EPRPair

formal source

  35/-! ## The EPR Setup -/
  36
  37/-- An EPR pair: two entangled particles A and B. -/
  38structure EPRPair where
  39  /-- State: |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 -/
  40  entangled : Bool := true
  41  /-- Location of particle A -/
  42  location_A : ℝ × ℝ × ℝ
  43  /-- Location of particle B (far away) -/
  44  location_B : ℝ × ℝ × ℝ
  45  /-- Separation distance -/
  46  separation : ℝ
  47
  48/-- A measurement on one particle of the pair. -/
  49structure Measurement where
  50  /-- Measurement axis (e.g., spin direction) -/
  51  axis : ℝ × ℝ × ℝ
  52  /-- Outcome: +1 or -1 -/
  53  outcome : Int
  54  /-- Time of measurement -/
  55  time : ℝ
  56
  57/-! ## Bell Nonlocality -/
  58
  59/-- Bell's theorem: No local hidden variable theory can reproduce quantum correlations.
  60
  61    For measurements at angles θ_A and θ_B:
  62    E(θ_A, θ_B) = -cos(θ_A - θ_B)
  63
  64    This violates the Bell inequality:
  65    |E(a,b) - E(a,b')| + |E(a',b) + E(a',b')| ≤ 2
  66
  67    Quantum mechanics gives 2√2 ≈ 2.83, violating the bound! -/
  68noncomputable def quantumCorrelation (θ_A θ_B : ℝ) : ℝ :=