F alpha at optimizer
f_alpha_at_optimizer
Project documentation
Sub-lemma for Step 1b: the conj of H_hat by σ^{−γ} simplifies to (ρ.M.conj (σ^γ).mat)^{α−1}. This uses σ^{−γ} · σ^γ = identity (on support) to cancel the outer σ^γ factors. -/ theorem H_hat_conj_sigma (hα : 1 < α) (ρ σ : MState d) : let γ := (1 - α) / (2 * α) (H_hat α ρ σ).conj (σ.M ^ (-γ)).mat = (ρ.M.conj (σ.M ^ γ).mat) ^ (α - 1) := by intro γ have hγ :...
Exact Lean statement
theorem f_alpha_at_optimizer (hα : 1 < α) (ρ σ : MState d) :
f_alpha α (H_hat α ρ σ) ρ σ = Q̃_ α(ρ‖σ)Formal artifact
Lean source
theorem f_alpha_at_optimizer (hα : 1 < α) (ρ σ : MState d) : f_alpha α (H_hat α ρ σ) ρ σ = Q̃_ α(ρ‖σ) := by have h_inner : ⟪ρ.M, H_hat α ρ σ⟫_ℝ = ((ρ.M.conj (σ.M ^ ((1 - α) / (2 * α))).mat) ^ α).trace := by exact inner_rho_H_hat hα ρ σ have h_conj : (H_hat α ρ σ).conj (σ.M ^ ((α - 1) / (2 * α))).mat = (ρ.M.conj (σ.M ^ ((1 - α) / (2 * α))).mat) ^ (α - 1) := by convert H_hat_conj_sigma ( hα := hα ) ( ρ := ρ ) ( σ := σ ) using 1 ring_nf! unfold f_alpha sandwichedTraceFunctional simp_all [ sub_div ] rw [ ← HermitianMat.rpow_mul ] ; norm_num [ show α ≠ 0 by positivity, show α - 1 ≠ 0 by linarith ]; · rw [ mul_div_cancel₀ _ ( by linarith ) ] ; ring; · apply_rules [ HermitianMat.conj_nonneg ]; exact ρ.nonneg- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:453-465
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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.