Quantum Information-Theoretical Size Bounds for Conjunctive Queries with Functional Dependencies

Abstract

Deriving formulations to estimate worst-case size bounds for conjunctive queries under various constraints has been at the core of theoretical database research. If the problem has no constraints or has a single functional dependency, tight worst-case size bounds are computable. If the problem has more than one functional dependency, computing tight bounds can be difficult in practice and may even require an infinite number of linear inequalities in its optimization formulation. While these challenges have been addressed with varying methods, no prior research has employed quantum information theory to address this problem. In this work, we establish a connection between earlier classical information theory-based works and quantum information theory. We propose replacing the classical Shannon entropy formulation with the quantum R\'enyi entropy of order $\alpha \in (0,1)$ whose entropy cone is characterized simply by non-negativity. The first key result is to express the bound in terms of optimizing over quantum states and R\'enyi entropy. Optimizing with respect to quantum states rather than classical distributions transfers the hardness into the problem of characterizing classical states, yielding a sound but generally not tight upper bound. We further quantify this hardness explicitly by proposing a dichotomy theorem: if the query satisfies a head absorption rule, then the R\'enyi program value is at most one and $|Q(D)| \leq \mathrm{rmax}(D)$. Otherwise, the program is unbounded.