Research proof index

Shannon entropy of a distribution is at most ln d.

The partition function Z for an ideal gas.

For Antitone functions, the InfRegularized is the supremum of values.

Under the support condition σ.M.ker ≤ ρ.M.ker (i.e., supp(ρ) ⊆ supp(σ)), conjugation by σ^γ and σ^{-γ} preserves the inner product: ⟪ρ.M, H⟫ = ⟪σ^γ ρ σ^γ, σ^{-γ} H σ^{-γ}⟫. This holds because the kernel condition ensures ρ is supported on supp(σ), where σ^γ σ^{-γ} acts as the identity.

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 :...

The intermediate operator inequality: ρ_AB ⊗ σ_C⁻¹ ≤ (ρ_A ⊗ σ_BC⁻¹).reindex(assoc⁻¹). This is derived from W_mat_sq_le_one by algebraic manipulation (conjugation and simplification).