Research proof index

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.

The sandwiched Rényi divergence is monotone under partial trace for α > 1. This follows from monotonicity of the trace functional together with the fact that D̃_α = log(Q̃_α) / (α - 1) and both log and 1/(α-1) are order-preserving for α > 1.

The trace functional is strictly positive when the kernel condition holds. Under σ.M.ker ≤ ρ.M.ker (i.e. supp(ρ) ⊆ supp(σ)), the sandwich σ^γ ρ σ^γ ≠ 0 because ρ has support inside σ's support. -/ theorem sandwichedTraceFunctional_pos (ρ σ : MState d) (hker : σ.M.ker ≤ ρ.M.ker) : 0 < Q̃_ α(ρ‖σ) := by rw [sandwichedTraceFunctional] apply HermitianMat...

The trace functional Q̃_α is jointly convex for α > 1. This is Proposition 3 of the paper, originally from Frank–Lieb. The proof uses the variational formula: Q̃_α(ρ‖σ) = sup_{H ≥ 0} f_α(H, ρ, σ) where f_α(H, ρ, σ) = α · Tr[ρ H] - (α-1) · Tr[(σ^{-γ} H σ^{-γ})^{α/(α-1)}] is jointly convex in (ρ, σ) for fixed H (since the first term is linear an...