Prove2Me

`BookProof.DensitySpectral.isDensityMatrix_of_unitary_diagonal` (U : Matrix.unitaryGroup n ℂ) (d : n → ℝ) (hd : ∀ i, 0 ≤ d i) (hsum : ∑ i, d i = 1) : IsDensityMatrix ((U : Matrix n

Proved
BookProof.DensitySpectral.isDensityMatrix_of_unitary_diagonal

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDensitySpectral.

BookProof.DensitySpectral.isDensityMatrix_of_unitary_diagonal (U : Matrix.unitaryGroup n ℂ) (d : n → ℝ) (hd : ∀ i, 0 ≤ d i) (hsum : ∑ i, d i = 1) : IsDensityMatrix ((U : Matrix n n ℂ) * diagonal (RCLike.ofReal ∘ d) * (U : Matrix n n ℂ)ᴴ)

Formalization note: Lean 4 identifier BookProof.DensitySpectral.isDensityMatrix_of_unitary_diagonal.

Preamble
-- Generated from ChapterDensitySpectral.lean — theorem BookProof.DensitySpectral.isDensityMatrix_of_unitary_diagonal
import Mathlib
import Definitions.Def_ChapterDensitySpectral
import Definitions.Def_ChapterB4
import Definitions.Def_ChapterDensityMarginalConditional
open BookProof.ChapterB4
open BookProof.DensitySpectral



open Matrix
open scoped BigOperators ComplexOrder

variable {n : Type*} [Fintype n] [DecidableEq n]
Formal statement
theorem BookProof.DensitySpectral.isDensityMatrix_of_unitary_diagonal
    (U : Matrix.unitaryGroup n ℂ) (d : n → ℝ)
    (hd : ∀ i, 0 ≤ d i) (hsum : ∑ i, d i = 1) :
    IsDensityMatrix ((U : Matrix n n ℂ) * diagonal (RCLike.ofReal ∘ d)
      * (U : Matrix n n ℂ)ᴴ) := by sorry

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