Quantum Entanglement in Dephaze Physics

I. The Conventional Paradox

In standard quantum mechanics, entanglement is paradoxical because it assumes separable particles. To match observations, it introduces the mysterious "collapse of the wave function," a non-local process that lacks physical clarity.

II. Dephaze Approach

Dephaze starts from a unified, non-local phase field Ψ. Local excitations ("particles") are not independent, but correlated aspects of the same field.

III. Building the Bridge: PDE → Algebra

We derive the algebraic rules (C1, C2) from the Dephaze PDE:

• C1: normalized, unbiased state (ω(𝟙)=1, ω(A)=0)
• C2: two-outcome measurement as a reflector (A²=𝟙)

The key is the ergodic CP-semigroup generated by local measurement coupling, whose fixed-point algebra is isomorphic to ℂ², producing projectors P₊, P₋ and reflector A=2P₊−𝟙.

IV. Proofs & Stability

1. GNS and invariance

The coherence commutator induces a *-derivation, generating automorphisms αₜ; the 0-energy constraint ensures an invariant state ω. GNS gives ω(𝟙)=1.

2. Ergodic projector

The measurement semigroup (CP, unital) yields an ergodic average E, a conditional expectation onto Fix(R).

3. Bistability

The PDE has a double-well effective potential. LaSalle invariance → two attractors Ψ⁺, Ψ⁻. With small noise, the process splits into two ergodic components ⇒ Fix(R) ≅ ℂ².

4. Reflector & Bias

Define A=2P₊−𝟙 ⇒ A²=𝟙. Flip symmetry ensures ω(A)=0. Thus C1 & C2 hold.

V. Tsirelson Bound

With C1 and C2 established, the CHSH operator satisfies:

Therefore, in any Dephaze-consistent state, |S| ≤ 2√2. The entanglement "mystery" becomes a theorem.

VI. Conclusion

Entanglement is no longer paradoxical. In Dephaze physics, non-locality is natural; separability is the illusion. The Tsirelson bound arises as a structural theorem from the 0-point PDE and ergodic dynamics, not as an axiom.