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arXiv:0801.2429 · International Journal of Quantum Information

HELICITY ENTANGLEMENT OF MOVING BODIES

HONGBAO ZHANG, SHUXIN SHAO, SONG HE

Published Mar 1, 2009• 2 citations
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Abstract

We investigate the Lorentz transformation of the reduced helicity density matrix for a pair of massive spin 1/2 particles. The corresponding Wootters concurrence shows no invariant meaning, which implies that we can generate helicity entanglement simply by the transformation from one reference frame to another. The difference between the helicity and spin case is also discussed.

DOI
10.1142/s0219749909005225

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We investigate the Lorentz transformation of the reduced helicity density matrix for a pair of massive spin 1/2 particles. The corresponding Wootters concurrence shows no invariant meaning, which implies that we can generate helicity entanglement simply by the transformation from one reference frame to another. The difference between the helicity and spin case is also discussed.
Lorentz transformation of the reduced helicity matrix for a massive spin 1/2 particle is first investigated in the framwork of relativistic quantum information theory. The corresponding entangled helicity entropy is calculated and has no invariant meaning as that of spin. The variation of the entropy as the relative velocity of motion, however, differs significantly from that of spin due to their distinct transformation behaviors under the action of Lorentz group. The underlying physical explanations are also discussed.
For a massive spin 1/2 field, we present the reduced spin and helicity density matrix, respectively, for the same pure one particle state. Their relation has also been developed. Furthermore, we calculate and compare the corresponding entanglement entropy for spin and helicity within the same inertial reference frame. Due to the distinct dependence on momentum degree of freedom between spin and helicity states, the resultant helicity entropy is different from that of spin in general. In particular, we find that both helicity entanglement for a spin eigenstate and spin entanglement for a right handed or left handed helicity state do not vanish, and their Von Neumann entropy has no dependence on the specific form of momentum distribution, as long as it is isotropic.
Lorentz transformation of the reduced helicity density matrix for a massive spin- particle is investigated in the framework of relativistic quantum information theory for the first time. The corresponding helicity entropy is calculated, which shows no invariant meaning as that of spin. The variation of the helicity entropy with the relative speed of motion of inertial observers, however, differs significantly from that of spin due to their distinct transformation behaviors under the action of the Lorentz group. This novel and odd behavior unique to the helicity may be readily detected by high-energy physics experiments. The underlying physical explanations are also discussed.
We calculate the entanglement between a pair of polarization-entangled photon beams as a function of the reference frame, in a fully relativistic framework. We find the transformation law for helicity basis states and show that, while it is frequency independent, a Lorentz transformation on a momentum-helicity eigenstate produces a momentum-dependent phase. This phase leads to changes in the reduced polarization density matrix, such that entanglement is either decreased or increased, depending on the boost direction, the rapidity, and the spread of the beam.
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.