Research proof index

The average of f : Ket d → T on an ensemble that mixes to a pure state ψ is f ψ

The average of f : MState d → T on a coerced pure-state ensemble ↑e : MEnsemble d α is equal to averaging the restricted function over Kets f ∘ pure : Ket d → T on e. -/ theorem average_of_pure_ensemble {T : Type _} {U : Type*} [AddCommGroup U] [Module ℝ U] [inst : Mixable U T] (f : MState d → T) (e : PEnsemble d α) : average f (toMEnsemble e) = p...

Let $X_1',X_2'$ be independent copies of the $\tau$-minimizers $X_1,X_2$. Write $k=d[X_1;X_2]$ and $I_1=I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']$. Then the entropy of the four-variable sum obeys $H[X_1+X_2+X_1'+X_2']\le\tfrac12H[X_1]+\tfrac12H[X_2]+(2+\eta)k-I_1$.

entropic_PFR_conjecture: For two $G$-valued random variables $X^0_1, X^0_2$, there is some subgroup $H \leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \le 11 d[X^0_1;X^0_2]$.

In the project's entropic PFR package with parameter $\eta=1/9$, there is a subspace $H$ and a random variable $U$ uniformly distributed on $H$ such that each reference variable is within six times their mutual Ruzsa distance of $U$: $d(X_1,U),d(X_2,U)\le6d(X_1,X_2)$.

Among the first $n+1$ enumerated phases, enumΘ'ArgMax returns the smallest index $i$ at which the function $g$ attains its maximum at $x$. Equivalently, every index $j\le n$ has value at most the value at $i$, and every earlier index $j<i$ has strictly smaller value.