Research proof index

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...

If the kernels of the components are contained, then the kernel of the Kronecker product is contained.

If the kernel of a product state is contained in another, the left component kernel is contained.

If the kernel of a product state is contained in another, the right component kernel is contained.

The kernel condition σ.M.ker ≤ ρ.M.ker is preserved under partial trace. This follows because supp(ρ) ⊆ supp(σ) implies supp(Tr_B ρ) ⊆ supp(Tr_B σ): if v ∈ supp(Tr_B ρ), then ⟨v, (Tr_B ρ) v⟩ > 0, so for some basis vector e_b we have v ⊗ e_b ∈ supp(ρ) ⊆ supp(σ), hence ⟨v, (Tr_B σ) v⟩ ≥ ⟨v ⊗ e_b, σ (v ⊗ e_b)⟩ > 0.

A ket is a product state iff its components are cross-multiplicative.