Retrit States Violating the KCBS Inequality and Necessary Conditions for Maximal Contextuality

TL;DR

This paper investigates the conditions under which quantum states violate the KCBS inequality, exploring symmetries and rotations of the measurement setup to identify maximally contextual qutrit states.

Contribution

It introduces a detailed analysis of the symmetries and rotations affecting contextuality in the KCBS scenario, identifying conditions for maximal contextuality in real Hilbert space qutrits.

Findings

Identified rotation angles leading to contextuality violations.

Mapped the relationship between Euler angles and maximally contextual states.

Derived mathematical relations linking rotations to state parameters.

Abstract

Since violations of inequalities implied by non-contextual and local hidden variable theories are observed, it is essential to determine the set of (non-)contextual states. Along this direction, one should determine the conditions under which quantum contextuality is observed. It is also important to determine how one can find maximally contextual qutrits. In this work, we revisit the Klyachko-Can-Binicio\u{g}lu-Shumovsky (KCBS) scenario where we observe a five-measurement state-dependent contextuality. We investigate possible symmetries of the KCBS pentagram, i.e., the conservation of the contextual characteristic of a qutrit-system. For this purpose, the KCBS operator including five cyclic measurements is rotated around the Z-axis. We then check a set of rotation angles to determine the contextuality and non-contextuality regions for the eigenstates of the spin-1 operator for an arbitrary rotation. We perform the same operation for the homogeneous linear combination of the eigenstates with spin values +1 and -1. More generally, we work on the real subgroup of the three dimensional Hilbert space to determine the set of (non-)contextual states under certain rotations in the physical Euclidean space $\mathbb{E}^3$. Finally, we show data on Euler rotation angles for which maximally contextual retrits (qutrits of the real Hilbert space) are found, and derive mathematical relations through data analysis between Euler angles and qutrit states parameterized with spherical coordinates.