 Partial trace in a nonorthogonal product expansion

The vectors need not form orthonormal bases. With the inner product linear in its second entry, expand the rank-one operator $|\Psi ⟩⟨\Psi |$ and take the trace on the second factor. This gives the displayed reduced positive operator. For every observable $A$ on the first factor,Positivity also follows directly from $⟨v,S_{1}v⟩=\Vert {\sum }_j⟨v,{\xi }_j⟩{\eta }_j{\Vert }^2$. Thus the physical density operator is $S_1/\Vert \Psi {\Vert }^2$ when the nonzero source vector is not normalized. The Gram matrix of the second-factor vectors supplies the cross coefficients; replacing those coefficients by Kronecker deltas without checking orthonormality changes the subsystem state. This formula is useful when an efficient product expansion uses nonorthogonal vectors, even though the partial trace itself is basis-independent.