Bell state is not a join

In plain terms

The saying that entanglement means the whole is more than its parts is repeated so often that it has stopped meaning anything, and it deserves to be made into a theorem with real content, a precise inventory of what the whole contains that the parts don't.

In the arena picture, where CSD describes each system by a geometric space of possible configurations called its arena, combining two systems offers two candidate constructions. The first is mere pairing, in which every joint configuration is a configuration of the left system together with a configuration of the right one and nothing more, and the second is the genuine composite arena the theory demands. The theorem says the Bell state, the most famous entangled state in quantum mechanics, lives in the composite arena and is provably not any pairing, as no choice of a left configuration and a right configuration jointly produces it.

That's entanglement as an address, a strict gap between the paired construction and the composite one, with the Bell state exhibited inside the gap.

In CSD

The composite arena is built by concatenating the modes, the elementary degrees of freedom, of the two components, and the pairing of two component points is given by a standard construction, the Segre construction, which is the arena-level counterpart of the tensor product by which quantum mechanics combines systems. The theorem is the arena-side signature of the difference between combining by tensor product and combining by simple pairing, as the composite strictly exceeds the image of the pairing map, and the Bell state witnesses it.

What the excess does has its own answer. An observer with access to one side of the Bell state alone sees its reduced state, and that reduced state is the maximally mixed ensemble on the correlated patterns, equal weight on each possibility with no information about which one holds. Entanglement is therefore exactly what turns the weights a local context assigns to a definite state into the weights of a statistical mixture, while the labels of the remote pattern drop out entirely.

Mathematically

The theorem bell_not_join states that the Bell ray over distinct correlated configuration pairs isn't in the image of the arena join.

Companion results show that the join factorises densities as Kronecker products, with exact marginals and with joint expectations of product observables factoring on joins. That's a property of product states, held by every local composite, and it isn't local tomography, which is RecordLocallyTomographic in SigmaLayer/TensorTomography.lean. The composite algebras are generated by the mode- local subalgebras, with the reconstruction theorem consumed arena-natively.

The reduced density of the Bell ray is the equal mixture with local weights exactly one-half.

Related entries

• No-signallingNothing done to the far wing of an entangled pair changes the statistics seen at the near wing, and CSD proves this exactly, for every state, entangled ones included.
• Field arenaTake finitely many field modes, cap the number of quanta each can hold, and the whole toolbox of quantum field theory, propagators, light cones, Wick's theorem and thermal states, reappears as exact theorems on one finite arena.
• Peres criterionYou can test for entanglement with a pencil, as you transpose half of the joint description, and if a negative eigenvalue appears the pair is entangled.
• Kronecker productTwo quantum systems become one by multiplying their descriptions rather than adding them, which is why quantum resources multiply where classical ones merely add.
• Hong-Ou-Mandel effectWhen two identical photons meet at a beam splitter they always leave together and never one through each exit, so their indistinguishability shows up as a hole in the data.