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arXiv:quant-ph/0209164 · Physical Review A

Relativistic entanglement and Bell’s inequality

Doyeol Ahn, Hyuk‐Jae Lee, Young Hoon Moon, Sung Woo Hwang

Published Jan 10, 2003• 116 citations

Abstract

In this paper, the Lorentz transformation of entangled Bell states seen by a moving observer is studied. The calculated Bell observable for four joint measurements turns out to give a universal value, $〈\^a\ensuremath{\bigotimes}\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}〉+〈\^a\ensuremath{\bigotimes}{b}^{\ensuremath{'}}〉+〈{a}^{\ensuremath{'}}\ensuremath{\bigotimes}\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}〉\ensuremath{-}〈{a}^{\ensuremath{'}}\ensuremath{\bigotimes}{b}^{\ensuremath{'}}〉=(2/\sqrt{2\ensuremath{-}{\ensuremath{\beta}}^{2}})(1+\sqrt{1\ensuremath{-}{\ensuremath{\beta}}^{2}}),$ where $\^a,\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}$ are the relativistic spin observables derived from the Pauli-Lubanski pseudovector and $\ensuremath{\beta}=(v/c).$ We found that the degree of violation of the Bell's inequality is decreasing with increasing velocity of the observer and Bell's inequality is satisfied in the ultrarelativistic limit where the boost speed reaches the speed of light.

DOI: 10.1103/physreva.67.012103

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In this paper, the Lorentz transformation of entangled Bell states seen by a moving observer is studied. The calculated Bell observable for four joint measurements turns out to give a universal value,

〈\^a\ensuremath{\bigotimes}\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}〉+〈\^a\ensuremath{\bigotimes}{b}^{\ensuremath{'}}〉+〈{a}^{\ensuremath{'}}\ensuremath{\bigotimes}\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}〉\ensuremath{-}〈{a}^{\ensuremath{'}}\ensuremath{\bigotimes}{b}^{\ensuremath{'}}〉=(2/\sqrt{2\ensuremath{-}{\ensuremath{\beta}}^{2}})(1+\sqrt{1\ensuremath{-}{\ensuremath{\beta}}^{2}}),

where

\^a,\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}

are the relativistic spin observables derived from the Pauli-Lubanski pseudovector and

\ensuremath{\beta}=(v/c).

We found that the degree of violation of the Bell's inequality is decreasing with increasing velocity of the observer and Bell's inequality is satisfied in the ultrarelativistic limit where the boost speed reaches the speed of light.

Entanglement of tripartite spin states under Lorentz transformations is studied in the context of Bell’s inequality and positive partial transpose criterion. First the relativistic analogue of Bell’s inequality is discussed for three qubit states by explicit calculation of the Wigner rotation. We use the relativistic invariant spin operator which is related to the Pauli-Lubanski pseudo vector. For observers at rest the Bell’s inequality is speed-independent and maximally violated. For moving observers it’s shown that Bell’s inequality is violated and the amount of violation depends on the boost speed. We show that in ultrarelativistic limit Bell’s inequality is still maximally violated. We also obtained the critical value for satisfying Bell’s inequality. The critical value of boost speed for violation of inequality for particles moving in the center of mass frame is greater than that for particles moving with the same momentum. Second we investigate the entanglement distillability of tripartite mixed spin states under Lorentz transformations in the context of Werner states. We show that there are states that will change from distillable (entangled) into separable for a certain value of rapidity.

To understand the characteristics of quantum entanglement of massive particles under Lorentz boost, we first introduce a relevant relativistic spin observable, and evaluate its expectation values for the Bell states under Lorentz boost. Then we show that maximal violation of the Bell's inequality can be achieved by properly adjusting the directions of the spin measurement even in a relativistically moving inertial frame. Based on this we infer that the entanglement information is preserved under Lorentz boost as a form of correlation information determined by the transformation characteristic of the Bell state in use.

In order to understand the characteristics of quantum entanglement of massive particles under Lorentz boost, we first introduce a relevant relativistic spin observable, and evaluate its expectation values for the Bell states under Lorentz boost. Then we show that maximal violation of the Bell's inequality can be achieved by properly adjusting the directions of the spin measurement even in a relativistically moving inertial frame. Based on this we infer that the entanglement information is preserved under Lorentz boost as a form of correlation information determined by the transformation characteristic of the Bell state in use.

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.