  In this paper, we introduce a notion of quantum discrepancy, a
non-commutative version of combinatorial discrepancy which is defined for
projection systems, i.e. finite sets of orthogonal projections, as
non-commutative counterparts of set systems. We show that besides its natural
algebraic formulation, quantum discrepancy, when restricted to set systems, has
a probabilistic interpretation in terms of determinantal processes.
Determinantal processes are a family of point processes with a rich algebraic
structure. A common feature of this family is the local repulsive behavior of
points. Alishahi and Zamani (2015) exploit this repelling property to construct
low-discrepancy point configurations on the sphere.
  We give an upper bound for quantum discrepancy in terms of $N$, the dimension
of the space, and $M$, the size of the projection system, which is tight in a
wide range of parameters $N$ and $M$. Then we investigate the relation of these
two kinds of discrepancies, i.e. combinatorial and quantum, when restricted to
set systems, and bound them in terms of each other.
