`BookProof.ChapterIrreversible.entropy_bornDist_eq_zero_iff` (v : Fin n → ℂ) (hv : ∑ a, ‖v a‖ ^ 2 = 1) : entropy (bornDist v) = 0 ↔ IsDeterministicColumn v

Proved

BookProof.ChapterIrreversible.entropy_bornDist_eq_zero_iff

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterIrreversible.

BookProof.ChapterIrreversible.entropy_bornDist_eq_zero_iff (v : Fin n → ℂ) (hv : ∑ a, ‖v a‖ ^ 2 = 1) : entropy (bornDist v) = 0 ↔ IsDeterministicColumn v

Formalization note: Lean 4 identifier BookProof.ChapterIrreversible.entropy_bornDist_eq_zero_iff.

Preamble

-- Generated from ChapterIrreversible.lean — theorem BookProof.ChapterIrreversible.entropy_bornDist_eq_zero_iff
import Mathlib
import Definitions.Def_ChapterIrreversible
open BookProof.ChapterIrreversible


open scoped BigOperators
open Finset


variable {n : ℕ}

Formal statement

theorem BookProof.ChapterIrreversible.entropy_bornDist_eq_zero_iff (v : Fin n → ℂ) (hv : ∑ a, ‖v a‖ ^ 2 = 1) :
    entropy (bornDist v) = 0 ↔ IsDeterministicColumn v := by sorry

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