Quantum value versus the no-signalling benchmark for higher-cardinality updates

Determine whether, for every integer t≥2 and dimension d>2t in the balanced hypercube rendezvous problem, the finite-dimensional quantum value ω_Q(d,t) is strictly less than the no-signalling value ω_NS(d,t).

Background

The paper studies balanced rendezvous on the d-dimensional hypercube, where two agents receive vertices at Hamming distance 2t and each performs a single update of cardinality t. The no-signalling benchmark is computed explicitly, and finite-dimensional quantum attainment of that benchmark is characterized by the existence of a projective t-sphere resolution.

For t≥2 and d>2t, the paper proves that no finite-dimensional projective t-sphere resolution exists. Consequently, exact finite-dimensional attainment of the no-signalling value is impossible in this regime. However, the nonexistence of an attaining finite-dimensional strategy does not by itself establish a strict gap between the quantum supremum and the no-signalling value, because the quantum correlation set need not be closed and non-attaining strategies or limiting sequences could in principle approach the benchmark.