 Trace-adjoint identity for partial transpose

For bipartite operators $X,Y$ in a fixed product basis,Expand both operators in products of matrix units. On the transposed factor, the identity is $\mathrm{Tr}(AT)=\mathrm{Tr}(A^T{T}^T)$; bilinearity gives the bipartite result. It allows a negative partial transpose eigenvector to define an entanglement witness.