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In matrix theory, a well established relation $(AB)^{T}=B^{T}A^{T}$ holds forany 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 ofderiving the matrix equality $(AB)^{\Gamma}=A^{\Gamma}B^{\Gamma}$ for any $4\times 4$ matrices $A$ and $B$, where $\Gamma$ denote the partialtransposition. We found that, in general, $(AB)^{\Gamma}\neqA^{\Gamma}B^{\Gamma}$ holds for $4 \times 4$ matrices $A$ and $B$ but thereexist particular set of $4 \times 4$ matrices for which $(AB)^{\Gamma}=A^{\Gamma}B^{\Gamma}$ holds. We have exploited this matrix equality toinvestigate the separability problem. Since it is possible to decompose thedensity matrices $\rho$ into two positive semi-definite matrices $A$ and $B$ sowe are able to derive the separability condition for $\rho$ when$\rho^{\Gamma}=(AB)^{\Gamma}=A^{\Gamma}B^{\Gamma}$ holds. Due to thenon-uniqueness property of the decomposition of the density matrix into twopositive semi-definte matrices $A$ and $B$, there is a possibility togeneralise the matrix equality for density matrices lives in higher dimension.These results may help in studying the separability problem for higherdimensional and multipartite system.

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