`BookProof.DensitySpectral.density_iff_exists_unitary_diagonal` (ρ : Matrix n n ℂ) : IsDensityMatrix ρ ↔ ∃ (U : Matrix.unitaryGroup n ℂ) (d : n → ℝ), (∀ i, 0 ≤ d i) ∧ (∑ i,...

Proved

BookProof.DensitySpectral.density_iff_exists_unitary_diagonal

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDensitySpectral.

BookProof.DensitySpectral.density_iff_exists_unitary_diagonal (ρ : Matrix n n ℂ) : IsDensityMatrix ρ ↔ ∃ (U : Matrix.unitaryGroup n ℂ) (d : n → ℝ), (∀ i, 0 ≤ d i) ∧ (∑ i, d i = 1) ∧ ρ = (U : Matrix n n ℂ) * diagonal (RCLike.ofReal ∘ d) * (U : Matrix n n ℂ)ᴴ

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

Preamble

-- Generated from ChapterDensitySpectral.lean — theorem BookProof.DensitySpectral.density_iff_exists_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.density_iff_exists_unitary_diagonal (ρ : Matrix n n ℂ) :
    IsDensityMatrix ρ ↔
      ∃ (U : Matrix.unitaryGroup n ℂ) (d : n → ℝ),
        (∀ i, 0 ≤ d i) ∧ (∑ i, d i = 1) ∧
        ρ = (U : Matrix n n ℂ) * diagonal (RCLike.ofReal ∘ d) * (U : Matrix n n ℂ)ᴴ := by sorry

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