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, $(AB)^{Gamma} eq A^{Gamma}B^{Gamma}$ 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.