GPU-accelerated semidefinite programming for causal games

The process matrix formalism describes quantum correlations in scenarios without a fixed causal order between local laboratories. Operational signatures of such correlations can be investigated through causal games. A paradigmatic example is the Guess-Your-Neighbour's-Input game, in which two parties attempt to guess each other's inputs. Correlations compatible with any definite, or probabilistically mixed, causal order cannot achieve a winning probability exceeding 1/2. The best process-matrix strategy currently known attains a value of approximately 0.6218 using local dimension d=5, while the strongest known dimension-independent upper bound is 0.7592. In this work, we investigate whether increasing the local dimension beyond d = 5 can narrow this gap. To this end, we employ a see-saw optimization scheme in which each step is formulated as a semidefinite program. For scalability, we develop a custom implementation of the SCS solver in which the dominant computational cost, the projection onto the positive-semidefinite cone, is offloaded to a GPU, yielding a six-fold speedup. Using this implementation, we explore local dimensions up to d = 8, and we do not find significant improvements over the value at d=5. Our results suggest that either qualitatively different strategies are required to approach the known upper bound, or that the bound itself is not tight.