Classifying all mutually unbiased bases in Rel
Abstract
Finding all the mutually unbiased bases in various dimensions is a problem of fundamental interest in quantum information theory and pure mathematics. The general problem formulated in finite-dimensional Hilbert spaces is open. In the categorical approach to quantum mechanics one can find examples of categories which behave ``like'' the category of finite-dimensional Hilbert spaces in various ways but are subtly different. One such category is the category of sets and relations, Rel. One can formulate the concept of mutually unbiased bases here as well. In this note we classify all the mutually unbiased bases in this category by relating it to a standard question in combinatorics.
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