The Discrete Phase Space For 3-Qubit And 2-Qutrit Systems Based On Galois Field

Abstract

Generally, quantum states are abstract states that carry probabilistic information of position and momentum of any dynamical physical quantity in quantum system. E.P.Wigner (1932) had introduced a function that can determine the combination of position and momentum simultaneously, and it was the starting point to define a phase space probability distribution for a quantum mechanical system using density matrix formalism. This function named as Wigner Function. Recently, Wootters (1987) has developed a discrete phase space analogous to Wigner’s ideas. The space is based on Galois field or finite field. The geometry of the space is represented by N ´ N point, where N denoted the number of elements in the field and it must be a prime or a power of a prime numbers. In this work, we study the simplest way to compute the binary operations in finite field in order to form such a discrete space. We developed a program using Mathematica software to solve the binary operation in the finite field for the case of 3-qubit and 2-qutrit systems. The program developed should also be extendible for the higher number of qubit and qutrit. Each state is defined by a line aq + bp = c and parallel lines give equivalent states. The results show that, there are 9 set of parallel lines for the 3-qubit system and 10 sets of parallel lines for 2-qutrit system. These complete set of parallel lines called a ‘striation’.