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arXiv:2308.03482

Nine Lorentz covariant bitensors for two Dirac spinors to indicate all entangled states

Markus Johansson

Published Aug 7, 2023• 0 citations
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Abstract

The spinorial degrees of freedom of two spacelike separated Dirac particles are considered and a collection of nine locally Lorentz covariant bitensors is constructed. Four of these bitensors have been previously described in [Phys. Rev. A 105, 032402 (2022), arXiv:2103.07784]. The collection of bitensors has the property that all nine bitensors are simultaneously zero if and only if the state of the two particles is a product state. Thus this collection of bitensors indicates any type of spinor entanglement between two spacelike separated Dirac particles.

DOI
10.48550/arxiv.2308.03482

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•Jan 1, 2018•4 citations
The spinorial degrees of freedom of two spacelike separated Dirac particles are considered and a collection of nine locally Lorentz covariant bitensors is constructed. Four of these bitensors have been previously described in [Phys. Rev. A 105, 032402 (2022), arXiv:2103.07784]. The collection of bitensors has the property that all nine bitensors are simultaneously zero if and only if the state of the two particles is a product state. Thus this collection of bitensors indicates any type of spinor entanglement between two spacelike separated Dirac particles.
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-12\frac{1}{2}21​ 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.
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 behavior 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.
The effects of Lorentz boosts on the quantum entanglement encoded by a pair of massive spin-12\frac{1}{2}21​ particles are described according to the Lorentz covariant structure described by Dirac bispinors. The quantum system considered incorporates four degrees of freedom: two of them related to the bispinor intrinsic parity and the other two related to the bispinor spin projection, i.e., the Dirac particle helicity. Because of the natural multipartite structure involved, the Meyer-Wallach global measure of entanglement is preliminarily used for computing global quantum correlations, while the entanglement separately encoded by spin degrees of freedom is measured through the negativity of the reduced two-particle spin-spin state. A general framework to compute the changes on quantum entanglement induced by a boost is developed and then specialized to describe three particular antisymmetric two-particle states. According to the results obtained, two-particle spin-spin entanglement cannot be created by the action of a Lorentz boost in a spin-spin separable antisymmetric state. On the other hand, the maximal spin-spin entanglement encoded by antisymmetric superpositions is degraded by Lorentz boosts driven by high-speed frame transformations. Finally, the effects of boosts on chiral states are shown to exhibit interesting invariance properties, which can only be obtained through such a Lorentz covariant formulation of the problem.
The generation of entangled states and their degree of entanglement are studied in a relativistic formulation for the case of two interacting spin-1/2 charged particles. In the realm of quantum electrodynamics, we revisit the interaction that produces entanglement between the spin components of covariant Dirac spinors describing the two particles. In this way, we derive the relativistic version of the spin-spin interaction, widely used in the nonrelativistic regime. Following this consistent approach, the relativistic invariance of the generated entanglement is discussed.
In this paper we describe the transformation properties of quantum entanglement encoded in a pair of spin 1/2 particles described via Dirac bispinors. Due to the intrinsic parity-spin internal structure of the bispinors, the joint state is a four-qubit state exhibiting multipartite entanglement, and to compute global correlation properties we consider the averaged negativities over each possible bi-partition. We also consider specific bipartitions, such as the spin-spin and the particle-particle bipartitions. The particle-particle entanglement, between all degrees of freedom of one particle and all degrees of freedom of the other particle, is invariant under boosts if each particle has a definite momentum, although the spin-spin entanglement is degraded for high speed boosts. Correspondingly, the mean negativities are not invariant since the boost drives changes into correlations encoded in specific bipartitions. Finally, the results presented in the literature about spin-momentum entanglement are recovered by considering the projection of bispinorial states into positive intrinsic parity, and some striking diffrences between the appropriate approach for this case and the one usually treated in the literature are discussed.
In this paper we describe the transformation properties of quantum entanglement encoded in a pair of spin 1/2 particles described via Dirac bispinors. Due to the intrinsic parity- spin internal structure of the bispinors, the joint state is a four-qubit state exhibiting multipartite entanglement, and to compute global correlation properties we consider the averaged negativities over each possible bi-partition. We also consider specific bipartitions, such as the spin-spin and the particle-particle bipartitions. The particle-particle entanglement, between all degrees of freedom of one particle and all degrees of freedom of the other particle, is invariant under boosts if each particle has a definite momentum, although the spin-spin entanglement is degraded for high speed boosts. Correspondingly, the mean negativities are not invariant since the boost drives changes into correlations encoded in specific bipartitions. Finally, the results presented in the literature about spin-momentum entanglement are recovered by considering the projection of bispinorial states into positive intrinsic parity, and some striking differences between the appropriate approach for this case and the one usually treated in the literature are discussed.