`BookProof.ChapterTimeTranslation.trace_eq_iff_isDeterministic_pure` (U : Matrix (Fin n) (Fin n) ℂ) : (∀ (a : Fin n) (Ψ : Fin n → ℂ), (∑ k : Fin n, ‖Ψ k‖ ^ 2) = 1 → (diagPart (rho

Proved

BookProof.ChapterTimeTranslation.trace_eq_iff_isDeterministic_pure

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTimeTranslation.

BookProof.ChapterTimeTranslation.trace_eq_iff_isDeterministic_pure (U : Matrix (Fin n) (Fin n) ℂ) : (∀ (a : Fin n) (Ψ : Fin n → ℂ), (∑ k : Fin n, ‖Ψ k‖ ^ 2) = 1 → (diagPart (rho Ψ) * measOp U a).trace = (rho Ψ * measOp U a).trace) ↔ IsDeterministic U

Formalization note: Lean 4 identifier BookProof.ChapterTimeTranslation.trace_eq_iff_isDeterministic_pure.

Preamble

-- Generated from ChapterTimeTranslation.lean — theorem BookProof.ChapterTimeTranslation.trace_eq_iff_isDeterministic_pure
import Mathlib
import Definitions.Def_ChapterTimeTranslation
import Definitions.Def_ChapterReconstruct
open BookProof.ChapterReconstruct
open BookProof.ChapterTimeTranslation


open scoped BigOperators
open Finset Matrix
open BookProof.ChapterReconstruct


variable {n : ℕ}

Formal statement

theorem BookProof.ChapterTimeTranslation.trace_eq_iff_isDeterministic_pure (U : Matrix (Fin n) (Fin n) ℂ) :
    (∀ (a : Fin n) (Ψ : Fin n → ℂ), (∑ k : Fin n, ‖Ψ k‖ ^ 2) = 1 →
        (diagPart (rho Ψ) * measOp U a).trace = (rho Ψ * measOp U a).trace)
      ↔ IsDeterministic U := by sorry

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