Strength of the nonlocality of two-qubit entangled state and its applications

Anuma Garg, Satyabrata Adhikari

Published 2022-12-21Version 1

Nonlocality is a feature of quantum mechanics that cannot be explained by local realistic theory. It can be detected by the violation of Bell inequality. We have investigated the problem of detection of nonlocality with the help of an XOR game in which the players may share a two-qubit entangled state $\rho_{AB}^{ent}$. The shared state may generate a nonlocal correlation between the players which is related to the maximum probability $P^{max}$ of success of the game. For the detection of nonlocality of $\rho_{AB}^{ent}$, we have defined a quantity known as strength of nonlocality denoted by $S_{NL}(\rho_{AB}^{ent})$. We mainly deal with the situation when $P^{max}$ fails to detect the nonlocality of $\rho_{AB}^{ent}$. To study this problem, we have established a relation between $P^{max}$ and the expectation value of the CHSH witness operator $W_{CHSH}$ and hence between $S_{NL}(\rho_{AB}^{ent})$ and $W_{CHSH}$. We found that this relationship fails to detect nonlocality when $W_{CHSH}$ does not detect the state $\rho_{AB}^{ent}$. In this case, we developed a process by which we may estimate $S_{NL}(\rho_{AB}^{ent})$. We further obtain an upper bound of $S_{NL}(\rho_{AB}^{ent})$ in terms of optimal witness operator $W^{opt}$ when it detect the state $\rho_{AB}^{ent}$. We establish a linkage between the two-qubit nonlocality determined by the strength of nonlocality and the three-qubit nonlocality determined by the Svetlichny operator. Lastly, as an application, we have used the introduced measure of non-locality $S_{NL}(\rho_{23})$ to determine the upper bound of the power of the controller in the controlled quantum teleportation, where $\rho_{23}$ denote the reduced density operator of the pure three-qubit state $\rho_{123}$.