Frame representations of qudit quantum mechanics

Abstract

There exist many attempts to define a Wigner function for qudits, each of them coming with its advantages and limitations. The existing finite versions have simple definitions, but they are artificial in their construction and do not allow an intuitive state analysis. The continuous versions have more complicated definitions, but they are similar to the original Wigner function and allow a visualization of the quantum states. The version based on the concept of tight frame we present is finite, but it has certain properties and applications similar to those of continuous versions. Based on the frame representation, we present several graphical representations of qubit states, and define two new parameters concerning them. We show that, from a mathematical point of view, the qubit is the orthogonal projection of qutrit.