Optimal quantum violation of Clauser-Horne-Shimony-Holt like steering inequality

TL;DR

This paper determines the maximum quantum violation of a steering inequality analogous to CHSH, revealing state-dependent violations for different 2-qubit pure entangled states, unlike previous inequalities.

Contribution

It establishes the optimal quantum violation of a steering inequality and shows it varies with different entangled states, unlike other known inequalities.

Findings

Optimal violation matches Cirel'son bound for CHSH.

Violation amount varies across different 2-qubit pure states.

The inequality's violation is necessary and sufficient for steering.

Abstract

We study a recently proposed Einstein-Podolsky-Rosen steering inequality [arXiv- 1412.8178 (2014)]. Analogous to Clauser-Horne-Shimony-Holt (CHSH) inequality for Bell nonlocality, in the simplest scenario, i.e., 2 parties, 2 measurements per party and 2 outcomes per measurement, this newly proposed inequality has been proved to be necessary and sufficient for steering. In this article, using an equivalence between measurement incompatibility (non joint measurability) and steering, we find the optimal violation amount of this inequality in quantum theory. Interestingly, the optimal violation amount matches with optimal quantum violation of CHSH inequality, i.e., Cirel'son quantity. We further study the optimal violation of this inequality for different bipartite quantum states. To our surprise we find that optimal violation amount is different for different $2$-qubit pure entangled states, which is not the case for all other existing steering inequalities.