Research proof index

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...

A fibring identity in the $\tau$-minimizer setup. Let $X_1',X_2'$ be independent copies of $X_1,X_2$ and put $k=d[X_1;X_2]$. Then $d[X_1+X_2';X_2+X_1']+d[X_1\mid X_1+X_2';X_2\mid X_2+X_1']+I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']=2k$.

A uniform distribution on a set with doubling constant K has self Rusza distance at most log K.

If $Z_1, Z_2$ are independent, then $d[Z_1; Z_2]$ is equal to $d[\pi(Z_1);\pi(Z_2)] + d[Z_1|\pi(Z_1); Z_2 |\pi(Z_2)]$ plus $I( Z_1 - Z_2 : (\pi(Z_1), \pi(Z_2)) | \pi(Z_1 - Z_2) ).$

A Ruzsa-distance lower bound for sums of independent copies. Let $X_1',X_2'$ be independent copies of the $\tau$-minimizers $X_1,X_2$, and set $k=d[X_1;X_2]$. Then $d[X_1+X_1';X_2+X_2']\ge k-\frac{\eta}{2}\bigl(d[X_1;X_1]+d[X_2;X_2]\bigr).$