Candidate entanglement invariants for two Dirac spinors

TL;DR

This paper constructs and analyzes five Lorentz invariants for two Dirac particles, exploring their potential as measures of entanglement and their behavior under local unitary transformations.

Contribution

It introduces five Lorentz invariants for two Dirac spinors and studies their properties as potential entanglement measures under various evolutions.

Findings

Lorentz invariants are zero for product states

Invariants have invariant absolute values under certain local evolutions

Relations to Wootters concurrence are established

Abstract

We consider two spacelike separated Dirac particles and construct five invariants under the spinor representations of the local proper orthochronous Lorentz groups. All of the constructed Lorentz invariants are identically zero for product states. The behaviour of the Lorentz invariants under local unitary evolutions that act unitarily on any subspace with fixed particle momenta is studied. All of the Lorentz invariants have invariant absolute values on such subspaces if the evolutions are generated by local zero-mass Dirac Hamiltonians. Some of them also for the case of nonzero-mass. Therefore, they are considered potential candidates for describing spinor entanglement of two Dirac particles, with either zero or arbitrary mass. Furthermore, their relations to the Wootters concurrence is investigated and their representations in the Foldy-Wouthuysen picture is given.