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arXiv:1101.2565 · Physical Review A

Extraction of timelike entanglement from the quantum vacuum

S. Jay Olson, Timothy C. Ralph

Published Jan 4, 2012• 78 citations
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Abstract

An intriguing property of the massless quantum vacuum state is that it contains entanglement between both spacelike and timelike separated regions of space-time. The implications of timelike entanglement and its connection to standard entanglement, however, are unexplored. Here we show that timelike entanglement can be extracted from the massless Minkowski vacuum and converted into standard entanglement ``at a given time'' between two inertial, two-state detectors at the same spatial location: one coupled to the field in the past and the other coupled to the field in the future. The procedure used here demonstrates a time correlation as a requirement for extraction; e.g., if the past detector was active at a quarter to 12:00, then the future detector must wait to become active at precisely a quarter past 12:00 in order to achieve entanglement.

DOI
10.1103/physreva.85.012306

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An intriguing property of the massless quantum vacuum state is that it contains entanglement between both spacelike and timelike separated regions of space-time. The implications of timelike entanglement and its connection to standard entanglement, however, are unexplored. Here we show that timelike entanglement can be extracted from the massless Minkowski vacuum and converted into standard entanglement ``at a given time'' between two inertial, two-state detectors at the same spatial location: one coupled to the field in the past and the other coupled to the field in the future. The procedure used here demonstrates a time correlation as a requirement for extraction; e.g., if the past detector was active at a quarter to 12:00, then the future detector must wait to become active at precisely a quarter past 12:00 in order to achieve entanglement.
We note that massless fields within the future and past light cone may be quantized as independent systems. The vacuum is shown to be a nonseparable state of these systems, exactly mirroring the known entanglement between the spacelike separated Rindler wedges. This leads to a notion of timelike entanglement. We describe an inertial detector which exhibits a thermal response to the vacuum when switched on at t=0, due to this property. The feasibility of detecting this effect is discussed, with natural experimental parameters appearing at the scale of 100 GHz.
We demonstrate that the future and left Rindler wedges of Minkowski spacetime are entangled, leading to the Unruh effect. Similarly, the past and right Rindler wedges are also entangled. We propose a protocol to extract this entanglement using two two-state detectors located in the past and right Rindler wedges. By scaling the detector transition frequencies inversely with Minkowski time, entanglement from the quantum field is transferred to the detectors, suggesting they may support quantum teleportation via the vacuum. Our protocol can be implemented with current quantum systems, such as flux-tunable transmon qubits. This research provides new insights into the entanglement properties of spacetime and hints at practical applications for secure quantum information transfer using the vacuum state of a quantum field.
In this study, we establish a connection between time- and spacelike entanglement entropy. We show that timelike entanglement entropy is closely related to spacelike entanglement entropy and its temporal derivative. For a broad class of states, it can be uniquely determined by a linear combination of spacelike entanglement entropy and its first-order temporal derivative. This relation holds, for instance, in states conformally equivalent to the vacuum in two-dimensional conformal field theories. For more general states, we demonstrate that the relation can be constructed perturbatively. Our results suggest that timelike entanglement entropy is constrained by causality. Moreover, this relation provides a unified framework for time- and spacelike entanglement entropy, within which the imaginary component of timelike entanglement entropy can be understood as arising from the noncommutativity between the twist operator and its first-order temporal derivative.
We consider entanglement extraction into two two-level Unruh-DeWitt detectors from a vacuum of a neutral massless quantum scalar field in a four-dimensional space-time, where the general monopole coupling to the scalar field is assumed. Based on the reduced density matrix of the two detectors derived within the perturbation theory, we show that the single copy of the entangled pair of the detectors can be utilized in quantum teleportation, even when the detectors are separated acausally, while we observe no violation of the Bell--CHSH inequality. In the case of the Minkowski vacuum, in particular, we find that entanglement usable in quantum teleportation is extracted due to the special relativistic effect when the detectors are in a relative inertial motion, while it is not when they are comoving inertially and the switching of the detectors is executed adiabatically at infinite past and future.
We ask whether quantum teleportation from the outside to the inside of a horizon is possible, using entanglement extracted from a vacuum. We first calculate analytically, within the perturbation theory, entanglement extracted from the Minkowski vacuum into a pair of an inertial and an accelerated Unruh-DeWitt detectors, which are initially in the ground states and interact with a neutral massless scalar field for an infinitely long time. We find that entanglement can be extracted, but is "fragile", depending on adiabaticity of switching of the detectors at the infinite past and future. We then consider the standard scheme of quantum teleportation utilizing the extracted entanglement, and find that the standard teleportation is not superior to channels without entanglement.
We consider a pair of two-level Unruh-DeWitt detectors accelerated uniformly in the Minkowski vacuum of a massless neutral scalar field and analyze, within the perturbation theory, the entanglement extracted from the vacuum into the Unruh-DeWitt detectors when the switching of the detectors is performed adiabatically enough at the asymptotic past and future. We consider the cases where the detectors are accelerated parallelly, antiparallelly, and in differently orientated directions. We show that entanglement is extracted if they are accelerated antiparallelly and the ratios of the excitation energy to the magnitude of the acceleration coincide between the two detectors. On the other hand, we find that the detectors are not entangled when the detectors are accelerated parallelly or in orientated directions. We discuss these results from the viewpoint of the energy conservation associated with the timelike boost Killing vector fields tangent to the worldlines of the detectors.