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

TL;DR

This paper demonstrates that any independence system on a set of states in general probability theory can be realized through perfect distinguishability structures, expanding understanding of state discrimination.

Contribution

It proves that every independence system can be realized as a set of perfectly distinguishable states in general probability theory.

Findings

Any independence system can be realized in the described manner.

The structure of perfect distinguishability aligns with independence systems.

The result broadens the understanding of state discrimination structures.

Abstract

Let $s_1,s_2,\ldots s_n$ be states of a general probability theory, and $\mathcal A$ be the set of all subsets of indices $H \subset [n]\equiv\{1,2,\ldots n\}$ 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 $\textit{independence system}$ 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.