Research proof index

If a distribution has an element with probability 1, the distribution has a constant.

The expectation value of a random variable over α = Fin 2 is the same as Mixable.mix with probabiliy weight X.distr 0

Quantum relative entropy when σ has full rank -/ theorem qRelativeEnt_rank {ρ σ : MState d} [σ.M.NonSingular] : (𝐃(ρ‖σ) : EReal) = ⟪ρ.M, ρ.M.log - σ.M.log⟫ := by apply qRelativeEnt_ker simp [HermitianMat.nonSingular_ker_bot] section lowerSemicontinuous_1 variable {d : Type*} [Fintype d] [DecidableEq d] open scoped InnerProductSpace RealInnerProductSpace...

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,...