The "factor width" of a positive semidefinite matrix X is the least positive integer k for which X can be written as a convex combination of positive semidefinite matrices that are non-zero only on a single k-by-k principal submatrix (different k-by-k principal submatrices can be used in different terms in the convex combination). The set of matrices with factor width at most k forms an operator system, and we explore three natural ways in which this operator system arises in quantum information theory:
1. It is a "slice" of the set of matrices with Schmidt number at most k, in the sense that there is an orthogonal projection that, when applied to the set of matrices with Schmidt number k, produces exactly the set of matrices with factor width k.
2. It forms the set of "free states" in the resource theory of k-coherence, which classifies and quantifies how much superpositioning is present in a quantum state. (This is analogous to the set of separable matrices forming the set of free states in the resource theory of entanglement.)
3. It is possible to deterministically perform (n-k)-state exclusion on a set of n pure quantum states if and only if their Gram matrix has factor width at most k. For example, if k = 1 then it is possible to perform (n-1)-state exclusion (i.e., it is possible to distinguish the n pure states) if and only if their Gram matrix has factor width 1 (i.e., if and only if the Gram matrix is diagonal, which is equivalent to the pure states being mutually orthogonal).