Research proof index

Step 1c: H_hat is a maximizer: for all H ≥ 0, f_α(H) ≤ f_α(H_hat). This uses the trace Young inequality: for PSD A, B and conjugate exponents p, q > 1, ⟪A, B⟫ ≤ Tr[A^p]/p + Tr[B^q]/q. Applied with A = σ^γ ρ σ^γ, B = σ^{-γ} H σ^{-γ}, p = α, q = α/(α-1), the inner product identity ⟪ρ, H⟫ = ⟪A, B⟫ (under the support condition) yield...

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.

If A ≥ 0 and A ≤ 1, then each eigenvalue of A is in [0, 1].

Hermitian trace Hölder inequality: for PSD A, B and conjugate exponents p, q > 1, ⟪A, B⟫ ≤ Tr[A^p]^(1/p) * Tr[B^q]^(1/q).

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...

If for all i, ker(σs i) ≤ ker(ρs i), then ker(∑ w i • σs i) ≤ ker(∑ w i • ρs i), provided all weights are nonneg and all matrices are PSD.