bell_violation

formal source

  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
 103/-- The reduced density matrix of B is unaffected by Alice's measurement.
 104
 105    ρ_B = Tr_A(ρ_AB) = same before and after Alice's measurement
 106
 107    This is a mathematical fact of quantum mechanics. -/
 108theorem reduced_density_unchanged :