Towards Necessary and sufficient state condition for violation of a multi-settings Bell inequality

TL;DR

This paper develops a necessary and sufficient criterion based on state parameters for high-dimensional entangled states to violate a generalized Bell inequality, extending the Horodecki criterion beyond two-qubit systems.

Contribution

It introduces a Horodecki-like criterion for arbitrary two-qudit states to determine Bell inequality violations involving multiple measurement settings, generalizing known inequalities.

Findings

Criterion is sufficient for violation, necessary in specific cases

Derived measurement constraints for maximum violation

Applicable to various classes of states including Bell diagonal and non-decomposable states

Abstract

High dimensional quantum entanglement and the advancements in their experimental realization provide a playground for fundamental research and eventually lead to quantum technological developments. The Horodecki criterion determines whether a state violates Clauser-Horne-Shimony-Holt (CHSH) inequality for a two-qubit entangled state, solely from the state parameters. However, it remains a challenging task to formulate similar necessary and sufficient criteria for a high-dimensional entangled state for the violation of a suitable Bell inequality. Here, we develop a Horodecki-like criterion based on the state parameters of arbitrary two-qudit states to violate a two-outcome Bell inequality involving $2^{n-1}$ and $n$ measurement settings for Alice and Bob, respectively. This inequality reduces to the well-known CHSH and Gisin's elegant Bell inequalities for $n=2$ and $n=3$, respectively. While the proposed criterion is sufficient to violate the Bell inequality, it becomes necessary as well for the following cases; (i) $m$ copies of Bell diagonal states for arbitrary $n$, (ii) Non-decomposable states whose correlation matrix is diagonalized by local unitaries, and (iii) for any arbitrary two-qubit state when $n=3$, where the maximal value of the Bell functional is achieved with Bob's measurements being pairwise anticommuting. For any states, we derive the constraints on Alice's measurements in achieving the maximum quantum violation for this inequality.