`born_rule_normalized`

proved

show as: view math explainer →

module: IndisputableMonolith.Measurement.BornRule
domain: Measurement
line: 162 · github
papers citing: none yet

open explainer

Generate a durable explainer page for this declaration.

open lean source

IndisputableMonolith.Measurement.BornRule on GitHub at line 162.

browse module

All declarations in this module, on Recognition.

explainer page

Tracked in the explainer inventory; generation is lazy so crawlers do not trigger LLM jobs.

open explainer

depends on

used by

formal source

 159    simp [complementAmplitudeSquared, initialAmplitudeSquared, Real.cos_sq_add_sin_sq rot.θ_s]
 160
 161/-- Born rule normalized: from recognition costs to normalized probabilities -/
 162theorem born_rule_normalized (C₁ C₂ : ℝ) (α₁ α₂ : ℂ)
 163  (h₁ : Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₁‖ ^ 2)
 164  (h₂ : Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₂‖ ^ 2) :
 165  ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2 = 1 := by
 166  have hdenom : Real.exp (-C₁) + Real.exp (-C₂) ≠ 0 :=
 167    (add_pos (Real.exp_pos _) (Real.exp_pos _)).ne'
 168  calc ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2
 169      = Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) +
 170        Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) := by rw [← h₁, ← h₂]
 171      _ = (Real.exp (-C₁) + Real.exp (-C₂)) / (Real.exp (-C₁) + Real.exp (-C₂)) := by
 172        simpa using
 173          (add_div (Real.exp (-C₁)) (Real.exp (-C₂)) (Real.exp (-C₁) + Real.exp (-C₂))).symm
 174      _ = 1 := div_self hdenom
 175
 176end Measurement
 177end IndisputableMonolith

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.