operate on a classical probabilistic bit. Such a binary system is described by a two-dimensional vector of probabilities, corresponding to the probability p0 of being in 0 and p1 of being in 1. The operations I and X can then be represented on this system as the two-by-two matrices In group theoretic parlance, we say that these two matrices form a representation of the group, which effectively means that the multiplication among these matrices mimics the operation among the elements of the group that is being represented.