We describe a recently developed generalisation of the Poincar ′ e sphere method, to represent pure states of a three-level quantum system in a convenient geometrical manner. The construction depends on the properties of the group SU (3) and its generators in the defining representation, and uses geometrical objects and operations in an eight dimensional real Euclidean space. This construction is then used to develop a generalisation of the well known Pancharatnam geometric phase formula, for evolution of a three-level system along a geodesic triangle in state space.

We show that every Mueller matrix, that is a real 4 x 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L2KLl where L1 and L2 are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-I1 Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through : he G eigenvalues and G eigenvectors of the real symmetric N matrix N = MGM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.

No abstract

Canonical forms of two-qubits under the action of stochastic local operations and classical communications (SLOCC) offer great insight for understanding non-locality and entanglement shared by them. They also enable geometric picture of two-qubit states within the Bloch ball. It has been shown (Verstraete et.al. (Phys. Rev. A 64, 010101(R) ( 2001)) that an arbitrary two-qubit state gets transformed under SLOCC into one of the two different canonical forms. One of these happens to be the Bell diagonal form of two-qubit states and the other non-diagonal canonical form is obtained for a family of rank deficient two-qubit states. The method employed by Verstraete et.al. required highly non-trivial results on matrix decompositions in n dimensional spaces with indefinite metric.Here we employ an entirely different approach -inspired by the methods developed by Rao et. al., (J. Mod. Opt. 45, 955 (1998)) in classical polarization optics -which leads naturally towards the identification of two inequivalent SLOCC invariant canonical forms for two-qubit states. In addition, our approach results in a simple geometric visualization of two-qubit states in terms of their SLOCC canonical forms.