Research proof index

If $d[X_1;X_2]=0$, then there exists a subgroup $H \leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality.

If $d[X ;X]=0$, then there exists a subgroup $H \leq G$ such that $d[X ;U_H] = 0$.

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 :...

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...

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...

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...