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.
We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C
∗-algebras. Relatedly, we study entanglement in (infinite dimensional) C
∗-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C
∗-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C
∗-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C
∗-algebras. We show that the minimal and maximal tensor products of state spaces of C
∗-algebras agree precisely when one of the two C
∗-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C
∗-algebras.
Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C
∗-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C
∗-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C
∗-algebras.
We construct the tensor product for the property lattices of two entities. We give a physical interpretation for this tensor product. We show that the tensor product is never an orthocomplemented lattice if both entities are nontrivial. It also never satisfies the covering law. This is the reason why this tensor product does not exist in quantum mechanics and does not exist in quantum logic. We analyze the relation of the tensor product with the usual description of two entities in quantum mechanics. At the end, we give a mathematical way of constructing this tensor product.