Cantor correlations I. Operator systems and Cantor games
Georgios Baziotis, Alexandros Chatzinikolaou, Ivan G. Todorov, Lyudmila Turowska
TL;DR
The paper develops a unified operator-system framework for no-signalling correlations over Cantor spaces, capturing infinite products of finite non-local games. It constructs a universal Cantor operator system $\mathcal{S}_{X,A}$ as an inductive limit, linking Cantor correlations of various types to states on tensor products (max, c, min) of universal Cantor operator systems, and proving BW-closure results for qc and qa. A Cantor game theory is introduced, showing that one-shot game values converge along the Cantor construction: $\omega_t(\mathcal{G},\mu_{XY})=\lim_n \omega_t(\mathcal{G}_n,\mu_{X_nY_n})$ and the corresponding game-tensor norms converge $\|t_{\mathcal{G}}^{(n)}\|_{\tau}$, providing a bridge between finite and infinite parallel repetitions. The work connects to foundational questions like the Tsirelson problem and Connes Embedding Problem, with implications for device-independent cryptography and the study of asymptotic nonlocality in infinite-dimensional settings.
Abstract
We study no-signalling correlations over Cantor spaces, placing the product of infinitely many copies of a finite non-local game in a unified general setup. We define the subclasses of local, quantum spatial, approximately quantum and quantum commuting Cantor correlations and describe them in terms of states on tensor products of inductive limits of operator systems. We provide a correspondence between no-signalling (resp. approximately quantum, quantum commuting) Cantor correlations and sequences of correlations of the same type over the projections onto increasing number of finitely many coordinates. We introduce Cantor games, and associate canonically such a game to a sequence of finite input/output games, showing that the numerical sequence of the values of the games in the sequence converges to the corresponding value of the compound Cantor game.