It's difficult to follow. IIUC they are using the density matrix D, and calling the coefficients "probabilities". Only the diagonal has an interpretation as probabilities, the other parts are more difficult to interpret and can be negative (and are not probabilities).
We shall argue in this paper that a central piece of modern physics does not really belong to physics at all but to elementary probability theory... When we extend probability theory by allowing cases to count negatively, we find that the Hilbert space framework of quantum mechanics proper emerges...