Trace Functional eq i Sup f alpha
traceFunctional_eq_iSup_f_alpha
Plain-language statement
Step 1 (Variational formula): For α > 1, the trace functional equals the supremum of f_α over all PSD H: Q̃_α(ρ‖σ) = ⨆ (H : HermitianMat d ℂ) (_ : 0 ≤ H), f_alpha α H ρ σ. The optimizer is H_hat = σ^γ (σ^γ ρ σ^γ)^{α−1} σ^γ.
Exact Lean statement
theorem traceFunctional_eq_iSup_f_alpha (hα : 1 < α) (ρ σ : MState d) (hker : σ.M.ker ≤ ρ.M.ker) :
Q̃_ α(ρ‖σ) = ⨆ (H : {H : HermitianMat d ℂ // 0 ≤ H}), f_alpha α H.1 ρ σFormal artifact
Lean source
theorem traceFunctional_eq_iSup_f_alpha (hα : 1 < α) (ρ σ : MState d) (hker : σ.M.ker ≤ ρ.M.ker) : Q̃_ α(ρ‖σ) = ⨆ (H : {H : HermitianMat d ℂ // 0 ≤ H}), f_alpha α H.1 ρ σ := by rw [ @ciSup_eq_of_forall_le_of_forall_lt_exists_gt ]; · exact fun i => f_alpha_le_at_optimizer hα ρ σ i i.2 hker |> le_trans <| le_of_eq <| f_alpha_at_optimizer hα ρ σ; · intro w hw; exact ⟨ ⟨ H_hat α ρ σ, H_hat_nonneg ρ σ ⟩, hw.trans_le ( f_alpha_at_optimizer hα ρ σ ▸ le_rfl ) ⟩- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:638-643
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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.
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Plain-language statement
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Source project: quantumInfo
Person-level attribution pending.
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Plain-language statement
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Source project: quantumInfo
Person-level attribution pending.