F alpha jointly convex
f_alpha_jointly_convex
Plain-language statement
Step 3 (Convexity in σ): For fixed H ≥ 0 and ρ, and α > 1, the map σ ↦ f_alpha α H ρ σ is convex. The key is that for p = α/(α−1) > 1: • A ↦ Tr[A^p] is convex on PSD matrices (trace function convexity, Theorem 2.10 of Carlen), • σ ↦ σ^{−γ} H σ^{−γ} is concave in σ by Lieb concavity (since −γ = (α−1)/(2α) ∈ (0,½)), • The composition...
Exact Lean statement
theorem f_alpha_jointly_convex (hα : 1 < α) (H : HermitianMat d ℂ) (hH : 0 ≤ H)
{ι : Type*} [Fintype ι]
(w : ι → ℝ) (hw_nonneg : ∀ i, 0 ≤ w i) (hw_sum : ∑ i, w i = 1)
(ρs σs : ι → MState d) (ρ_mix σ_mix : MState d)
(hρ_mix : ρ_mix.M = ∑ i, w i • (ρs i).M)
(hσ_mix : σ_mix.M = ∑ i, w i • (σs i).M) :
f_alpha α H ρ_mix σ_mix ≤ ∑ i, w i * f_alpha α H (ρs i) (σs i)Formal artifact
Lean source
theorem f_alpha_jointly_convex (hα : 1 < α) (H : HermitianMat d ℂ) (hH : 0 ≤ H) {ι : Type*} [Fintype ι] (w : ι → ℝ) (hw_nonneg : ∀ i, 0 ≤ w i) (hw_sum : ∑ i, w i = 1) (ρs σs : ι → MState d) (ρ_mix σ_mix : MState d) (hρ_mix : ρ_mix.M = ∑ i, w i • (ρs i).M) (hσ_mix : σ_mix.M = ∑ i, w i • (σs i).M) : f_alpha α H ρ_mix σ_mix ≤ ∑ i, w i * f_alpha α H (ρs i) (σs i) := by convert f_alpha_convex_in_sigma hα H hH ρ_mix _ _ _ _ using 1; any_goals assumption; constructor <;> intro h; · exact fun σ_mix hσ_mix => f_alpha_convex_in_sigma hα H hH ρ_mix w hw_nonneg hw_sum σs σ_mix hσ_mix; · apply (h σ_mix hσ_mix).trans unfold f_alpha; simp [ hρ_mix ]; simp [ sum_inner, inner_smul_left, mul_sub, sub_mul, mul_comm, mul_left_comm, Finset.mul_sum] simp [ ← Finset.mul_sum, ← Finset.sum_mul, hw_sum ]- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:719-735
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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.