Frame representation of quantum systems with finite-dimensional Hilbert space

Abstract

There exist many attempts to define a Wigner function for quantum systems with finite-dimensional Hilbert space, each of them coming with its advantages and limitations. The existing finite versions have simple definitions, but they are based only on the existence of a formal analogy with the continuous-variable Wigner function and do not allow an intuitive state analysis. The continuous versions have more complicated definitions, but they are closer 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. It allows us to present a new graphical representation of qubit states, and to define new parameters concerning them. An important advantage of frame representation follows from the use of redundant information. The values taken by the frame version of Wigner function are not independent. They have to satisfy a large number of mathematical relations, useful in error detection and correction.