Canonical forms of two-qubit states under local operations

TL;DR

This paper introduces a new, simpler method for deriving canonical forms of two-qubit states under SLOCC, providing clearer geometric visualization and understanding of entanglement.

Contribution

It presents an alternative approach inspired by classical polarization optics to identify two inequivalent SLOCC invariant canonical forms for two-qubit states.

Findings

Identifies two SLOCC invariant canonical forms for two-qubit states.

Provides a simple geometric visualization of two-qubit states.

Offers a more straightforward method compared to previous matrix decomposition techniques.

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 {\em 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.