Sharing of Nonlocality of a single member of an Entangled Pair Is Not Possible by More Than Two Unbiased Observers on the other wing

TL;DR

This paper proves that nonlocality of a single entangled particle cannot be shared among more than two independent, sequential observers on the other wing, using a formalism based on POVMs and CHSH inequality violations.

Contribution

It introduces a formalism with POVMs to analyze the trade-off between information gain and disturbance, proving the impossibility of sharing nonlocality beyond two observers.

Findings

More than two Bobs cannot violate CHSH inequality simultaneously.

Optimal measurement disturbance trade-off is derived from POVMs.

Nonlocality sharing is fundamentally limited to two observers.

Abstract

We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin $1/2$ particles can be shared among multiple observers on the other wing who act sequentially and independently of each other [1]. We first show that the optimality condition for the trade-off between information gain and disturbance in the context of weak or non-ideal measurements emerges naturally when one employs a one-parameter class of positive operator valued measures (POVMs). Using this formalism we then prove analytically that it is impossible to obtain violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by more than two Bobs in one of the two wings using unbiased input settings with an Alice in the other wing.