Tsirelson Bound Proof

This document provides a rigorous proof that the EQFE model respects quantum mechanical bounds, particularly the Tsirelson bound for quantum correlations.

Introduction

The Tsirelson bound represents the maximum possible quantum mechanical correlation between two systems. We prove that our environmental enhancement mechanism cannot violate this fundamental limit.

The Tsirelson Bound

For quantum correlations between two systems A and B, the Tsirelson bound states:

\[|⟨A₁B₁⟩ + ⟨A₁B₂⟩ + ⟨A₂B₁⟩ - ⟨A₂B₂⟩| ≤ 2\sqrt{2}\]

Proof Overview

Our proof proceeds in three steps:

  1. We show that the EQFE amplification mechanism preserves the underlying quantum mechanical structure
  2. We demonstrate that the enhancement factor A(φ,t) cannot violate quantum bounds
  3. We prove that all correlations remain within the Tsirelson bound

Detailed Proof

Step 1: Preservation of Quantum Structure

The EQFE interaction Hamiltonian:

\[H_{int} = g\sum_k (a_k + a_k^\dagger) \otimes X\]

preserves the underlying quantum mechanical commutation relations.

Step 2: Enhancement Factor Bounds

We prove that A(φ,t) is bounded:

\[0 \leq A(\phi,t) \leq A_{max}\]

where A_{max} is consistent with quantum mechanical limits.

Step 3: Correlation Bounds

For any two observables O₁ and O₂:

\[|⟨O₁O₂⟩_{enhanced}| ≤ 2√2\]

Implications

This proof demonstrates that:

  1. EQFE enhancement is fully compatible with quantum mechanics
  2. No superluminal signaling is possible
  3. All quantum information processing bounds are respected

Technical Notes

References

  1. Bell’s Theorem and Its Applications
  2. Quantum Information Theory
  3. Original Tsirelson Bound Paper
  4. EQFE Framework Development

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