Trade-off Relations of Bell Violations among Pairwise Qubit Systems

TL;DR

This paper explores how nonlocal quantum correlations, measured by CHSH violations, are distributed among multiqubit systems, revealing fundamental trade-offs that restrict simultaneous violations in pairwise qubits.

Contribution

It introduces a new trade-off relation for CHSH violations in multiqubit systems, generalizing to arbitrary multiqubit configurations and illustrating restrictions on nonlocality distribution.

Findings

Maximal CHSH violation in one pair limits violations in others.

All pairs cannot violate CHSH simultaneously in a three-qubit system.

Trade-off relations are extended to multiqubit systems.

Abstract

We investigate the nonlocality distributions among multiqubit systems based on the maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality of reduced pairwise qubit systems. We present a trade-off relation satisfied by these maximal violations, which gives rise to restrictions on the distribution of nonlocality among the subqubit systems. For a three-qubit system, it is impossible that all pairs of qubits violate the CHSH inequality, and once a pair of qubits violates the CHSH inequality maximally, the other two pairs of qubits must both obey the CHSH inequality. Detailed examples are given to illustrate the trade-off relations, and the trade-off relations are generalized to arbitrary multiqubit systems.