Qutrit witness from the Grothendieck constant of order four

TL;DR

This paper proves that the Grothendieck constant of order four exceeds that of order three, using a branch-and-bound algorithm, with implications for quantum dimension witnessing and black-box quantum information tasks.

Contribution

It establishes the inequality $K_G(3)<K_G(4)$, providing a new lower bound for $K_G(4)$ and demonstrating the algorithm's potential in quantum information applications.

Findings

$K_G(3) extless K_G(4)$ proven.

New lower bound $K_G(4) extgreater 1.4841$.

Algorithm applicable to quantum information tasks.

Abstract

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 $K_G(3)\le 1.4644$. Here we prove that $K_G(4)\ge 1.4841$, 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.