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Commutative $K$-theory is introduced by Adem-Gomez-Lind-Tillmann. as a generalized cohomology theory obtained from topological $K$-theory. The construction uses classifying spaces for commutativity, first introduced by Adem-Cohen-Torres Giese. In this paper we are interested in a $d$-torsion version of this construction: Let $G$ be a topological group. The aforementioned classifying space $B(\mathbb{Z}/d,G)$ is assembled from tuples of pairwise commuting elements in $G$ whose order divide $d$. We will describe the homotopy type of this space when $G$ is the stable unitary group, following the ideas of Gritschacher-Hausmann. The corresponding generalized cohomology theory will be called the commutative $d$-torsion $K$-theory, and will be denoted by $k\mu_d$. Our motivation for studying this cohomology theory comes from applications to operator-theoretic problems that arise in quantum information theory. For this we introduce another spectrum obtained from $k\mu_d$ and show that a famous construction from the study of quantum contextuality, known as Mermin's square, corresponds to a non-trivial class in this generalized cohomology theory.