Generating mutually unbiased bases and discrete wigner function for three-qubit system - Aliakbarzadeh, Mojtaba (2011)
In this research, we construct Wigner functions on discrete phase spaces to represent quantum states for the special case of 3-qubit system. For determining this discrete phase space, we label the axes of phase space with finite field (Galois field) having eight elements. Based on this labeling, we developed a program in Matlab software for 3-qubit system which is also extendible for higher dimensions and more number of cubits. Results of this program lead to nine sets of parallel lines which is named as striations. Equivalently we label the horizontal and vertical axes of our phase space by quantum states of 3 cubits. This labeling and the use of suitable translation vectors on our striations produce the nine mutually unbiased bases for the Hilbert space. For calculating discrete Wigner function, we have determined appropriate quantum net Q(¸). There are 89 different choices for defining quantum nets but by using some unitary operators we reduce our choices to just eight different choices of quantum nets (eight similarity classes). We developed another program in Maple, which gives us eight different similarity classes of quantum nets. This program also gives us the phase-space point operators A® related to each class of quantum nets (these A® operators are used in calculating Wigner function). Based on one of these similarity classes, we calculated the Wigner functions for GHZ and W states and also an embedded Bell state which is not completely entangled. We discussed some of the properties of these Wigner functions