The possibility of constructing a complete, continuous Wigner function for any quantum system has been a subject of investigation for over 50 years. A key system that has served to illustrate the difficulties of this problem has been an ensemble of spins. Here we present a general and consistent framework for constructing Wigner functions exploiting the underlying symmetries in the physical system at hand. The Wigner function can be used to fully describe any quantum system of arbitrary dimension or ensemble size.

In a previous paper [1] an Euler angle parametrization for SU (4) was given. Here we present the derivation of a generalized Euler angle parametrization for SU (N ). The formula for the calculation of the Haar measure for SU (N ) as well as its relation to Marinov's volume formula for SU (N ) [2, 3] will also be derived. As an example of this parametrization's usefulness, the density matrix parametrization and invariant volume element for a qubit/qutrit, three qubit, and two three-state systems, also known as two qutrit systems [4], will also be given. *

In this paper we give an explicit parametrization for all two qubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU (4) and all possible two qubit density matrices. The important group-theoretical properties of such a description are then manifest. We thus obtain the correct Haar (Hurwitz) measure and volume element for SU (4) which follows from this parametrization. In addition, we study the role of this parametrization in the Peres-Horodecki criteria for separability and its corresponding usefulness in calculating entangled two qubit states as represented through the parametrization. *

It has recently been shown that it is possible to represent the complete quantum state of any system as a phase-space quasi-probability distribution (Wigner function) [Phys Rev Lett 117, 180401]. Such functions take the form of expectation values of an observable that has a direct analogy to displaced parity operators. In this work we give a procedure for the measurement of the Wigner function that should be applicable to any quantum system. We have applied our procedure to IBM's Quantum Experience five-qubit quantum processor to demonstrate that we can measure and generate the Wigner functions of two different Bell states as well as the five-qubit Greenberger-Horne-Zeilinger (GHZ) state. As Wigner functions for spin systems are not unique, we define, compare, and contrast two distinct examples. We show how using these Wigner functions leads to an optimal method for quantum state analysis especially in the situation where specific characteristic features are of particular interest (such as for spin Schrödinger cat states). Furthermore we show that this analysis leads to straightforward, and potentially very efficient, entanglement test and state characterisation methods.