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arXiv:1801.00758 · Physical Review A

Global Dirac bispinor entanglement under Lorentz boosts

Victor A. S. V. Bittencourt, Alex E. Bernardini, Massimo Blasone A

Published Mar 13, 2018• 25 citations
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Abstract

The effects of Lorentz boosts on the quantum entanglement encoded by a pair of massive spin-$\frac{1}{2}$ 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.

DOI
10.1103/physreva.97.032106

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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.
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.
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.
Abstract We describe in this paper the effects of Lorentz boost on the quantum entanglement encoded in two-particle Dirac bispinor Bell-like states. Each particle composing the system described in this formalism has three degrees of freedom: spin, chirality, and momentum, and the joint state can be interpreted as a 6 qubit state. Given the transformation law of bispinor under boosts, we compute the change of the Meyer-Wallach global measure of quantum entanglement due to the frame transformation and study its equivalence to the results obtained for the relativistic spin 1/2 Bell-like states, constructed in the framework of the irreducible representations of the Lorentz group. We verify that the monotonic increase of the global entanglement under boosts for ultra-relativistic states is solely due to an increasing of the entanglement associated with the spins subsystems. For such ultra-relativistic states, the entanglement related to the chirality degrees of freedom is invariant, and the variation of the global entanglement of bispinor states is the same as the one calculated for relativistic spin 1/2 states. We also show that the particle-particle entanglement is invariant under boosts for any Bell-like state.
Abstract In this paper we study the single particle entanglement induced by a Lorentz boost. We described the particle as a solution of the free Dirac equation, a Dirac bispinor, and compare the induced momentum-spin entanglement with the result obtained in the widely considered framework of relativistic spin 1/2 states. The spin linear entropy for both approaches agree in the ultra relativistic limit. We also verify that the spin-momentum entanglement differers from the spin entropy for the bispinorial case, indicating a true multipartite entanglement involving the degrees of freedom of a Dirac bispinor state: momentum, spin and intrinsic parity. The fact that Dirac bispinors belongs to an irreducible representation of the complete Lorentz group, which also included parity as a symmetry, is the ultimate reason behind such non-trivial structure.
In this paper, we calculate the helicity rotation angle induced by Lorentz boosts. This is relevant for the study of Lorentz boost effects on quantum entanglement encoded in pairs of massive fermions, which are described in terms of positive energy solutions of the Dirac equation with definite helicity. A Lorentz boost describing the change to an inertial frame moving at uniform speed will in general rotate the particle’s helicity. We obtain the coefficients of the helicity superposition in the boosted frame and specialize our results for a perpendicular boost geometry. We verify that the helicity rotation angle can be obtained in terms of the Wigner rotation angle for spin [Formula: see text] states, bridging the framework considered in our previous works to the one of the Wigner rotations. Finally, we calculate the boost-induced spin-parity entanglement for a single particle.
Dirac bispinors belong to an irreducible representation of the complete Lorentz group, which includes parity as a symmetry yielding two intrinsic discrete degrees of freedom: chirality and spin. For massive particles, chirality is not dynamically conserved, which leads to chiral oscillations. In this contribution, we describe the effects of this intrinsic structure of Dirac bispinors on the quantum entanglement encoded in a lepton-antineutrino pair. We consider that the pair is generated through weak interactions, which are intrinsically chiral, such that in the initial state the lepton and the antineutrino have definite chirality but their spins are entangled. We show that chiral oscillations induce spin entanglement oscillations and redistribute the spin entanglement to chirality-spin correlations. Such a phenomenon is prominent if the momentum of the lepton is comparable with or smaller than its mass. We further show that a Bell-like spin observable exhibits the same behavior of the spin entanglement. Such correlations do not require the knowledge of the full density matrix. Our results show novel effects of the intrinsic bispinor structure and can be used as a basis for designing experiments to probe chiral oscillations via spin correlation measurements.