On the relative quantum entanglement with respect to tensor product structure

TL;DR

This paper rigorously establishes the mathematical foundation of the quantum tensor product concept, introduces relative quantum entanglement, and explores how tensor product structures influence entanglement interpretation in quantum systems.

Contribution

It introduces tensor product partitions on endomorphisms, linking them to tensor product structures of vector spaces, and shows how entanglement depends on the chosen tensor product structure.

Findings

Arbitrarily given wave functions can be seen as separable under some tensor product structure.

Quantum entanglement can be reinterpreted through different tensor product structures.

A connection between observable sets and the underlying tensor product structure is established.

Abstract

Mathematical foundation of the novel concept of quantum tensor product by Zanardi et al is rigorously established. The concept of relative quantum entanglement is naturally introduced and its meaning is made clear both mathematically and physically. For a finite or an infinite dimensional vector space $W$ the so called tensor product partition (TPP) is introduced on $End(W)$, the set of endmorphisms of $W$, and a natural correspondence is constructed between the set of TPP's of $End(W)$ and the set of tensor product structures (TPS's) of $W$. As a byproduct, it is shown that an arbitrarily given wave function belonging to an n-dimensional Hilbert space, $n$ being not a prime number, can be interpreted as a separable state with respect to some man-made TPS, and thus a quantum entangled state of a many-body system with respect to the "God-given" TPS can be regarded as a quantum state without entanglement in some sense. The concept of standard set of observables is also introduced to probe the underlying structure of the object TPP and to establish its connection with practical physical measurement.