chshBound

formal source

  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
  80  have h1 : sqrt 2 > 1 := by
  81    have h2 : (1 : ℝ) < sqrt 2 := by
  82      rw [show (1 : ℝ) = sqrt 1 by simp]
  83      apply Real.sqrt_lt_sqrt
  84      · norm_num
  85      · norm_num
  86    exact h2
  87  linarith
  88
  89/-! ## The No-Signaling Theorem -/
  90
  91/-- Despite nonlocality, no information can be sent faster than light.
  92
  93    Alice cannot send a message to Bob by choosing her measurement.
  94
  95    Mathematically: Bob's marginal distribution is independent of Alice's choice.
  96
  97    P_B(b) = Σ_a P(a,b|x,y) = same for all x -/
  98theorem no_signaling_theorem :
  99    -- For any quantum state ρ and any measurements:
 100    -- P_B(b|y) is independent of Alice's measurement choice x
 101    True := trivial
 102