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arXiv:2308.00896

Lorentz invariant polynomials as entanglement indicators for Dirac particles

Markus Johansson

Published Aug 2, 2023• 0 citations
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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.

DOI
10.48550/arxiv.2308.00896

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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 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 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.
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.
It is a recent observation that entanglement classification for qubits is closely related to local SL(2,C)-invariants including the invariance under qubit permutations [Dür, et al., Phys. Rev. A 62, 062314 (2000); Osterloh, A. and Siewert, J., Phys. Rev. A 72, 012337 (2005); O. Chterental and D. Ž. Ðoković, Linear Algebra Research Advances (Nova Science, Hauppauge, N.Y., 2007), Chap. 4, p. 133], which has been termed SL∗ invariance. In order to single out the SL∗ invariants, we analyze the SL(2,C)-invariants of four (five) qubits and decompose them into irreducible modules for the symmetric group S4 (S5) of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of SL∗-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced by Osterloh and Siewert [Phys. Rev. A 72, 012337 (2005); Int. J. Quant. Inf. 4, 531 (2006)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the Ω-process [Luque, J.-G. and Thibon, J.-Y., J. Phys. A 39, 371 (2005)]. As the degrees of invariants increase, the alternative method proves to be particularly efficient.
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.