Acquisition of delocalized information via classical and quantum carriers

We investigate the information-theoretic power of spatial superposition by analyzing tasks in which infor- mation is locally encoded at multiple distant sites and must be acquired by a single information carrier, such as a particle. Within an operational framework, we systematically compare the statistical correlations that can be generated in such tasks using classical particles, quantum particles in spatial superposition, and more general "second-order interference" resources. We bound classical strategies via convex polytopes and present a study of their symmetry, demonstrating that the vertices are inherently connected to K-juntas as defined in the classical theory of Boolean functions, while their facet inequalities are in one-to-one correspondence with oracle games. We then analyze the violation of the "fingerprinting inequality" achievable by the use of one quantum particle, and we study the dependence of this violation on the dimension d of the particle's internal degree of freedom. In particular, we show that the case of d = 2 can achieve a higher violation of the inequality than the previously investigated case of d = 1. We also provide analytic and numerical evidence that this violation cannot be further increased for larger d > 2. Finally, we find that both quantum and any other (generalized) second-order interference models exhibit the same asymptotic scaling in violating the fingerprinting inequality. Our results thereby further articulate quantum interference and spatial superposition as a resource for information processing.