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 Measurement Result of the measurement is random : X = x 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 Measurement Result of the measurement is random : X = x The quantum state of a system encodes the probabilities of its measurable ”observables”. That is, the probability of obtaining each of the possible outcomes 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 operator) ρ on a complex Hilbert space H (called the space of states) s.t.: Self adjoint: ρ ∗ = ρ , 1 Positif: � ψ, ρψ � ≥ 0, for all ψ ∈ H , 2 Trace 1: Tr( ρ ) = 1. 3 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 = L 2 ( R ) = | f ( x ) | 2 dx < ∞ f : R → C , • A particular orthonormal basis : the Fock basis 1 : 1 H n ( x ) e − x 2 / 2 . ψ n ( x ) = � √ π 2 n n ! • In the Fock basis, a state is described by an infinite density matrix ρ = [ ρ j , k ] j , k ∈ N called a density matrix . 1 H n ( x )= n th 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 dim H � X = x a P a , a where • the eigenvalues { x a } a of the observable X are real, • P a is the projection onto the one dimensional space generated by the eigenvector of X corresponding to the eigenvalue x a . 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 { x a } a of the observable X s.t.: • the probability distribution is P ρ ( X = x a ) = Tr ( P a ρ ) , • the expectation function E ρ ( X ) = Tr ( X ρ ) . 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 oscillator. 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 observables X φ := 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 I 2 additional laser of high intensity | z | >> 1 ( local oscillator (LO)). detector I 1 − I 2 √ 2 η | z | ∼ p η ρ ( x | φ ) vacuum 2 • The phase Φ of the LO is choosen s.t. Φ ∼ U [0 , π ] . I 1 vacuum 1 • Split the mixing in 2 beams and each signal beam is measured by a photodetector beam splitter detector which gives reps. integrated currents I 1 and I 2 proportional to the number of local photons. oscilator z = | z | e iφ • Result of the measurement of X φ is a r.v. X | Φ = φ of probability density p ρ ( ·| φ ). 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 X φ is a r.v. X | Φ = φ of probability density p ρ ( ·| φ ) s.t.: F 1 [ p ρ ( ·| φ ]( t ) = Tr ( ρ e it X φ ) := � W ρ ( t cos φ, t sin φ ) , where W ρ is the Wigner function, an equivalent representation for a quantum state ρ �� W ρ : R 2 → R and W ρ ( q , p ) dqdp = 1 W ρ plays the role of a ”quasi-probability density” of ( Q , P ). Its Radon transform is always a probability density and is s.t. : � ∞ p ρ ( x | φ ) := ℜ [ W ρ ]( x , φ ) = W ρ ( x cos φ + t sin φ, x sin φ − t cos φ ) dt . −∞ Therefore the result of an ideal measurement : ( X , Φ) ∼ p ρ ( x , φ ) = 1 π ℜ [ W ρ ]( 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 • W ρ Wigner function Observable ↔ X φ := Q cos φ + P sin φ Φ ∼ U [0 , π ] Result of an ideal ↔ r.v. ( X , Φ) of prob. density p ρ ( x , φ ) = 1 measurement by QHT π ℜ [ W ρ ]( 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. ( Z ℓ , Φ ℓ ) ℓ =1 ,..., n i.i.d. such that � Z ℓ = X ℓ + (1 − η ) / (2 η ) ξ ℓ • 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 ) / (4 η ) ∈ [0 , 1 / 4[ and the ideal setting : γ = 0. 2 Note 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. ( Z ℓ , Φ ℓ ) ℓ =1 ,..., n i.i.d. such that Z ℓ = X ℓ + √ 2 γ ξ ℓ of probability density: � 1 � p γ π ℜ [ W ρ ]( · , φ ) ∗ N γ ρ ( z , φ ) = ( z ) 1 l [0 ,π ] ( φ ) , N γ ( · ) = N (0 , 2 γ ) and γ := (1 − η ) / (4 η ) ∈ [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. ( Z ℓ , Φ ℓ ) ℓ =1 ,..., n i.i.d. such that Z ℓ = X ℓ + √ 2 γ ξ ℓ of probability density: � 1 � p γ π ℜ [ W ρ ]( · , φ ) ∗ N γ ρ ( z , φ ) = ( z ) 1 l [0 ,π ] ( φ ) , N γ ( · ) = N (0 , 2 γ ) and γ := (1 − η ) / (4 η ) ∈ [0 , 1 / 4[ and the ideal setting : γ = 0. Goal: Reconstruct the quantum state from ( Z ℓ , Φ ℓ ) 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 1 1 0.2 0.8 0.8 0.15 0.6 0.6 0.1 0.4 0.4 0.05 0.2 0.2 0 0 0 0 0 0 5 5 5 20 20 20 10 10 10 15 15 15 15 15 15 10 10 10 20 20 20 5 5 5 25 25 25 1 1 π e − q 2 − p 2 . π e − ( q − a ) 2 − p 2 . 1 π (2 q 2 + 2 p 2 + 1) e − q 2 − p 2 . Lounici, Meziani and Peyr´ e Wigner function estimation in QHT
Introduction to quantum mechanics Statistical part Main result A realistic class of quantum states 3 For C ≥ 1, B > 0 and 0 < r ≤ 2, R ( C , B , r ) := { ρ quantum state : | ρ j , k | ≤ C exp( − B ( j + k ) r / 2 ) } . In [Aubry, Butucea and M.(2009)], these decrease condition on ρ j , k has been translated on the corresponding Wigner function : W ρ . 3 The 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
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