Tensor-product operator

For linear operators $A$ on $V$ and $B$ on $W$, the tensor product operator is characterized byFor bounded linear operators on Hilbert spaces, it extends continuously to the Hilbert tensor product. If both factors are positive semidefinite operators, so is their tensor product operator. In finite dimensions an orthonormal eigenbasis in each factor gives a tensor-product basis with nonnegative eigenvalues $a_i{b}_j$.

The positivity assertion also holds without an eigenbasis. Take the positive square root of an operator of each factor and put $C=A^{1/2}\otimes B^{1/2}$. Then $A\otimes B=C^{†}C$, so its quadratic form is $\Vert Cz{\Vert }^2\ge 0$. This is the positivity used in the necessary direction of the positive partial transpose criterion, where each product term in a separable density matrix remains positive after transposing one factor.