Interaction of Multiple Tensor Product Operators of the Same Type: an Introduction

TL;DR

This paper explores the properties and interactions of multiple tensor product operators on finite-dimensional Hilbert spaces, highlighting their impact on state separability, locality, and the structure of multipartite quantum systems.

Contribution

It introduces the concept that multiple tensor product operators of the same type exist and can be related by unitary transformations, affecting state separability and locality in quantum systems.

Findings

Different tensor product operators can represent different parts of a multipartite system.

Changing the tensor product operator can localize nonlocal operators and alter state separability.

The relationship between tensor product operators is characterized by unitary transformations.

Abstract

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.