Irreducible Lie su(2) algebra representations in terms of probability distribution functions

Abstract

Using spin-1/2 state (qubit state) description by the conditional probability distributions of dichotomic random variables, we construct the irreducible Lie su(2) algebra representations by means of the probability distribution functions. We obtain explicit 2×2-matrices of the Lie algebra elements and express them in terms of Pauli matrices. Then we discuss the possibility to obtain the probability representation of arbitrary Lie algebra, using the symmetry Lie group of quantum system states. Finally, we write Pauli matrices in the probability representation of dichotomic random variables.