Realization of an arbitrary structure of perfect distinguishability of states in general probability theory

Abstract

Let $s_1,s_2,\dots s_n$ be states of a general probability theory, and $\mathcal{A}$ be the set of all subsets of indices $H\subset [n]\equiv \{1,2,\dots n\}$ such that the states $(s_j{)}_{j\in H}$ are jointly perfectly distinguishable. All subsets with a single element are of course in $\mathcal{A}$, and since smaller collections are easier to distinguish, if $H\in \mathcal{A}$ and $L\subset H$ then $L\in \mathcal{A}$; in other words, $\mathcal{A}$ is a so-called $\text{independence system}$ on the set of indices $[n]$. In this paper it is shown that every independence system on $[n]$ can be realized in the above manner.