Canonical forms of two-qubit states under local operations

Abstract

Canonical forms of two-qubits under the action of stochastic local operations and classical communications (SLOCC) offer great insight for understanding nonlocality and entanglement shared by them. They also enable geometric picture of two-qubit states within the Bloch ball. It has been shown [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 a nondiagonal canonical form is obtained for a family of rank deficient two-qubit states. The method employed by Verstraete et al. [Phys. Rev. A 64, 010101(R) (2001)] required highly nontrivial results on matrix decompositions in $n$-dimensional spaces with an 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.