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arXiv:0902.1242 · Journal of Physics A Mathematical and Theoretical

Entangled states, Lorentz transformations and spin precession in magnetic fields

Amitabha Chakrabarti

Published May 28, 2009• 14 citations
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Abstract

Two positive mass, spin- particles created in an entangled state are studied in the presence of a constant magnetic field inducing distinct precessions, depending on the respective momenta, of the two spins. The charge and anomalous magnetic moment of each particle are taken into account. Consequences for entanglement and, more generally, on correlations, are derived. We start, however, with a compact derivation of the effects of Lorentz transformations on such entangled states, although that has been studied by several authors. Our formalism displays conveniently the analogies and the differences between the two cases. Moreover, combining the two, one obtains the case of constant, orthogonal electric and magnetic fields. More general perspectives are evoked in the concluding remarks.

DOI
10.1088/1751-8113/42/24/245205

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Two positive mass, spin- particles created in an entangled state are studied in the presence of a constant magnetic field inducing distinct precessions, depending on the respective momenta, of the two spins. The charge and anomalous magnetic moment of each particle are taken into account. Consequences for entanglement and, more generally, on correlations, are derived. We start, however, with a compact derivation of the effects of Lorentz transformations on such entangled states, although that has been studied by several authors. Our formalism displays conveniently the analogies and the differences between the two cases. Moreover, combining the two, one obtains the case of constant, orthogonal electric and magnetic fields. More general perspectives are evoked in the concluding remarks.
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This paper was concerned with the spin-momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigen states (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin-momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not observer-dependent quantity in inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and has shown that entanglement decreases with respect to the increasing of observer velocity. Finally, we argue that, Wigner rotation is induced by Lorentz transformations can be realized as controlling operator.
Simple examples are presented of Lorentz transformations that entangle the spins and momenta of two particles with positive mass and spin 1∕21∕21∕2. They apply to indistinguishable particles, produce maximal entanglement from finite Lorentz transformations of states for finite momenta, and describe entanglement of spins produced together with entanglement of momenta. From the entanglements considered, no sum of entanglements is found to be unchanged.
In this work we study the entanglement properties under a Lorentz boost of a pair of spin- 1 massive particles, with spin and momentum as the sole degrees of freedom of the system. Different cases for entanglement between spins and momenta are considered as initial states and it is shown that entanglement between the different degrees of freedom is, in general, not invariant. We analyze the entanglement change in terms of the Wigner rotation induced by the transformation and show that, for the case of spin-1 particles, the relative entanglement change between different states depends strongly on the boost velocity.
Lorentz boosts applied to particles with spin and momentum degrees of freedom induce momentum-dependent rotations. As, in general, different particles have different momenta, the transformation of the whole state is not a representation of the rotation group. Here we identify the group that acts on a two-particle system and, for the case where the momenta of the particles are correlated, find invariant subspaces that have interesting properties for quantum information processes in relativistic scenarios. A basis of states for the study of transformations of spin states under Lorentz boosts is proposed, which is a good candidate for use in building quantum communication protocols in relativistic scenarios.
The entanglement dependence of the noncyclic geometric phase is analyzed. A pair of noninteracting spin-12\frac{1}{2}21​ particles prepared in an arbitrarily entangled state and precessing in an external time-independent uniform magnetic field is considered. It is shown that the geometric phase reduces to a sum of one-particle geometric phases for product states and takes on the two values corresponding to the phase factors $\ifmmode\pm\else\textpm\fi{}1$ for maximally entangled states. If only one of the particles is affected by the magnetic field it is demonstrated that the influence of entanglement on the geometric phase may be interpreted as an effective reduction of the degree of polarization of the affected particle. The generalization to more than two precessing spin-12\frac{1}{2}21​ particles, in the particular case where Schmidt decompositions exists, is briefly outlined. The geometric phase for a pair of spin-12\frac{1}{2}21​ particles with a spin-spin interaction is calculated. In this model we show that the noncyclic geometric phase for a certain class of states may be interpreted solely in terms of the solid angle enclosed by the geodesically closed curve on a two-sphere parametrized by the evolving Schmidt coefficients. This suggests a geometric interpretation of Schmidt decompositions for spin-12\frac{1}{2}21​ pairs analogous to that of the Poincar\'e sphere for a single spin 12.\frac{1}{2}.21​.