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

• URL: http://arxiv.org/abs/2301.06553v1
• Date: Mon, 16 Jan 2023 18:33:39 GMT
• Title: Realization of an arbitrary structure of perfect distinguishability of states in general probability theory
• Authors: Mih\'aly Weiner
• Abstract summary: All subsets with a single element are of course in $mathcal A$, and since smaller collections are easier to distinguish, if $Hin mathcal A$ and $L subset H$ then $Lin mathcal A$; in other words, $mathcal A$ is a so-called $textitindependence system$ on the set of indices $[n]$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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