Research proof index

The identity channel on D dimensional space achieves a rate of log2(D).

A channel cannot achieve a rate greater than log2(D), where D is the output dimension.

If $X_1, X_2$ is a $\phi$-minimizer, then $d[X_1;X_2] = 0$.

Let $T_1,T_2,T_3$ be measurable random variables in a finite abelian group with $T_1+T_2+T_3=0$, and set $\delta=I[T_1:T_2]+I[T_1:T_3]+I[T_2:T_3]$. For any measurable $Y_1,\ldots,Y_n$ and any $\alpha>0$, there is a measurable random variable $U$ such that $d[U;U]+\alpha\sum_{i=1}^n d[Y_i;U]\le\left(2+\frac{\alpha n}{2}\right)\delta+\alpha\sum_{i=1}^n d[Y_i;T_2].$

Doubly stochastic Hölder inequality: for nonneg a, b, doubly stochastic w, and conjugate p, q > 1: ∑{ij} a_i * b_j * w{ij} ≤ (∑ a_i^p)^{1/p} * (∑ b_j^q)^{1/q}.

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