`born_rule_normalized`

proved

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module: IndisputableMonolith.Measurement.BornRuleLight
domain: Measurement
line: 15 · github
papers citing: none yet

open explainer

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open lean source

IndisputableMonolith.Measurement.BornRuleLight on GitHub at line 15.

browse module

All declarations in this module, on Recognition.

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depends on

used by

formal source

  12open Real Complex
  13
  14/-- Born rule normalized: from recognition weights to normalized probabilities. -/
  15theorem born_rule_normalized (C₁ C₂ : ℝ) (α₁ α₂ : ℂ)
  16  (h₁ : Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₁‖ ^ 2)
  17  (h₂ : Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₂‖ ^ 2) :
  18  ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2 = 1 := by
  19  have hdenom : Real.exp (-C₁) + Real.exp (-C₂) ≠ 0 :=
  20    (add_pos (Real.exp_pos _) (Real.exp_pos _)).ne'
  21  calc ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2
  22      = Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) +
  23        Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) := by
  24          rw [← h₁, ← h₂]
  25      _ = (Real.exp (-C₁) + Real.exp (-C₂)) / (Real.exp (-C₁) + Real.exp (-C₂)) := by
  26          rw [div_add_div_same]
  27      _ = 1 := div_self hdenom
  28
  29end Measurement
  30end IndisputableMonolith

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