New characterization of two-state normal distribution

In this article we give a purely noncommutative criterion for the characterization of two-state normal distribution. We prove that families of two-state normal distribution can be described by relations which is similar to the conditional expectation in free probability, but has no classical analogue. We also show a generalization of Bozejko, Leinert and Speicher's formula (relating moments and noncommutative cumulants).