Quantum Discrepancy: A Non-Commutative Version of Combinatorial Discrepancy

TL;DR

This paper introduces quantum discrepancy, a non-commutative extension of combinatorial discrepancy for projection systems, linking algebraic, probabilistic, and geometric aspects, and providing bounds and relations between the two discrepancy notions.

Contribution

It defines quantum discrepancy for projection systems, establishes bounds, and explores its relation to classical combinatorial discrepancy, extending discrepancy theory into the non-commutative setting.

Findings

Provided an upper bound for quantum discrepancy in terms of space dimension and system size.

Showed the tightness of bounds in a wide parameter range.

Bound the relation between combinatorial and quantum discrepancies.

Abstract

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.