  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.
