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arXiv:quant-ph/0308156 · New Journal of Physics

Quantum entanglement under Lorentz boost

Daeho Lee, Ee Chang-Young

Published Jun 28, 2004• 62 citations
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Abstract

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.

DOI
10.1088/1367-2630/6/1/067

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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.
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 paper, the Lorentz transformations of entangled Bell states with general momentum not necessarily orthogonal to the boost direction and spin are studied. We extend quantum correlations and Bell's inequality to the relativistic regime by considering normalized relativistic observables. It is shown that quantum information along the direction perpendicular to the boost is eventually lost, and Bell's inequality is not always violated for entangled states in special relativity. This could impose restrictions on certain quantum information processing, such as quantum cryptography using massive particles.
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.
One of the most fundamental phenomena of quantum physics is entanglement. It describes an inseparable connection between quantum systems, and properties thereof. In a quantum mechanical description even systems far apart from each other can share a common state. This entanglement of the subsystems, although arising from mathematical principles, is no mere abstract concept, but can be tested in experiment, and be utilized in modern quantum information theory procedures, such as quantum teleportation. In particular, entangled states play a crucial role in testing our understanding of reality, by violating Bell inequalities. While the role of entanglement is well studied in the realm of nonrelativistic quantum mechanics, its significance in a relativistic quantum theory is a relatively new field of interest. In this work the consequences of a relativistic description of quantum entanglement are discussed. We analyze the representations of the symmetry groups of special relativity, i.e. of the Lorentz group, and the Poincaré group, on the Hilbert space of states. We describe how unitary, irreducible representations of the Poincaré group for massive spin 1/2 particles are constructed from representations of Wigner's little group. We then proceed to investigate the role of the Wigner rotations in the transformation of quantum states under a change of inertial reference frame. Considering different partitions of the Hilbert space of 2 particles, we find that the entanglement of the quantum states appears different in different inertial frames, depending on the form of the states, the chosen inertial frames, and the particular choice of partition. It is explained, how, despite of this, the maximally possible violation of Bell inequalities is frame independent, when using appropriate spin observables, which are related to the Pauli-Ljubanski vector, a Casimir operator of the Poincaré group.
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.