Researchers have employed GPU-accelerated semidefinite programming to explore the limits of quantum correlations in scenarios without a fixed causal order. Using the process matrix formalism and the "Guess-Your-Neighbour's-Input" game, they sought to determine if increasing the local dimension beyond d=5 could improve the success probability in these types of games, where classical correlations or those with a defined causal order do not exceed 1/2.

The "Guess-Your-Neighbour's-Input" game is a paradigmatic example for studying the operational signatures of quantum correlations without a fixed causal order. The best strategy known to date, using a local dimension d=5, achieves a winning probability of approximately 0.6218. The dimension-independent theoretical upper bound is 0.7592. The goal of this work was to investigate whether increasing the local dimension could narrow this gap between the experimental value and the theoretical limit.

To address the problem, the researchers developed a "see-saw" optimization scheme where each step was formulated as a semidefinite program. For scalability, they implemented a custom version of the SCS solver, offloading the dominant computational cost (the projection onto the positive-semidefinite cone) to a GPU. This optimization resulted in a six-fold speedup, allowing exploration of local dimensions up to d=8. However, the results did not show significant improvements over the value obtained with d=5.

These findings suggest two possibilities: either qualitatively different strategies are required to approach the known upper bound, or the upper bound itself is not as tight as previously thought. The study highlights the complexity of characterizing quantum correlations in the absence of a well-defined causal order and the need for new theoretical or experimental approaches to fully explore their potential.