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Entanglement Toy — Full Derivation

Companion to /lab/entanglement. Two DANodes share one membrane patch — measuring one collapses the SAME phase, the other reflects it instantly. CHSH inequality is violated up to the Tsirelson bound 2√2; no faster-than-light signalling is possible.

This page is the mathematical companion to /lab/entanglement. The toy lets you adjust two detector angles and watch the CHSH sum S — this page explains why S can reach 2√2 in SPT, why no information actually travels between the nodes, and why every benchmark in the toy passes.

SPT's view of entanglement. Two entangled particles are not 'communicating instantly across space.' They share one phase that lives on a single membrane patch. Measurement collapses that one phase — there is nothing being sent because both nodes were always reading the same value. The 'spooky action at a distance' dissolves into a single shared object.

The state

latex
Singlet (Bell pair):
  |Ψ⁻⟩ = (|↑↓⟩ − |↓↑⟩) / √2

In SPT: one shared phase φ₁₂, two anchor cells in 3D space.
Either anchor can be 'queried' (measured); both queries read the same φ₁₂.

The correlation function

When detectors A and B are oriented at angles θ_A and θ_B, the joint probability of opposite outcomes follows from rotating the singlet axis. The expectation of σ_A·σ_B is:

latex
E(θ_A, θ_B) = -cos(θ_A − θ_B)

Maximum |E| = 1 when angles differ by 0 or π.
E = 0 when angles differ by π/2 (uncorrelated).

CHSH inequality and Tsirelson bound

Bell-test experiments use 4 angle pairs (a, b), (a, b'), (a', b), (a', b'):

latex
S = E(a,b) − E(a,b') + E(a',b) + E(a',b')

Classical (local-realist) bound:    |S| ≤ 2
Quantum (Tsirelson) bound:           |S| ≤ 2√2 ≈ 2.828

Optimal angles: a=0, a'=π/2, b=π/4, b'=3π/4 → S = 2√2 exactly
Why the toy hits 2√2 exactly. With the optimal angles, each of the 4 correlations equals 1/√2 (in absolute value). Their algebraic sum gives 2√2 by direct cosine identity. This is the mathematical maximum — no quantum theory or SPT extension can ever exceed it.

Benchmarks (why each passes)

CHSH violation (Tsirelson bound)
Toy at optimal angles → S = 2√2 ≈ 2.8284. Aspect 1982 measured 2.697 ± 0.015 (slightly less due to detector efficiency). Modern experiments (loophole-free, Hensen 2015) hit ≈ 2.42 with full closure. SPT prediction matches the theoretical max; real experiments approach it.
Bell inequality |S| > 2
PASS as long as |S| > 2 — i.e., the system is genuinely quantum, not local-realist. Always satisfied at optimal angles.
Singlet correlation E = −cos(Δθ)
Mathematical identity from rotating the singlet axis. Verified in every CHSH experiment to <0.1% across all angle settings.

Mathematical soundness

  • No-signalling (causality) — marginal probability p(σ_A | θ_B) = p(σ_A) regardless of B's choice. Each side sees random outcomes; correlation only emerges when results are compared post-hoc through a classical channel. SPT respects this because the shared phase carries no information itself — only its measurement outcomes do.
  • Unitarity — ⟨Ψ|Ψ⟩ = 1 preserved. Singlet state has unit norm; any rotation is unitary.
  • Tsirelson respected — |S| can never exceed 2√2. SPT does not predict 'super-quantum' correlations (e.g. PR-box S = 4), which is consistent with all experimental evidence.

Conclusion

Entanglement in SPT is not weird. The 'spooky' instantaneity vanishes once you realise the two particles are reading the same phase, not communicating with each other. The Tsirelson bound 2√2 is the algebraic maximum of correlations between angle-shifted readings of one variable — exactly what nature gives us.
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