Trivariate analysis of two qubit symmetric separable state

Abstract

One of the main problems of quantum information theory is developing the separability criterion which is both necessary and sufficient in nature. Positive partial transposition test (PPT) is one such criterion which is both necessary and sufficient for $2\times 2$ and $2\times 3$ systems but not otherwise. We express this strong PPT criterion for a system of 2-qubit symmetric states in terms of the well-known Fano statistical tensor parameters and prove that a large set of separable symmetric states are characterised by real statistical tensor parameters only. The physical importance of these states are brought out by employing trivariate representation of density matrix wherein the components of ${\mathbf{J}}$, namely $J_{x}$, $J_{y}$, $J_{z}$ are considered to be the three variates. We prove that this set of separable states is characterised by the vanishing average expectation value of $J_{y}$ and its covariance with $J_{x}$ and $J_{z}$. This allows us to identify a symmetric separable state easily. We illustrate our criterion using 2-qubit system as an example.