Generation of maximally entangled states with subluminal Lorentz boosts

Abstract

Recent work [1] has studied entanglement between the spin and momentum components of a single spin-1/2 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.Introduction.-Quantum entanglement is widely held to be the crucial feature that discriminates between quantum and classical physics; it is also at the heart of quantum information theory. While most of the theory of entanglement is non-relativistic, a complete account of entanglement requires that we understand its behavior in the relativistic regime.Studies in relativistic quantum information have found that single and two particle entanglement becomes an observer dependent phenomenon when viewed from different Lorentz boosted frames [2-10]. Recent work [1] has also investigated entanglement between the spin and momentum components of a single particle and showed that it reaches a maximum value only when boosts approach the speed of light. In this paper, however, we demonstrate that maximal entanglement can be obtained for realistic quantum states with boosts less than the speed of light. We furthermore show that this behavior can be given a natural geometric explanation.Properties of Wigner rotation.-We start by reviewing some of the properties of Wigner rotation that are key to our analysis. Wigner rotation arises from the fact that the subset of Lorentz boosts does not form a subgroup of the Lorentz group. Consider three inertial observers O, O and O where O has velocity v 1 relative to O and O has v 2 relative to O . Then the combination of two canonical boosts Λ(v 1 ) and Λ(v 2 ) that relates O to O is in general a boost and a rotation,

Maps Generated by Entangled Momenta: Exploring Spin Entanglement in Relativity

“…the angle between two boosts as well as the the momenta of the system and the observer. As an interesting implication, note that this means that states whose spin and momentum are separable for observer O may display spin-momentum entanglement to the moving observer O Λ , see for example [3,17]. The peculiar dependency of spin on momentum can be viewed in terms of an analogy from quantum information theory [2,3].…”

“…Therefore, one of the central challenges of relativistic quantum information is the characterization of entanglement in relativity. The first studies on the subject appeared more than a decade ago [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the overall conclusion emerging is that relativistic entanglement in both inertial and accelerated frames is observer dependent [18].…”

“…Analyzing the situation reveals that these findings commonly involve different momentum states and boost angles, or geometries. The study of the simplest system, the single particle [17], which can be viewed as playing the role of a relativistic qubit, also suggests that geometry has an important role to play. Along the same lines, the literature on the Wigner rotation is quite clear about the fact that its nature is highly geometric, yet except from a few cases [24] there is little work that systematically takes this into account.…”

Complete complementarity relations and their Lorentz invariance

“…As pointed out by Palge & Dunningham [21], the main aspect to be noticed here is that many of these apparently conflicting results involve systems containing different particle states and boost geometries. Therefore, entanglement under Lorentz boosts is highly dependent on the boost scenario in question [22]. For single-particle states, a spin-momentum product state can be transformed into an entangled state.…”

Complete complementarity relations in curved spacetimes

“…These apparently conflicting results involve systems containing different particle states and boost geometries [13]. Therefore, entanglement under Lorentz boosts is highly dependent on the boost scenario in question [14], which led to a rich variety of works exploring these different scenarios by several researchers [15][16][17][18][19][20][21][22]. More generally, the entanglement for observers constantly accelerated in a flat space-time was considered in [23][24][25].…”