Discrete Wigner Function Reconstruction and Compressed Sensing

Wigner function(WF), a quasi-probability distribution in phase space, was first introduced to describe quantum state in quantum mechanics by E. P. Wigner[1]. And later, it was extended to classical optics and signal processing. Since its birth, a great number of applications have been conducted in different fields. As for the original quantum case, like a continuous onedimensional quantum system, the Wigner function is defined as