Acquisition of Delocalized Information via Classical and Quantum Carriers

TL;DR

The paper develops an operational framework to compare information acquisition using classical particles, quantum particles in spatial superposition, and generalized second-order interference. It shows that classical strategies form convex polytopes with vertices tied to $K$-juntas and facet inequalities corresponding to oracle games, while quantum strategies can violate fingerprinting-type inequalities; the maximal quantum violation is achieved with a two-dimensional internal space ($d=2$) under symmetric encoding, and the violation scales as $\mathcal{O}(N^{-3})$, matching the limits set by second-order interference models $\mathcal{J}_{N,2}$. The results articulate spatial superposition as a resource for information processing and establish a tight relationship between classical polytope geometry, quantum violations, and higher-order interference. The work suggests avenues for further exploration, including multi-particle setups and potential applications to MAC capacities and cryptographic protocols.

Abstract

We investigate the information-theoretic power of spatial superposition by analyzing tasks in which information 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.

Figures (4)

• Figure 1: Examples of our setup for $N=4$ bits featuring respectively three classical particles and one quantum particle.
• Figure 2: An octahedron ($\mathcal{C}_{2,1}$) as the convex hull of two squares ($\mathcal{C}_{1,1}$). The vertices correspond to boolean 1-juntas in 2 variables.
• Figure 3: Numerically obtained maximal violations $|\delta|$ of the fingerprinting inequality \ref{['fingerprint_ineq']} for different quantum protocols. The blue cross ($\times$) corresponds to the maximal violation calculated in previous works horvat_quantum_2021zhang_building_2022. The improved violation represented by the dashed line corresponds to the result of our Theorem \ref{['thm:max_qm_uniform']}.
• Figure 4: Sketch of the sets $\mathcal{C}_{N,N-1}$, $\mathcal{Q}_{N,1}^d$ and $\mathcal{J}_2$. See main text for details.

Theorems & Definitions (6)

• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Conjecture 3.4
• Theorem 3.5