Maximal Quantum Violation of the CGLMP Inequality on Its Both Sides

TL;DR

This paper analyzes the maximum quantum violations of the CGLMP inequality in high dimensions, revealing finite limits, asymmetry between positive and negative violations, and their relation to entanglement levels.

Contribution

It provides the first comprehensive numerical analysis of maximal violations for large dimensions and compares these states with maximally entangled states.

Findings

Max violations tend to finite limits as dimension increases.

Negatively maximal violations are slightly stronger than positive ones.

Entanglement degree of violating states approaches a nonmaximal value.

Abstract

We investigate the maximal violations for both sides of the $d$-dimensional CGLMP inequality by using the Bell operator method. It turns out that the maximal violations have a decelerating increase as the dimension increases and tend to a finite value at infinity. The numerical values are given out up to $d=10^6$ for positively maximal violations and $d=2\times 10^5$ for negatively maximal violations. Counterintuitively, the negatively maximal violations tend to be a little stronger than the positively maximal violations. Further we show the states corresponding to these maximal violations and compare them with the maximally entangled states by utilizing entangled degree defined by von Neumann entropy. It shows that their entangled degree tends to some nonmaximal value as the dimension increases.