The star-algebra \( M_2 \otimes M_2 \) models a pair of qubits. We show in detail that \( M_3 \oplus \mathbb{C} \) models an unordered pair of qubits. Then we use the late 19th century Schur-Weyl duality, to characterize the star-algebra that models an unordered n-tuple of d-level quantum systems.
With more elementary Representation Theory and Number Theory, we characterize the quantum cycles. We finish with a characterization of the von Neumann algebra that models the unordered words.