  Tensor product operators on finite dimensional Hilbert spaces are studied.
The focus is on bilinear tensor product operators. A tensor product operator on
a pair of Hilbert spaces is a maximally general bilinear operator into a target
Hilbert space. By 'maximally general' is meant every bilinear operator from the
same pair of spaces to any Hilbert space factors into the composition of the
tensor product operator with a uniquely determined linear mapping on the target
space. There are multiple distinct tensor product operators of the same type;
there is no "the" tensor product. Distinctly different tensor product operators
can be associated with different parts of a multipartite system without
difficulty. Separability of states, and locality of operators and observables
is tensor product operator dependent. The same state in the target state space
can be inseparable with respect to one tensor product operator but separable
with respect to another, and no tensor product operator is distinguished
relative to the others; the unitary operator used to construct a Bell state
from a pair of |0>'s being highly tensor product operator-dependent is a prime
example. The relationship between two tensor product operators of the same type
is given by composition with a unitary operator. There is an equivalence
between change of tensor product operator and change of basis in the target
space. Among the gains from change of tensor product operator is the
localization of some nonlocal operators as well as separability of inseparable
states. Examples are given.
