Relation of quasiprobabilities to Bargmann representation of states

A relation of the Wigner quasiprobability to the Bargmann representation of pure and mixed states by convolution is derived and generalized to the main class of quasiprobabilities and its inversion is given. The derivation uses a realization of the abstract group SU(1, 1) by second-order differentiation and multiplication operators for a pair of complex conjugated variables and disentanglement of exponential functions of these operators by group-theoretical methods. Examples for the calculation of the Wigner quasiprobability via the Bargmann representation of states demonstrate the action of this relation. A short collection of different basic representations of the Wigner quasiprobability is given. An appendix presents results for the disentanglement of SU(1, 1)-group operators by products of special operators in different ordering.