What information about a wave function is given by ``measuring'' it with the help of the tomographic method of recovering the Wigner distribution?
The article concerns the theory of measurement in non-relativistic quantum mechanics. The tomographic method has been developed by physicists in order to recover the Wigner distribution of a quantum state from projections. The authors give arguments indicating that mathematical difficulties cannot be escaped in this approach. By solving this inverse problem, arbitrary small errors in the measured values of marginal distributions may give rise to arbitrary large errors in the wave function. These distributions being obtained on a finite number of points; this fact introduces errors when Radon theorem is applied. Furthermore, the compactness hypothesis which is used by the numerical algorithms for inversion is not compatible with Paley-Wiener theorem. For that reason, the authors propose to solve this inversion problem from a well posed mathematical point of view. In some sense, they succeed partly to do that by considering a smoothing procedure of the wave function by Gaussian distribution. The correspondence of projections and these smoothed objects can be derived with mathematical rigor, establishing connections in the estimates between the number of points of evaluations of projections and parameters of the smoothed procedure. This is the main result of the paper.
scientific article; zbMATH DE number 1300658