Relativistic Effects in Quantum Entanglement

TL;DR

This paper explores how relativistic effects influence quantum entanglement, showing that entanglement varies with inertial frames but Bell inequality violations remain invariant when using proper observables.

Contribution

It provides a detailed analysis of relativistic transformations of entangled states and demonstrates frame-independent Bell inequality violations with suitable spin measurements.

Findings

Entanglement depends on the inertial frame and state partition.

Wigner rotations affect the transformation of quantum states.

Bell inequality violations are frame-independent with proper spin observables.

Abstract

One of the most fundamental phenomena of quantum physics is entanglement. It describes an inseparable connection between quantum systems, and properties thereof. In a quantum mechanical description even systems far apart from each other can share a common state. This entanglement of the subsystems, although arising from mathematical principles, is no mere abstract concept, but can be tested in experiment, and be utilized in modern quantum information theory procedures, such as quantum teleportation. In particular, entangled states play a crucial role in testing our understanding of reality, by violating Bell inequalities. While the role of entanglement is well studied in the realm of nonrelativistic quantum mechanics, its significance in a relativistic quantum theory is a relatively new field of interest. In this work the consequences of a relativistic description of quantum entanglement are discussed. We analyze the representations of the symmetry groups of special relativity, i.e. of the Lorentz group, and the Poincar\'e group, on the Hilbert space of states. We describe how unitary, irreducible representations of the Poincar\'e group for massive spin 1/2 particles are constructed from representations of Wigner's little group. We then proceed to investigate the role of the Wigner rotations in the transformation of quantum states under a change of inertial reference frame. Considering different partitions of the Hilbert space of 2 particles, we find that the entanglement of the quantum states appears different in different inertial frames, depending on the form of the states, the chosen inertial frames, and the particular choice of partition. It is explained, how, despite of this, the maximally possible violation of Bell inequalities is frame independent, when using appropriate spin observables, which are related to the Pauli-Ljubanski vector, a Casimir operator of the Poincar\'e group.