Qutrit witness from the Grothendieck constant of order four
Péter Diviánszky, Erika Bene, Tamás Vértesi · Physical Review A · 2017
In this paper, we prove that ${K}_{G}(3)<{K}_{G}(4)$, where ${K}_{G}(d)$ denotes the Grothendieck constant of order $d$. To this end, we use a branch-and-bound algorithm commonly used in the solution of NP-hard problems. It has recently been proven that ${K}_{G}(3)\ensuremath{\le}1.4644$. Here we prove that ${K}_{G}(4)\ensuremath{\ge}1.4841$, which has implications for device-independent witnessing dimensions greater than two. Furthermore, the algorithm with some modifications may find applications in various black-box quantum information tasks with large number of inputs and outputs.