Lorentz invariant polynomials as entanglement indicators for Dirac particles

TL;DR

This paper develops Lorentz invariant polynomial indicators for detecting spinor entanglement in Dirac particles, extending previous methods and analyzing their properties and dynamical evolution.

Contribution

It introduces new mixed polynomial entanglement indicators for Dirac particles that are invariant under Lorentz transformations and can detect complex entanglement structures.

Findings

Constructed mixed polynomials for two and three Dirac particles.

Identified polynomials that detect entanglement not seen by homogeneous invariants.

Discussed the dynamical evolution of these polynomial indicators.

Abstract

The spinorial degrees of freedom of two or more spacelike separated Dirac particles are considered and a method for constructing mixed polynomials that are invariant under the spinor representations of the local proper orthochronous Lorentz groups is described. The method is an extension of the method for constructing homogeneous polynomials introduced in [Phys. Rev. A 105, 032402 (2022), arXiv:2103.07784] and [Ann. Phys. (N. Y.) 457, 169410 (2023), arXiv:2105.07503]. The mixed polynomials constructed by this method are identically zero for all product states. Therefore they are considered indicators of the spinor entanglement of Dirac particles. Mixed polynomials can be constructed to indicate spinor entanglement that involves all the particles, or alternatively to indicate spinor entanglement that involves only a proper subset of the particles. It is shown that the mixed polynomials can indicate some types of spinor entanglement that involves all the particles but cannot be indicated by any homogeneous locally Lorentz invariant polynomial. For the case of two Dirac particles mixed polynomials of bidegree (2,2) and bidegree (3,1) are constructed. For the case of three Dirac particles mixed polynomials of bidegree (2,2), bidegree (3,1) and bidegree (3,3) are constructed. The relations of the polynomials constructed for two and three Dirac particles to the polynomial spin entanglement indicators for two and three non-relativistic spin-$\frac{1}{2}$ particles are described. Moreover, the constructed polynomial indicators of spinor entanglement are in general not invariant under local time evolutions of the particles but evolve dynamically and we discuss how to describe this dynamical evolution.