Bayesian nonparametric estimation for Quantum Homodyne Tomography

Zacharie Naulet (1) , Eric Barat (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LM2S (CEA, LIST) - Laboratoire Modélisation et Simulation de Systèmes (CEA, LIST)

We estimate the quantum state of a light beam from results of quantum homodyne tomography noisy measurements performed on identically prepared quantum systems. We propose two Bayesian nonparametric approaches. The first approach is based on mixture models and is illustrated through simulation examples. The second approach is based on random basis expansions. We study the theoretical performance of the second approach by quantifying the rate of contraction of the posterior distribution around the true quantum state in the L-2 metric.

Mots clés: Bayesian nonparametric estimation; inverse problem; nonparametric estimation; quantum homodyne tomography; Radon transform; Wigner distribution; mixture prior; Wilson bases; rate of contraction; NOISY DATA; POSTERIOR DISTRIBUTIONS; CONVERGENCE-RATES; MODULATION SPACES; WIGNER FUNCTION; MINIMAX; and phrases: Bayesian nonparametric estimation; inverse prob- lem; rate of con- traction

Domaines: Traitement du signal et de l'image [eess.SP]; Physique Quantique [quant-ph]; Analyse numérique [math.NA]; Applications [stat.AP]

Zacharie Naulet, Eric Barat. Bayesian nonparametric estimation for Quantum Homodyne Tomography. Electronic Journal of Statistics , 2017, 11 (2), pp.3595 - 3632. ⟨10.1214/17-EJS1322⟩. ⟨hal-01881075⟩