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.
Particle systems admit a variety of tensor product structures (TPSs) depending on the complete system of commuting observables chosen for the analysis. Different notions of entanglement are associated with these different TPSs. Global symmetry transformations and dynamical transformations factor into products of local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs and corresponding measures of entanglement are defined for particle scattering systems.
This is Paper 005 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops the emergence of tensor-product structure and entanglement within Binary Mechanics. Building on the preceding treatment of effective Hilbert-space representation, it asks why composite quantum systems are represented by tensor products and why nonseparable, entangled states naturally appear. The central claim is that tensor products are not primitive assumptions. They arise when a finite binary-mechanical boundary problem can be decomposed into separately addressable boundary regions whose completion labels can be tracked independently. In that case, the joint completion structure is represented by the tensor product of the corresponding effective Hilbert spaces. Product states require a stronger condition: the underlying resolved graph histories and their phase counts must also factor across the chosen decomposition. Entanglement appears when completion labels remain separately addressable, but the underlying history ensemble does not factor. This non-factorization may be caused by cross-boundary graph structure, shared closure constraints, polarity consistency conditions, or other finite boundary restrictions. In this view, entanglement is the effective Hilbert-space expression of a shared finite history structure rather than a mysterious additional quantum postulate. The paper also distinguishes exact tensor products from physically effective tensor-product decompositions. Exact factorization is an ideal case; in physical applications, tensor-product structure is expected to hold when cross-boundary history constraints are negligible at the scale of description. The result is a finite, graph-history account of composite quantum structure: distinguishable completion channels give effective Hilbert bases, independently addressable labels give tensor-product spaces, factorizing histories give product states, and non-factorizing histories give entanglement.