Research proof index

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.

For PSD A and γ ≠ 0, the product A^γ * A^{-γ} equals the support projection of A. This is because x^γ * x^{-γ} = if x = 0 then 0 else 1 for x ≥ 0.

For α > 1, the map H ↦ f_α(H, ρ, σ) is concave (for fixed ρ, σ), so the optimal H is a maximizer.

The Data Processing Inequality for the Sandwiched Rényi relative entropy (α > 1). Every CPTP map Φ satisfies D̃_α(Φρ‖Φσ) ≤ D̃_α(ρ‖σ). The proof uses the Stinespring representation (see CPTPMap.exists_purify): every CPTP map can be written as ancilla preparation + unitary conjugation + partial trace. Since the sandwiched Rényi divergence is invariant...

The Data Processing Inequality for the Sandwiched Rényi relative entropy (α > 1). Every CPTP map Φ satisfies D̃_α(Φρ‖Φσ) ≤ D̃_α(ρ‖σ). The proof uses the Stinespring representation (see CPTPMap.exists_purify): every CPTP map can be written as ancilla preparation + unitary conjugation + partial trace. Since the sandwiched Rényi divergence is invariant...

Monotonicity of the sandwiched Rényi divergence under traceLeft for α > 1. Follows from sandwichedRenyiEntropy_mono_traceRight' + SWAP invariance.