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Relativistic Effects on Quantum Bell States of Massive Spin 1 Particles

Janardan Prasad Singh

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

We examine the behaviour of the maximally entangled Bell state of two spin 1 massive particles under relativistic transformations. On the basis of explicit calculations of the Wigner rotation and the use of transformation properties of Dirac spinors, we establish that the constituent particles of the Bell state undergo momentum dependent rotation of the spin orientations characterized by the Wigner angle φW = tan −1 sinhsinh τ cosh � +cosh τ . However, since local unitarity is retained in the process, the corresponding entanglement fidelity is not lost. c

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We examine the behaviour of the maximally entangled Bell state of two spin 1 massive particles under relativistic transformations. On the basis of explicit calculations of the Wigner rotation and the use of transformation properties of Dirac spinors, we establish that the constituent particles of the Bell state undergo momentum dependent rotation of the spin orientations characterized by the Wigner angle φW = tan −1 sinhsinh τ cosh � +cosh τ . However, since local unitarity is retained in the process, the corresponding entanglement fidelity is not lost. c
Abstract: We examine the behaviour of the maximally entangled Bell state of two spin 12 massive particles under relativistic transformations. On the basis of explicit calculations of the Wigner rotation and the use of transformation properties of Dirac spinors, we establish that the constituent particles of the Bell state undergo momentum dependent rotation of the spin orientations characterized by the Wigner angle φW = tan−1 sinh sinh τcosh+cosh τ. However, since local unitarity is retained in the process, the corresponding entanglement fidelity is not lost.
We study the transformation of maximally entangled states under the action of Lorentz transformations in a fully relativistic setting. By explicit calculation of the Wigner rotation, we describe the relativistic analog of the Bell states as viewed from two inertial frames moving with constant velocity with respect to each other. Though the finite dimensional matrices describing the Lorentz transformations are non-unitary, each single particle state of the entangled pair undergoes an effective, momentum dependent, local unitary rotation, thereby preserving the entanglement fidelity of the bipartite state. The details of how these unitary transformations are manifested are explicitly worked out for the Bell states comprised of massive spin 1/21/21/2 particles and massless photon polarizations. The relevance of this work to non-inertial frames is briefly discussed.
We have studied the relativistic effect on the spin state of a massive Dirac particle under an arbitrary Lorentz transformation using the Foldy-Woutheysen representation. We have obtained the transformation matrix from the state of a massive Dirac particle with an initial momentum to a state with a transformed momentum under an arbitrary homogeneous Lorentz transformation. We have shown moving observers obtain the consistent monotonic behavior between the concurrence and the maximum value of Bell parameter for the spin state of the pair of particles.
We describe the spin and momentum degrees of freedom of a system of two massive spin-12\frac{1}{2}21​ particles as a four-qubit system. Then we explicitly show how the entanglement changes between different partitions of the qubits, when considered by different inertial observers. Although the two particle entanglement corresponding to a partition into Alice's and Bob's subsystems is, as often stated in the literature, invariant under Lorentz boosts, the entanglement with respect to other partitions of the Hilbert space on the other hand, is not. It certainly does depend on the chosen inertial frame and on the initial state considered. The change of entanglement arises, because a Lorentz boost on the momenta of the particles causes a Wigner rotation of the spin, which in certain cases entangles the spin with the momentum states. We systematically investigate the situation for different classes of initial spin states and different partitions of the four-qubit space. Furthermore, we study the behavior of Bell inequalities for different observers and demonstrate how the maximally possible degree of violation, using the Pauli-Lubanski spin observable, can be recovered by any inertial observer.
In this thesis, entanglement under fully relativistic settings are discussed. The thesis starts with a brief review of the relativistic quantum mechanics. In order to describe the effects of Lorentz transformations on the entangled states, quantum mechanics and special relativity are merged by construction of the unitary irreducible representations of Poincar{é} group on the infinite dimensional Hilbert space of state vectors. In this framework, the issue of finding the unitary irreducible representations of Poincar{é} group is reduced to that of the little group. Wigner rotation for the massive particles plays a crucial role due to its effect on the spin polarization directions. Furthermore, the physical requirements for constructing the correct relativistic spin operator is also studied. Then, the entanglement and Bell type inequalities are reviewed. Special attention has been devoted to two historical papers, by EPR in 1935 and by J.S. Bell in 1964. The main part of the thesis is based on the Lorentz transformation of the Bell states and the Bell inequalities on these transformed states. It is shown that entanglement is a Lorentz invariant quantity. That is, no inertial observer can see the entangled state as a separable one. However, it was shown that the Bell inequality may be satisfied for the Wigner angle dependent transformed entangled states. Since the Wigner rotation changes the spin polarization direction with the increased velocity, initial dichotomous operators can satisfy the Bell inequality for those states. By choosing the dichotomous operators taking into consideration the Wigner angle, it is always possible to show that Bell type inequalities can be violated for the transformed entangled states.
In this work we analyse the amount of entanglement associated with the spin and momentum degrees of freedom of a single massive spin- particle from a relativistic perspective. The effect of a Lorentz boost introduces a Wigner rotation that correlates the spin and momentum degrees of freedom. We show that the natural basis to discuss the geometrical effects of the boost are the helicity eigenstates in the rest frame. In the mid-relativistic regime (where the Wigner rotation angle is limited by ) we prove for states with equal helicity that the entanglement with respect to the Wigner rotation angle is monotonically decreasing, however, in the ultra-relativistic regime ( ) the entanglement is increasing. If the states are prepared as a superposition of unequal helicity eigenstates, the monotonic behaviour is inverted. This implies that in the ultra-relativistic regime a geometrical setup can be found such that the amount of entanglement exhibits local maxima or minima. This shows a counter-intuitive behaviour of the relative amount of entanglement, an effect due to the internal and external geometrical configuration space, and points towards the difficulties in achieving a Lorentz invariant formulation of entanglement in general.