`Measurement`

definition

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module: IndisputableMonolith.Quantum.NonlocalityNoSignaling
domain: Quantum
line: 49 · github
papers citing: none yet

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IndisputableMonolith.Quantum.NonlocalityNoSignaling on GitHub at line 49.

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formal source

  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 :=
  69  -Real.cos (θ_A - θ_B)
  70
  71/-- The CHSH Bell inequality. -/
  72noncomputable def chshBound : ℝ := 2
  73
  74/-- The quantum (Tsirelson) bound. -/
  75noncomputable def tsirelsonBound : ℝ := 2 * sqrt 2
  76
  77/-- **THEOREM**: Quantum mechanics violates Bell inequality. -/
  78theorem bell_violation : tsirelsonBound > chshBound := by
  79  unfold tsirelsonBound chshBound

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