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Open Systems & Information Dynamics

Maps Generated by Entangled Momenta: Exploring Spin Entanglement in Relativity

Veiko Palge, Jacob Dunningham, S. Groote, Hannes Liivat

Published Mar 1, 2019• 1 citation
Publisher

Abstract

We study relativistic entanglement of a bipartite system consisting of massive spin-1/2 particles with momenta. The spin state is described by the maximally entangled Bell state and momenta are given by entangled Gaussian distributions. We conceptualize the dependency between spin and momentum in relativity along the lines of controlled operations in quantum information theory. This leads to a systematic study of maps generated by Wigner rotations on the spin degree of freedom of the total system in different boost scenarios. We use a visualization tool from quantum information theory in order to get better insight into how and why the entanglement changes in different boost geometries.

DOI
10.1142/s123016121950001x

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We study relativistic entanglement of a bipartite system consisting of massive spin-1/2 particles with momenta. The spin state is described by the maximally entangled Bell state and momenta are given by entangled Gaussian distributions. We conceptualize the dependency between spin and momentum in relativity along the lines of controlled operations in quantum information theory. This leads to a systematic study of maps generated by Wigner rotations on the spin degree of freedom of the total system in different boost scenarios. We use a visualization tool from quantum information theory in order to get better insight into how and why the entanglement changes in different boost geometries.
In this paper we explore the entanglement of two relativistic spin-1/21/21/2 particles with continuous momenta. The spin state is described by the Bell state and the momenta are given by Gaussian distributions of product and entangled form. Transformations of the spins are systematically investigated in different boost scenarios by calculating the orbits and concurrence of the spin degree of freedom. By visualizing the behavior of the spin state we get further insight into how and why the entanglement changes in different boost situations.
In this paper we explore the entanglement of two relativistic spin-1/21/21/2 particles with continuous momenta. The spin state is described by the Bell state and the momenta are given by Gaussian distributions of product form. Transformations of the spins are systematically investigated in different boost scenarios by calculating the orbits and concurrence of the spin degree of freedom. By visualizing the behavior of the spin state we get further insight into how and why the entanglement changes in different boost situations.
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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.
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.
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.