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ScienceWeek
QUANTUM PHYSICS: ON ATOM QUANTUM TELEPORTATION
Notes by ScienceWeek:
Quantum
teleportation is the transmission and reconstruction over arbitrary
distances of the state of a quantum system, an effect first suggested
by Bennett et al in 1993 (Phys. Rev. Lett. 70:1895). The achievement of
the effect depends on the phenomenon of entanglement, an essential
feature of quantum mechanics. Entanglement is unique to quantum
mechanics, and involves a relationship (a "superposition of states")
between the possible quantum states of two entities such that when the
possible states of one entity collapse to a single state as a result of
suddenly imposed boundary conditions, a similar and related collapse
occurs in the possible states of the entangled entity no matter where
or how far away the entangled entity is located.
The following points are made by M. Riebe et al (Nature 2004 429:734):
1)
Teleportation of a quantum state encompasses the complete transfer of
information from one particle to another. The complete specification of
the quantum state of a system generally requires an infinite amount of
information, even for simple two-level systems (qubits). Moreover, the
principles of quantum mechanics dictate that any measurement on a
system immediately alters its state, while yielding at most one bit of
information. The transfer of a state from one system to another (by
performing measurements on the first and operations on the second)
might therefore appear impossible. However, it has been shown(1) that
the entangling properties of quantum mechanics, in combination with
classical communication, allow quantum-state teleportation to be
performed. Teleportation using pairs of entangled photons has been
demonstrated(2-5), but such techniques are probabilistic, requiring
post-selection of measured photons.
2) Teleportation of a state
from a source qubit to a target qubit requires three qubits: the
sender's source qubit and an ancillary qubit that is maximally
entangled with the receiver's target qubit, providing the strong
quantum correlation. Once these states have been prepared, a quantum
mechanical measurement is performed jointly on the source qubit and the
ancilla qubit (specifically, a Bell-state measurement, which projects
the two qubits onto a basis of maximally entangled states). In this
process, the two qubits are projected onto one of four equally likely
outcomes. At the same time, the non-local properties of quantum
mechanics cause the target qubit to be projected onto one of four
corresponding states, each related to the original state of the source
qubit, even though no measurement was performed on this qubit.
Knowledge of the result of the source-ancilla measurement allows one to
choose a simple a priori operation to be carried out on the target
qubit, resulting in reconstruction of the original quantum state.
3)
Because each of the four results on the source-ancilla measurement are
equally likely, regardless of the nature of the teleported state, no
information about the state is obtained (thus the no-cloning theorem is
not violated). Further, as classical communication of the measurement
outcome is required to complete the state reconstruction, a state
cannot be teleported faster than the speed of light. Teleportation does
demonstrate a number of fascinating fundamental properties of quantum
theory, in particular the non-local property of entangled states, which
allows the projective measurement of the source-ancilla pair to create
a definite pure state in the target qubit. Furthermore, teleportation
has considerable implications for the nascent technology of quantum
information processing. Besides being a compelling benchmark algorithm
for a three-qubit quantum computer, teleportation is a possible
primitive for large-scale devices.
4) In summary: The authors
report deterministic quantum-state teleportation between a pair of
trapped calcium ions. Following closely the original proposal(1), the
authors create a highly entangled pair of ions and perform a complete
Bell-state measurement involving one ion from this pair and a third
source ion. State reconstruction conditioned on this measurement is
then performed on the other half of the entangled pair. The measured
fidelity is 75%, demonstrating unequivocally the quantum nature of the
process.
References (abridged):
1. Bennett, C. H. et al.
Teleporting an unknown quantum state via dual classical and EPR
channels. Phys. Rev. Lett. 70, 1895-1899 (1993)
2. Bouwmeester, D. et al. Experimental quantum teleportation. Nature 390, 575-579 (1997)
3.
Boschi, D., Branca, S., DeMartini, F., Hardy, L. & Popescu, S.
Experimental realization of teleporting an unknown pure quantum state
via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev.
Lett. 80, 1121-1125 (1998)
4. Pan, J.-W., Daniell, M.,
Gasparoni, S., Weihs, G. & Zeilinger, A. Experimental demonstration
of four-photon entanglement and high-fidelity teleportation. Phys. Rev.
Lett. 86, 4435-4438 (2001)
5. Marcikic, I., de Riedmatten, H.,
Tittel, W., Zbinden, H. & Gisin, N. Long-distance teleportation of
qubits at telecommunication wavelengths. Nature 421, 509-513 (2003)
Nature http://www.nature.com/nature
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Related Material:
ON QUANTUM TELEPORTATION AND ENTANGLEMENT SWAPPING
Notes by ScienceWeek:
In
general, a "Hilbert space" is a linear vector space that can have an
infinite number of dimensions, the concept important because in quantum
mechanics the state of a system is represented by a vector in Hilbert
space. The dimension of the Hilbert space is not related to the
physical dimension of the system. The concept is named after the
mathematician David Hilbert (1862-1943).
The following points are made by D.W. Berry and B.C. Sanders {New Journal of Physics 2002 4:8):
1)
Quantum teleportation enables disembodied transport of the state of a
system to a distant system through (i) a shared entanglement resource,
(ii) a classical communication channel between the sender and receiver
[1] and (iii) an experimentally established isomorphism between the
Hilbert spaces of the sender and receiver [2]. Quantum teleportation is
significant in several areas, including transmission of quantum states
in noisy environments [1], sharing states in distributed quantum
networks [3] and implementation of quantum computation using resources
prepared offline [4,5]. Teleportation was initially proposed for
discrete-variable systems, where the state to be teleported has
finite-N levels [1], and a continuous-variable version has been adapted
for squeezed light experiments. The authors discuss quantum
teleportation of a quantum state in an arbitrary but finite
N-dimensional Hilbert space , realized physically as a spin system,
thereby generalizing the recent spin quantum teleportation proposal by
Kuzmich and Polzik (Phys. Rev. Lett. 2000 85:5639), which is only valid
in the infinite-N limit.
2) "Entanglement swapping" is closely
related to quantum teleportation. Whereas quantum teleportation enables
the state of a system (e.g. a particle or collection of particles) to
be teleported to an independent physical system via classical
communication channels and a shared entanglement resource, the purpose
of entanglement swapping is to instill entanglement between systems
that hitherto shared no entanglement. An entanglement resource is
required for entanglement swapping to occur; indeed the nomenclature
"entanglement swapping" describes the transfer of entanglement from a
priori entangled systems to a priori separable systems.
3) A
connection between entanglement swapping and quantum teleportation can
be seen as follows. Consider quantum teleportation of the state of one
particle, which is initially entangled with a second particle, but the
state of the second particle does not undergo quantum teleportation. In
perfect quantum teleportation, the state of the first particle is
faithfully transferred to a third particle that was initially
independent of the first two particles. Thus, subsequent to the quantum
teleportation, the second and third particles are entangled, perfectly
replacing the a priori entanglement of the first and second particles.
The entanglement resource inherent in quantum teleportation devices
enables this entanglement swapping to occur; thus, equivalence between
optimal entanglement resources for quantum teleportation and
entanglement swapping might be expected, but the authors demonstrate
that the optimal entanglement resources differ between quantum
teleportation and entanglement swapping for finite-N spin systems.
References (abridged):
1. Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A and Wootters W K 1993 Phys. Rev. Lett. 70:1895
2. van Enk S J 2001 J. Mod. Opt. 48:2049
3. Cirac J I, Ekert A K, Huelga S F and Macchiavello C 1999 Phys. Rev. A 59:4249
4. Gottesman D and Chuang I L 1999 Nature 402:390
5. Knill E, Laflamme R and Milburn G J 2001 Nature 409:46
New Journal of Physics http://www.iop.org/EJ/njp
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