On a matrix equality involving partial transposition and its relation to the separability problem

In matrix theory, a well established relation $(AB)^{T}=B^{T}A^{T}$ holds for any two matrices $A$ and $B$ for which the product $AB$ is defined. Here $T$ denote the usual transposition. In this work, we explore the possibility of deriving the matrix equality $(AB)^{\Gamma}=A^{\Gamma}B^{\Gamma}$ for any $4 \times 4$ matrices $A$ and $B$, where $\Gamma$ denote the partial transposition. We found that, in general, $(AB)^{\Gamma}\neq A^{\Gamma}B^{\Gamma}$ holds for $4 \times 4$ matrices $A$ and $B$ but there exist particular set of $4 \times 4$ matrices for which $(AB)^{\Gamma}= A^{\Gamma}B^{\Gamma}$ holds. We have exploited this matrix equality to investigate the separability problem. Since it is possible to decompose the density matrices $\rho$ into two positive semi-definite matrices $A$ and $B$ so we are able to derive the separability condition for $\rho$ when $\rho^{\Gamma}=(AB)^{\Gamma}=A^{\Gamma}B^{\Gamma}$ holds. Due to the non-uniqueness property of the decomposition of the density matrix into two positive semi-definte matrices $A$ and $B$, there is a possibility to generalise the matrix equality for density matrices lives in higher dimension. These results may help in studying the separability problem for higher dimensional and multipartite system.