born_rule_nonneg

formal source

 148  ‖ψ.amplitudes i‖^2
 149
 150/-- **THEOREM (Born Rule)**: Probabilities are non-negative. -/
 151theorem born_rule_nonneg {n : ℕ} (ψ : QuantumState n) (i : Fin n) :
 152    measurementProbability ψ i ≥ 0 := by
 153  unfold measurementProbability
 154  exact sq_nonneg _
 155
 156/-- **THEOREM (Born Rule Normalization)**: Probabilities sum to 1. -/
 157theorem born_rule_normalized {n : ℕ} (ψ : QuantumState n) :
 158    (Finset.univ.sum fun i => measurementProbability ψ i) = 1 := by
 159  unfold measurementProbability
 160  exact ψ.normalized
 161
 162/-! ## Ledger Commitment = Wavefunction Collapse -/
 163
 164/-- The norm of a normalized amplitude is 1.
 165    |z / |z|| = |z| / |z| = 1 for z ≠ 0. -/
 166theorem norm_div_norm_eq_one : ∀ (z : ℂ), z ≠ 0 → ‖z / ‖z‖‖ = 1 := by
 167  intro z hz
 168  rw [norm_div]
 169  have h1 : ‖(‖z‖ : ℂ)‖ = ‖z‖ := by simp [Complex.norm_real]
 170  rw [h1]
 171  exact div_self (norm_ne_zero_iff.mpr hz)
 172
 173/-- Commit a ledger to a specific outcome.
 174    This is the formal model of wavefunction collapse. -/
 175noncomputable def commit {n : ℕ} (L : UncommittedLedger n) (i : Fin n)
 176    (_h : ∃ b ∈ L.branches, b.outcome = i) : CommittedLedger n :=
 177  let b := L.branches.find? (fun b => b.outcome = i)
 178  match b with
 179  | some branch =>
 180      if hz : branch.amplitude ≠ 0 then
 181        ⟨i, branch.amplitude / ‖branch.amplitude‖, norm_div_norm_eq_one branch.amplitude hz⟩