Completely Random Measures (CRM) are distributions over space of measures (random measure) We let,

1 (Ω, F, P) be a probability space, and 2 (E, E) be a measurable space.

Zacharie Naulet BNP estimation for QHT 9th June 2015 21 / 36

Completely Random Measures

Completely Random Measures (CRM) are distributions over space of measures (random measure) We let,

1 (Ω, F, P) be a probability space, and 2 (E, E) be a measurable space.

Definition

A random measure Q : Ω × E → [0, ∞] is a CRM if Q(·, A1), . . . , Q(·, An) are independent random variables whenever A1, . . . , An ∈ E are disjoint sets.

Zacharie Naulet BNP estimation for QHT 9th June 2015 21 / 36

Completely Random Measures

Completely Random Measures (CRM) are distributions over space of measures (random measure) We let,

1 (Ω, F, P) be a probability space, and 2 (E, E) be a measurable space.

Definition

A random measure Q : Ω × E → [0, ∞] is a CRM if Q(·, A1), . . . , Q(·, An) are independent random variables whenever A1, . . . , An ∈ E are disjoint sets. Kingman, 1967b result : A CRM can be decomposed into three parts

1 an atomic measure with random atom locations and random mass, 2 an atomic measure with fixed atom locations and random mass, and 3 a deterministic measure.

Zacharie Naulet BNP estimation for QHT 9th June 2015 21 / 36

Completely Random Measures

Completely Random Measures (CRM) are distributions over space of measures (random measure) We let,

1 (Ω, F, P) be a probability space, and 2 (E, E) be a measurable space.

Definition

A random measure Q : Ω × E → [0, ∞] is a CRM if Q(·, A1), . . . , Q(·, An) are independent random variables whenever A1, . . . , An ∈ E are disjoint sets. Kingman, 1967b result : A CRM can be decomposed into three parts

1 an atomic measure with random atom locations and random mass, 2 an atomic measure with fixed atom locations and random mass, and 3 a deterministic measure.

We consider here only CRM satisfying item 1.

Zacharie Naulet BNP estimation for QHT 9th June 2015 21 / 36

Completely Random Measures (continued)

CRM have independent increments and are infinitely divisible, thus by Lévy-Khintchine theorem, a CRM assigns to all A ∈ E a random variable with characteristic function, E[eitQ(A)] = exp

• (0,∞)×A
• eitβ −1
• ν(dβdx)
• ,

where ν is a measure on R+ × E, called the control measure of the CRM, and satisfies

• (0,∞)×E

min(1, β) ν(dβdx) < +∞.

Zacharie Naulet BNP estimation for QHT 9th June 2015 22 / 36

Completely Random Measures (continued)

CRM have independent increments and are infinitely divisible, thus by Lévy-Khintchine theorem, a CRM assigns to all A ∈ E a random variable with characteristic function, E[eitQ(A)] = exp

• (0,∞)×A
• eitβ −1
• ν(dβdx)
• ,

where ν is a measure on R+ × E, called the control measure of the CRM, and satisfies

• (0,∞)×E

min(1, β) ν(dβdx) < +∞. Problem : CRM are real positive-valued, and we need a prior over complex-valued measures.

Zacharie Naulet BNP estimation for QHT 9th June 2015 22 / 36

Extended CRM

• Let ν be a σ-finite measure on G × R+ × [0, 2π] satisfying
• G×R+×[0,2π]

min(1, r) ν(dxdrdθ) < +∞

• Let P be a Completely Random measure on the product space

G × [0, 2π] with control measure ν. Then, Q(A) =

• A×[0,2π]

eiθ P(dxdθ)

a.s.

≡

• j∈J

rj eiθj ✶A(xj), is a (complex) extended completely random measure with control measure ν(dxdrdθ). Remark : this may be view as a generalization of the process of symmetrization of random measures that lead to signed measures.

Zacharie Naulet BNP estimation for QHT 9th June 2015 23 / 36

Canonical example of ECRM : Gamma process

Definition (of Gamma ECRM)

The complex Gamma ECRM with parameters α, η and base distribution F(dx) is the ECRM with Lévy measure ν(dxdrdθ) = αr −1 e−rη dr dθ 2π F(dx), where F a probability measure on (G, B(G)) and α, η > 0.

Lemma

Let PU be the uniform distribution on [0, 2π], DP(α, PU × F) be a Dirichlet process with base measure PU × F, Q ∼ DP(α, PU × F), and T ∼ Γ(α, η). Then the random measure Q such that Q(A) := T

• A×[0,2π] eiθ d

Q(x, θ) for all Borel sets A ⊂ G is distributed as a complex Gamma ECRM with parameters α, η and base distribution F.

Zacharie Naulet BNP estimation for QHT 9th June 2015 24 / 36

Gamma ECRM : Alternative constructions

From the previous lemma , the Gamma ECRM is easily built from the Dirichlet Process. This is convenient since the DP has been extensively studied and many representations of the DP are known Then we can,

• construct a stick-breaking representation of the Gamma ECRM
• construct a Chinese Restaurant Process representation of the Gamma

ECRM

• (put your favorite Dirichlet Process representation here)

Remark : these representations are very classical and are easily found in the literature.

Zacharie Naulet BNP estimation for QHT 9th June 2015 25 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 25 / 36

Summary

Finally, Yi|Xi, ξi, η = √ηXi +

• (1 − η)/2ξi,

i = 1, . . . , n ξi

iid

∼ N (0, 1), i = 1, . . . , n Xi|φi

iid

∼ R(W (ψ))(·, φi), i = 1, . . . , n ψ(u) = ϕ(u)/ϕ2 ϕ(u) =

• R2 π−1/4 exp
• −1

2(u − q)2 + ip(u − q 2)

• Q(dqdp)

Q ∼ ComplexGammaCRM(α, η)

Zacharie Naulet BNP estimation for QHT 9th June 2015 26 / 36

Summary

Finally, Yi|Xi, ξi, η = √ηXi +

• (1 − η)/2ξi,

i = 1, . . . , n ξi

iid

∼ N (0, 1), i = 1, . . . , n Xi|φi

iid

∼ R(W (ψ))(·, φi), i = 1, . . . , n ψ(u) = ϕ(u)/ϕ2 ϕ(u) =

• R2 π−1/4 exp
• −1

2(u − q)2 + ip(u − q 2)

• Q(dqdp)

Q ∼ ComplexGammaCRM(α, η) Remark : From the definition of Q and properties of CRM, ψ has almost-surely the series expression, ψ(u) =

∞

• k=1

rk eiθk exp

• −1

2(u − qk)2 + ipk(u − qk 2 )

• .

Zacharie Naulet BNP estimation for QHT 9th June 2015 26 / 36

Summary

Finally, Yi|Xi, ξi, η = √ηXi +

• (1 − η)/2ξi,

i = 1, . . . , n ξi

iid

∼ N (0, 1), i = 1, . . . , n Xi|φi

iid

∼ R(W (ψ))(·, φi), i = 1, . . . , n ψ(u) = ϕ(u)/ϕ2 ϕ(u) =

• R2 π−1/4 exp
• −1

2(u − q)2 + ip(u − q 2)

• Q(dqdp)

Q ∼ ComplexGammaCRM(α, η) Remark : From the definition of Q and properties of CRM, ψ has almost-surely the series expression, ψ(u) =

∞

• k=1

rk eiθk exp

• −1

2(u − qk)2 + ipk(u − qk 2 )

• .

Obviously the posterior distribution is analytically intractable, and we need to use a sampling procedure (mostly MCMC) to draw samples from the posterior.

Zacharie Naulet BNP estimation for QHT 9th June 2015 26 / 36

Posterior computations

How to sample from Π(df |Y1, . . . , Yn) ? Usual approaches from density mixtures models not directly applicable here, because we do not observe i.i.d samples available for allocation to measure components. Alternatives:

• Reversible-jumps MCMC Wolpert, Clyde, Tu, et al., 2011
• SIR algorithm Erhardsson, 2008

If the mixing measure is a Gamma ECRM:

• Approximation of the mixing measure + Measure-valued Markov chain

(This talk). The approximation is based on a result from Favaro, Guglielmi, Walker, et al., 2012, allowing to use a slightly modified version of algorithm 8 in Neal,

• 2000. See details in Zacharie Naulet and Eric Barat (2015). “Adaptive

Bayesian nonparametric regression using mixtures of kernels”. In: arXiv preprint arXiv:1504.00476.

Zacharie Naulet BNP estimation for QHT 9th June 2015 26 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 26 / 36

Simulation results

• We observe data Xi|φi, i = 1, . . . , N with density gψ (no noise situation)
• We simulated 1000 observations Xi|φ from the non-noisy model from

1 a Schrödinger cat state, and 2 a Fock state with 1 photon, and 3 a Fock state with 5 photons.

• We modeled ψ as a mixture superposition of coherent states by a

(complex) Gamma ECRM, and we use the algorithm from Naulet and Barat, 2015 with m = 200 particles, α = 1, and F a uniform distribution

• n sufficiently large balls of R2.

Zacharie Naulet BNP estimation for QHT 9th June 2015 27 / 36

Simulation results (continued)

(a) (b) (a). Scatter plot of the simulated observations (b). Wigner transform of the estimated (after 2000 burning iterations and 5000 iterations of the algorithm) wave-function ψ

Zacharie Naulet BNP estimation for QHT 9th June 2015 28 / 36

Simulation results (continued)

In blue, the true Wigner distribution. In red, the BNP estimate with 95% credible bands drawn with shading.

Zacharie Naulet BNP estimation for QHT 9th June 2015 29 / 36

Simulation results (continued)

Zacharie Naulet BNP estimation for QHT 9th June 2015 30 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 30 / 36

Asymptotics of Bayes procedure

The prior presented here is quite natural for QHT, but it is a possible inspiration for other regression problems (direct or indirect).

• Goal. Frequentist validation of the Bayes procedure.

The model(s) Yi|Xi

iid

∼ f (Xi) + ǫi, i = 1, . . . , n where ǫi|σ2 ∼ N(0, σ2) are iid and independent of (Xi)n

i=1. We consider both

cases where (Xi)n

i=1 are fixed, spread uniformly, and known ahead of time

(Fixed design) and (Xi)n

i=1 are iid from distribution PX (Random design). We

assume here that σ is known.

Zacharie Naulet BNP estimation for QHT 9th June 2015 31 / 36

Asymptotics of Bayes procedure : results

See Naulet and Barat, 2015 for details. Uǫ,n(f0) = {f | n−1 n

i=1 |Re(f0(zi) − f (zi)|2 < ǫ2}.

Theorem

Let f0 ∈ L2(Rd, dz), Φ be the coherent states kernel, α > 0, γ(dx) be a probability measure with supp(γ) = Rd × Rd, and assume that there exists C, C ′ > 0 such that, γ

• {(q, p) ∈ Rd × Rd||q|d + |p|d > n}
• ≤ C e−C′n .

Let ν(dxdθdr) = αγ(dx)dθ/(2π)HΓ(dr) be the Lévy measure of a Gamma C-ECRM. Assume f0 is the true response function and σ2 is known.

1 Fixed design. Let P∞

denote the joint distribution of (Yi)∞

i=1 given the covariates

(zi)∞

i=1, then under assumption NRD,

Π(Uc

ǫ,n(f0)|ν, Φ, Y1, . . . , Yn, z1, . . . , zn) → 0

P∞

0 −a.s. 2 Random design. Let P∞

denote the joint distribution of (Yi, Zi)∞

i=1, then under

assumption RD, Π(Uc

ǫ,n(f0)|ν, Φ, (Y1, Z1), . . . , (Yn, Zn)) → 0

P∞

0 −a.s.

Zacharie Naulet BNP estimation for QHT 9th June 2015 32 / 36

Asymptotics of Bayes procedure : results

• A similar result hold for wavelets kernels.
• For wavelets, more can be said about rates of convergence (current

work).

Zacharie Naulet BNP estimation for QHT 9th June 2015 33 / 36

Asymptotics of Bayes procedure : results

• A similar result hold for wavelets kernels.
• For wavelets, more can be said about rates of convergence (current

work).

What about QHT ?

Zacharie Naulet BNP estimation for QHT 9th June 2015 33 / 36

Asymptotics of Bayes procedure : method

Schwartz’s theorem for non iid observations (Ghosal, Van Der Vaart, et al., 2007) : If the following conditions are met for n large enough (K > 0 universal constant) :

• Existence of exponentially consistent test, (φi)n

i=1 such that for all f1 ∈ F with

d(f1, f0) > ǫ, P(n)

f0 φn ≤ e−Knǫ2,

sup

f ∈F,d(f ,f1)<ǫ

P(n)

f

(1 − φn) ≤ e−Knǫ2,

• Existence of sets Fn ⊂ F such that,

Π(F\Fn) ≤ e−Knǫ2 sup

ǫ>ǫn

log N(ǫ, Fn, d) ≤ nǫ2

n

• Prior positivity of Kullback-Leibler neighborhoods

Π

• f ∈ Fn
• Pf0 log dPf0

dPf < ǫ2

n,

• Pf0
• log dPf0

dPf 2 < ǫ2

n

• ≥ e−Knǫ2

n,

Then, for every Mn → ∞, Π({f ∈ F|d(f , f0) > Mnǫn}|(X1, Y1), . . . , (Xn, Yn)) → 0 P∞

f0 -a.s.

Zacharie Naulet BNP estimation for QHT 9th June 2015 33 / 36

Asymptotics of Bayes procedure : QHT ?

The difficult parts are :

• Prior positivity of Kullback-Leibler neighborhoods
• Existence of tests with sufficiently rapid decreasing Type I and Type II

errors. First item is about to be solved for QHT (very fresh result, to take with caution !). Second item is still to be investigated at the moment.

Zacharie Naulet BNP estimation for QHT 9th June 2015 34 / 36

Outline

1 Quantum Homodyne Tomography Quantum Mechanics : Short introduction Quantum Homodyne Tomography Canonical examples of quantum states 2 BNP estimate Mixtures models Square integrable representations and coherent states Completely random measures Summary and sampling Simulation results 3 Asymptotics results ? 4 Conclusion

Zacharie Naulet BNP estimation for QHT 9th June 2015 34 / 36

... one last thing before concluding

We focused the presentation on estimating pure states.

Zacharie Naulet BNP estimation for QHT 9th June 2015 35 / 36

... one last thing before concluding

We focused the presentation on estimating pure states. But, the space of states of is convex, and according to the Hilbert-Schmidt theorem on the canonical decomposition for compact self-adjoint operators, every state ρ can be decomposed as ρ =

N

• k=1

ckPψk,

N

• k=1

ck = 1, with eventually N = ∞, where cn > 0 are non-zero eigenvalues of ρ and {ψk}N

k=1 is an orthonormal set.

Zacharie Naulet BNP estimation for QHT 9th June 2015 35 / 36

... one last thing before concluding

We focused the presentation on estimating pure states. But, the space of states of is convex, and according to the Hilbert-Schmidt theorem on the canonical decomposition for compact self-adjoint operators, every state ρ can be decomposed as ρ =

N

• k=1

ckPψk,

N

• k=1

ck = 1, with eventually N = ∞, where cn > 0 are non-zero eigenvalues of ρ and {ψk}N

k=1 is an orthonormal set.

Thus, the problem of estimating a general state can be solved using a nonparametic mixture of the prior presented in this talk (for example, a Dirichlet Process Mixture of Complex-ECRM mixtures).

Zacharie Naulet BNP estimation for QHT 9th June 2015 35 / 36

Conclusion

Thank You.

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Appendix

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Dirichlet Process

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Pólya urn

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Chinese Restaurant Process

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Poisson Random Measures

Let,

• (E, E) be a measurable, locally compact and separable space.
• ν be a σ-finite measure on (E, E).

A random measure N : Ω × E → N ∪ {+∞} is a Poisson random measure with mean ν if

1 For each A ∈ E, N(·, A) ∼ Po(ν(A)) 2 If A1, . . . , An are pairwise disjoint sets in E then N(·, A1), . . . , N(·, An)

are independent Poisson distributed random variables. When ν(E) < +∞, there is a convenient way to construct and interpret

• PRM. Indeed, start with the probability measure π(·) = ν(·)/ν(E) on (E, E)

and let K ∼ Po(ν(E)), Xk

iid

∼ π(·) for 1 ≤ k ≤ K. From this sample form the following measure, N(A) =

K

• i=1

✶A(Xi), A ∈ E. (1)

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Poisson Random Measures

• When ν(E) = +∞ we can find a disjoint partition {Ei}∞

i=1 of E such

that ν(Ei) < +∞ for all i = 1, . . . , ∞.

• Define Ni the PRM with mean ν(Ei) on the subset Ei, then Ni is

almost-surely purely atomic.

• Let N(A) = ∞

i=1 Ni(A ∩ Ei) for all A ∈ E.

• As Ni(A ∩ Ei) are independent Poisson random variables with means

ν(Ei), it follows that N(A) is a Poisson random variable with mean ν(E) = +∞ and hence N is a Poisson random measure with mean ν. At the end, N(A) =

• i=I

✶A(Xi), A ∈ E. is a Poisson Random Measure with mean ν.

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Favaro’s result

Theorem (Favaro, Guglielmi, Walker, et al., 2012)

For any n > 0, define Qn =

1

n

i=1 ξi

n

i=1 ξiδzi, where

ξ1, . . . , ξn

iid

∼ Gamma(1, 1). Independently let zi = (θi, xi) ∈ [0, 2π] × G and (z1, . . . , zn) be a Pólya urn sequence with parameter α and base distribution PU × F. Then Qn → Q almost-surely, where Q ∼ DP(α, PU × F).

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Posterior computations

From the theorem of Favaro, Guglielmi, Walker, et al., 2012, the approximated model fm(·) = T

• G

Φ(x; ·)eiθ dQm(x, θ) Qm = 1 n

i=1 ξi m

• i=1

ξiδzi T

ind

∼ Gamma(α, η) ξi, . . . , ξm

iid

∼ Gamma(1, 1),

• ind. of (z1, . . . , zm)

(z1, . . . , zm) ∼ Pólya(α, PU × F), satisfies,

• Rd |fm(z) − f (z)|dz −

→ 0 a.s., where f has the distribution of the random function with mixing measure the complex Gamma ECRM with base distribution F.

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

Posterior computations : algorithm

Fix m ∈ N large enough, and Let K = {K1, . . . , Km} s.t. Ki = k iff zi = Zk = (x⋆

k , θ⋆ k) where Z = Z1, . . .

stand for unique values of z(m). At each iteration, successively sample from :

1 (Ki|K−i, y, Z, ξ, T) for 1 ≤ i ≤ n: let nk,i = #1≤l≤n l=i

{Kl = k}, κ(n) the number of distinct Zk values and κ0 a chosen natural, Ki

ind

∼

κ(m)

• k=1

nk,i Lk,i(Z, ξ, T, y) δk(·) + α κ0

κ0

• k=1

Lk+κ(m),i(Z, ξ, T, y) δk+κ(m)(·), where Lk,i(Z, ξ, T, y) stands for the likelihood under hypothesis that particle i is allocated to component k.

2 (Z|K, y, ξ, T): Random Walk Metropolis Hastings on parameters. 3 (ξi|ξ−i, K, y, Z, T) for 1 ≤ i ≤ m: Independent Metropolis Hastings with

prior Gamma(1, 1) taken as i.i.d. candidate distribution for ξi.

4 (T|K, y, Z, ξ): Random Walk Metropolis Hastings on scale parameter.

Zacharie Naulet BNP estimation for QHT 9th June 2015 36 / 36

References I

Abramovich, F, T Sapatinas, and BW Silverman (2000). “Stochastic expansions in an overcomplete wavelet dictionary”. In: Probability Theory and Related Fields 117.1, pp. 133–144 (cit. on p. 27). Butucea, Cristina, Mădălin Guţă, and Luis Artiles (2007). “Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data”. In: The Annals of Statistics 35.2,

• pp. 465–494 (cit. on p. 20).

Erhardsson, Torkel (2008). “Non-parametric Bayesian Inference for Integrals with respect to an Unknown Finite Measure”. In: Scandinavian Journal of Statistics 35.2, pp. 369–384 (cit. on p. 52). Escobar, Michael D and Mike West (1995). “Bayesian density estimation and inference using mixtures”. In: Journal of the american statistical association 90.430, pp. 577–588 (cit. on p. 27). Favaro, Stefano, Alessandra Guglielmi, Stephen G Walker, et al. (2012). “A class of measure-valued Markov Chains and Bayesian Nonparametrics”. In: Bernoulli 18.3, pp. 1002–1030 (cit. on pp. 52, 76, 77).

Zacharie Naulet BNP estimation for QHT 9th June 2015 37 / 36

References II

Ghosal, Subhashis, Aad Van Der Vaart, et al. (2007). “Convergence rates of posterior distributions for noniid observations”. In: The Annals

• f Statistics 35.1, pp. 192–223 (cit. on p. 63).

Grangier, Philippe (2011). “Make It Quantum and Continuous”. In: Science 332.6027, pp. 313–314 (cit. on p. 22). Grossmann, Alex, Jean Morlet, and T Paul (1985). “Transforms associated to square integrable group representations. I. General results”. In: Journal of Mathematical Physics 26.10, pp. 2473–2479 (cit. on

• p. 32).

Kingman, JFC (1967a). “Completely random measures”. In: Pacific Journal of Mathematics 21.1 (cit. on p. 29). Kingman, John (1967b). “Completely random measures”. In: Pacific Journal of Mathematics 21.1, pp. 59–78 (cit. on pp. 38–42). Kingman, John Frank Charles (1992). Poisson processes. Vol. 3. Oxford university press (cit. on p. 29). Leonhardt, Ulf and H Paul (1995). “Measuring the quantum state of light”. In: Progress in quantum electronics 19.2, pp. 89–130 (cit. on

• p. 20).

Zacharie Naulet BNP estimation for QHT 9th June 2015 38 / 36

References III

Naulet, Zacharie and Eric Barat (2015). “Adaptive Bayesian nonparametric regression using mixtures of kernels”. In: arXiv preprint arXiv:1504.00476 (cit. on pp. 27, 29, 52, 54, 60). Neal, Radford M (2000). “Markov chain sampling methods for Dirichlet process mixture models”. In: Journal of computational and graphical statistics 9.2, pp. 249–265 (cit. on p. 52). Pillai, Natesh S (2008). “Lévy random measures: Posterior consistency and applications”. PhD thesis. Duke University (cit. on pp. 27, 29). Pillai, Natesh S et al. (2007). “Characterizing the function space for Bayesian kernel models”. In: Journal of Machine Learning Research 8,

• pp. 1769–1797 (cit. on p. 27).

Rajput, Balram S and Jan Rosinski (1989). “Spectral representations of infinitely divisible processes”. In: Probability Theory and Related Fields 82.3, pp. 451–487 (cit. on p. 29). Rasmussen, Carl Edward (2004). “Gaussian processes in machine learning”. In: Advanced Lectures on Machine Learning. Springer,

• pp. 63–71 (cit. on p. 27).

Zacharie Naulet BNP estimation for QHT 9th June 2015 39 / 36

References IV

Wolpert, Robert L, Merlise A Clyde, Chong Tu, et al. (2011). “Stochastic expansions using continuous dictionaries: Lévy adaptive regression kernels”. In: The Annals of Statistics 39.4, pp. 1916–1962 (cit. on pp. 27, 29, 52).

Zacharie Naulet BNP estimation for QHT 9th June 2015 40 / 36