Research proof index

Let $1<q\le2$, let $q'$ be its Hölder conjugate, and suppose the linearized nontangential operators have the required uniform $L^2$ bound. For bounded measurable $F,G$ and measurable $f$ with $\lVert f(x)\rVert\le\mathbf 1_F(x)$, the supremum over all truncated scale intervals $-B\le s_1\le s_2\le B$ satisfies

$\int_G^+\sup_{s_1\le s_2}\lVert T_{s_1,s_2}f(x)\rVert\,dx\le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

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 Sandwiched Renyi Relative entropy is additive for α=1 (standard relative entropy). -/ private theorem sandwichedRelRentropy_additive_alpha_one (ρ₁ σ₁ : MState d₁) (ρ₂ σ₂ : MState d₂) : D̃_ 1(ρ₁ ⊗ᴹ ρ₂‖σ₁ ⊗ᴹ σ₂) = D̃_ 1(ρ₁‖σ₁) + D̃_ 1(ρ₂‖σ₂) := by by_cases h1 : σ₁.M.ker ≤ ρ₁.M.ker <;> by_cases h2 : σ₂.M.ker ≤ ρ₂.M.ker · simp only [SandwichedRelRentropy,...

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.