Twirling hermitian entry
twirling_hermitian_entry
Project documentation
The trace functional is invariant under 1_A ⊗ V conjugation. -/ theorem sandwichedTraceFunctional_conj_tensorUnitary (ρ σ : MState (dA × dB)) (V : Matrix.unitaryGroup dB ℂ) : Q̃_ α(ρ.conjTensorUnitary V‖σ.conjTensorUnitary V) = Q̃_ α(ρ‖σ) := by exact sandwichedTraceFunctional_conj_unitary_MState _ ρ σ section twirling variable {dA dB : Type*} variable [...
Exact Lean statement
lemma twirling_hermitian_entry
(κ : Type) [Fintype κ] (V : κ → Matrix.unitaryGroup dB ℂ)
(hV : ∀ (X : HermitianMat dB ℂ),
(Fintype.card κ : ℝ)⁻¹ • ∑ i : κ, X.conj (V i : Matrix dB dB ℂ) =
(X.trace / Fintype.card dB) • (1 : HermitianMat dB ℂ))
(X : HermitianMat dB ℂ) (b₁ b₂ : dB) :
∑ i : κ, ((V i).val * X.val * (V i).val.conjTranspose) b₁ b₂ =
(X.val.trace / (Fintype.card dB : ℂ)) * (Fintype.card κ : ℂ) *
(if b₁ = b₂ then 1 else 0)Formal artifact
Lean source
lemma twirling_hermitian_entry (κ : Type) [Fintype κ] (V : κ → Matrix.unitaryGroup dB ℂ) (hV : ∀ (X : HermitianMat dB ℂ), (Fintype.card κ : ℝ)⁻¹ • ∑ i : κ, X.conj (V i : Matrix dB dB ℂ) = (X.trace / Fintype.card dB) • (1 : HermitianMat dB ℂ)) (X : HermitianMat dB ℂ) (b₁ b₂ : dB) : ∑ i : κ, ((V i).val * X.val * (V i).val.conjTranspose) b₁ b₂ = (X.val.trace / (Fintype.card dB : ℂ)) * (Fintype.card κ : ℂ) * (if b₁ = b₂ then 1 else 0) := by replace hV := congr_arg ( fun s => s.val b₁ b₂ ) ( hV X ) ; simp_all [ div_eq_inv_mul ] ; convert congr_arg ( fun x : ℂ => x * Fintype.card κ ) hV using 1 <;> ring_nf · by_cases h : Fintype.card κ = 0 <;> simp_all [ HermitianMat.conj ]; · rw [ Fintype.card_eq_zero_iff ] at h simp_all only [Finset.univ_eq_empty, Finset.sum_empty] · classical induction ( Finset.univ : Finset κ ) using Finset.induction · simp_all [ Matrix.mul_assoc ] · simp_all [ Matrix.mul_assoc ] rfl · simp [ Matrix.one_apply, mul_assoc, mul_comm ]; simp [ Matrix.trace, HermitianMat.trace ]; congr! 2; exact Finset.sum_congr rfl fun _ _ => by simp [ Complex.ext_iff ] ;- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:1134-1155
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Conj Transpose isometry mul isometry le one
conjTranspose_isometry_mul_isometry_le_one
Project documentation
The operator norm of the conjugate transpose is equal to the operator norm. -/ theorem Matrix.opNorm_conjTranspose_eq_opNorm {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (A : Matrix m n 𝕜) : Matrix.opNorm Aᴴ = Matrix.opNorm A := by unfold Matrix.opNorm rw [← ContinuousLinearMap.adjoint.norm_map (toEuclideanLin A).toContinuousLine...
Source project: quantumInfo
Person-level attribution pending.
Convex roof of pure
convex_roof_of_pure
Plain-language statement
The convex roof extension of g : KetUpToPhase d → ℝ≥0 applied to a pure state ψ is g (KetUpToPhase.mk ψ).
Source project: quantumInfo
Person-level attribution pending.
Id achieves Rate log dim
CPTPMap.id_achievesRate_log_dim
Plain-language statement
The identity channel on D dimensional space achieves a rate of log2(D).
Source project: quantumInfo
Person-level attribution pending.