Research proof index

Fix a nonempty finite set $A$ in an elementary abelian $2$-group. For any measurable random variables $Y_1,Y_2$, there are a subspace $H$ and a random variable $U$ uniformly distributed on $H$ such that the source's $\rho[\,\cdot\,\#A]$ functional satisfies $2\rho[U\#A]\le\rho[Y_1\#A]+\rho[Y_2\#A]+8d[Y_1;Y_2]$.

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 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...