We consider the inverse (ill-posed) problem of Quantum Homodyne Tomography (see Artilès, 2005) from a fully bayesian point of view. In our context, functions of interest are complex valued wave-functions belonging to L2(R). We use an approach inspired by the stochastic expansion over continuous dictionnary introduced by Abramovith et al. (2000) and generalized later by Wolpert et al. (2011). The basic idea is to represent the function of interest as a weighted sum of building blocks, generally a kernel function with arbitrary parameters. We propose a method relying on group representations to build efficient kernels, allowing us to have expansions dense in large families of well-known Banach spaces. Hence our model could be useful for many other applications. As noticed by Wolpert et al., for computational purpose, an approximation is needed concerning the prior, because no one-to-one link between the components of the expansion and the observated data is possible. They propose a RJMCMC algorithm relying on the truncation of components whose weights are lower than a specified threshold, with general Lévy random fields as a prior. In the case of Lévy random fields built from the Gamma process, we propose an inference scheme based on the Dirichlet process (up to a Gamma distributed scale), for which a bunch of efficient algorithms for posterior sampling have been proposed (see Neal 2000, Ishwaran & James 2001). Concerning the approximation, instead of specifying a threshold for small components we believe the particle approximation of Favaro et al. (2012) might be a better candidate. Going back to our initial problem, preliminary results shows that our model does pretty good job, even in what we believe being the worst case, that is wave-functions being Hermite polynomials.

This is a joint work with Eric Barat, Judith Rousseau and Trong T. Truong