Qutrit witness from the Grothendieck constant of order four

In this paper, we prove that $K_G\left(3\right)<K_G\left(4\right)$, where $K_G\left(d\right)$ 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 $K_G\left(3\right)\le 1.4644$. Here we prove that $K_G\left(4\right)\ge 1.4841$, 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.

Figure 1

Time required to compute $L\left(M\right)$ as a function of the matrix dimension $n$ of $M$. The plot has a log-log scale. The purple square and blue circle markers correspond to the curves with parameter $k=0$ and $k=n/4$, respectively. The red line is for the sake of comparison.

Figure 2

The functions $f\left(q\right)$ and $\tilde{f}\left(q\right)$ are displayed in blue and green dots, respectively. An (irrelevant) multiplicative constant $c=2200$ is introduced for better comparison of the two plots. The feasible $q$ values on the $x$ axis correspond to the relation $q={\vec{a}}_i·{\vec{b}}_j$.