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arXiv:1109.5972 · Physical Review A

Behavior of entanglement and Cooper pairs under relativistic boosts

Veiko Palge, Vlatko Vedral, Jacob Dunningham

Published Oct 14, 2011• 22 citations
arXivHTMLPDF

Abstract

Recent work [J. A. Dunningham, V. Palge, and V. Vedral, Phys. Rev. A 80, 044302 (2009)] has shown how single-particle entangled states are transformed when boosted in relativistic frames for certain restricted geometries. Here we extend that work to consider completely general inertial boosts. We then apply our single-particle results to multiparticle entanglements by focusing on Cooper pairs of electrons. We show that a standard Cooper pair state consisting of a spin-singlet acquires spin-triplet components in a relativistically boosted inertial frame, regardless of the geometry. We also show that, if we start with a spin-triplet pair, two out of the three triplet states acquire a singlet component, the size of which depends on the geometry. This transformation between the different singlet and triplet superconducting pairs may lead to a better understanding of unconventional superconductivity.

DOI
10.1103/physreva.84.044303

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Recent work has argued that the concepts of entanglement and nonlocality must be taken seriously even in systems consisting of only a single particle. These treatments, however, are nonrelativistic, and, if single-particle entanglement is fundamental, it should also persist in a relativistic description. Here, we consider a spin-1/2 particle in a superposition of two different velocities as viewed by an observer in a relativistically boosted inertial frame and show that the entanglement between the two velocity modes survives right up to the speed of light. We also discuss how quantum gates could be implemented in this way and apply our results to the case of a superconductor. In particular, we show that an sss-wave superconductor would have ppp-wave components for a boosted observer.
Recent work [J. Dunningham, V. Palge, and V. Vedral, Phys. Rev. A 80, 044302 (2009)] studied entanglement between the spin and momentum components of a single spin-1/21/21/2 particle and showed that maximal entanglement is obtained only when boosts approach the speed of light. Here we extend the boost scenario to general geometries and show that, intriguingly, maximal entanglement can be achieved with boosts less than the speed of light. Boosts approaching the speed of light may even decrease entanglement. We also provide a geometric explanation for this behavior.
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.
It is well known that entanglement under Lorentz boosts is highly dependent on the boost scenario in question. For single-particle states, a spin-momentum product state can be transformed into an entangled state. However, entanglement is just one of the aspects that completely characterizes a quantum system. The other two are known as the wave-particle duality. Although the entanglement entropy does not remain invariant under Lorentz boosts, and neither do the measures of predictability and coherence, we show here that these three measures taken together, in a complete complementarity relation (CCR), are Lorentz invariant. Peres et al. (Peres et al. 2002 Phys. Rev. Lett. 88 , 230402. ( doi:10.1103/PhysRevLett.88.230402 )) realized that even though it is possible to formally define spin in any Lorentz frame, there is no relationship between the observable expectation values in different Lorentz frames. Analogously, one can, in principle, define complementary relations in any Lorentz frame, but there is no obvious transformation law relating complementary relations in different frames. However, our result shows that the CCRs have the same value in any Lorentz frame, i.e. there is a transformation law connecting the CCRs. In addition, we explore relativistic scenarios for single and two-particle states, which helps in understanding the exchange of different aspects of a quantum system under Lorentz boosts.