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arXiv:0912.4863 · Physical Review A

Relativistic entanglement of two massive particles

Nicolai Friis, Reinhold A. Bertlmann, Marcus Huber, Beatrix C. Hiesmayr

Published Apr 30, 2010• 113 citations
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Abstract

We describe the spin and momentum degrees of freedom of a system of two massive spin-$\frac{1}{2}$ 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.

DOI
10.1103/physreva.81.042114

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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 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.
We analyze the entanglement change, as seen by different relativistic observers, for a system consisting of two spin-1 particles, considering different partitions of the Hilbert space, which has spin and momentum degrees of freedom. We show that there exists a complete set of states of the spin subspace in which the entanglement change of any state in the set is zero for all partitions and all values of the Wigner angle. Moreover, these states only change by a global phase factor under the Lorentz boost. Within this basis, maximally entangled invariant states, interesting for quantum information purposes, are explicitly obtained. On the other hand, the entanglement in the particle-particle partition isisis Lorentz invariant, thus protecting the consistency of quantum correlations and teleportation results. We show how our results may be generalized to arbitrary spin.
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.
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.
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.
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