`BookProof.ChapterTimeTranslation.trace_rho_measOp` (U : Matrix (Fin n) (Fin n) ℂ) (a : Fin n) (Ψ : Fin n → ℂ) : (rho Ψ * measOp U a).trace = (∑ k : Fin n, (starRingEnd ℂ) (U k a)

Proved

BookProof.ChapterTimeTranslation.trace_rho_measOp

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTimeTranslation.

BookProof.ChapterTimeTranslation.trace_rho_measOp (U : Matrix (Fin n) (Fin n) ℂ) (a : Fin n) (Ψ : Fin n → ℂ) : (rho Ψ * measOp U a).trace = (∑ k : Fin n, (starRingEnd ℂ) (U k a) * Ψ k) * (∑ b : Fin n, (starRingEnd ℂ) (Ψ b) * U b a)

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

Preamble

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


open scoped BigOperators
open Finset Matrix
open BookProof.ChapterReconstruct


variable {n : ℕ}

Formal statement

theorem BookProof.ChapterTimeTranslation.trace_rho_measOp (U : Matrix (Fin n) (Fin n) ℂ) (a : Fin n) (Ψ : Fin n → ℂ) :
    (rho Ψ * measOp U a).trace =
      (∑ k : Fin n, (starRingEnd ℂ) (U k a) * Ψ k) *
        (∑ b : Fin n, (starRingEnd ℂ) (Ψ b) * U b a) := by sorry

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