Three-particle Bell-like inequalities under Lorentz transformations

TL;DR

This paper investigates how Lorentz transformations affect the violation of Bell-like inequalities in three-particle entangled states, revealing that relativistic effects diminish non-locality signals, with implications for relativistic quantum information.

Contribution

It introduces a detailed analysis of three-particle Bell inequalities under Lorentz boosts, comparing Pauli and Czachor's relativistic spin operators, and clarifies their impact on non-locality in relativistic regimes.

Findings

Violation of Svetlichny's inequality decreases with increasing boost velocity.

Mermin's and Collins' inequalities predict non-locality behavior consistent with the spin state.

Pauli spin operator results align better with relativistic spin behavior than Czachor's operator.

Abstract

We study the effects of Lorentz transformations on three-particle non-local system states (GHZ and W) of spin 1/2 particles, using the Pauli spin operator and a three-particle generalization of Bell's inequality, introduced by Svetlichny. In our setup, the moving and laboratory frames used the (same) set of measurement directions that maximally violate Svetlichny's inequality in the laboratory frame. We also investigate the behavior of Mermin's and Collins' inequalities. We find that, regardless of the particles' type of entanglement, violation of Svetlichny's inequality in the moving frame is decreased by increasing the boost velocity and the energy of particles in the laboratory frame. In the relativistic regime Svetlichny's inequality is a good criterion to investigate the non-locality of the GHZ state. We also find that Mermin's and Collins' inequalities lead to reasonable predictions, in agreement with the behavior of the spin state, about non-locality of the W state in the relativistic regime. Then, comparing our results with those in which Czachor's relativistic spin is used instead of the Pauli operator, we find that the results obtained by considering the Pauli spin operator are in better agreement with the behavior of spin state of the system in the relativistic information theory.