ar X iv : m at h - ph / 0 20 20 02 v 3 1 6 A pr 2 00 2 SU ( 4 ) Euler Angle Parameterization and Bipartite Density Matrices

In quantum mechanics, sets of density matrices are important for numerous reasons. For example, their compact notation make them useful for describing decoherence and entanglement properties of multi-particle quantum systems. In particular, two two-state density matrices, otherwise known as two qubit density matrices, are important for their role in explaining quantum teleportation, dense coding, and computation theorems. The aim of this paper is to show an explicit parameterization for the Hilbert space of all two qubit density matrices. Such a parameterization would be extremely useful for numerical calculations concerning entanglement and other quantum information parameters. We would also like to know the properties of such parameterized two qubit density matrices; in particular, the representation of their convex sets, their sub-sets, and their set boundaries in terms of our parameterization. Here we present a generalized Euler angle parameterization for SU(4) and all possible two qubit density matrices as well as the corrected Haar Measures for SU(3) and SU(4) from this parameterization. The role of the parameterization in the Peres-Horodecki criteria will also be introduced as well as its usefulness in calculating entangled two qubit states. ∗[email protected] †Present address: Lash Miller Chemical Laboratories, 80 St. George Street, University of Toronto, Toronto, Ontario, CANADA M5S 3H6. Email: [email protected] ‡[email protected]