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is entanglement purification (see Section 3.2.2), which is one ingredient for a
quantum repeater and is also based on the application of elementary quan-
tum gates such as a controlled NOT (CNOT) gate (see Section 3.2.3). Another
promising line of development involves entanglement in higher dimensions,
which might allow further advances such as quantum communication with a
higher resistance against noise (see Section 3.2.4). A very recent development
is the real-world application of entanglement-based quantum cryptography
(see Section 3.2.5). This is linked to the research on distributing entanglement
over long distances, which aims at the establishment of a quantum commu-
nication network (see Section 3.2.6), eventually on a global scale by using
satellites.
3.2 Advanced Quantum Communication
Schemes
3.2.1 Scalable Teleportation and Entanglement
Swapping
Teleportation of quantum states [12] is an intriguing concept within quan-
tum physics and a striking application of quantum entanglement. Besides
its importance for quantum computation [13,14], teleportation is at the heart
of the quantum repeater [15], a concept eventually allowing the distribution
of quantum entanglement over arbitrary distances and thus enabling quan-
tum communication over large distances and even networking on a global
scale.
The purpose of quantum teleportation is to transfer an arbitrary quan-
tum state to a distant location, e.g., from Alice to Bob, without transmit-
ting the actual physical object carrying the state. Classically this is an im-
possible task, since Alice cannot obtain the full information of the state to
be teleported without previous knowledge about its preparation. Quantum
physics, however, provides a working strategy. Suppose, Alice and Bob share
an ancilla entangled pair in advance. Alice then performs a Bell state mea-
surement between the teleportee particle and her shared ancilla, i.e., she
projects the two particles into the basis of Bell states. The four possible out-
comes of this measurement provide her with two bits of classical informa-
tion, which is sufficient to reconstruct the initial quantum state at Bob's side.
After communicating the classical result to Bob, he can perform one out of
four unitary operations to obtain the original state to be teleported. In de-
tail: suppose photon 1, which Alice wants to teleport to Bob, is in a general
polarization state
1 (unknown to Alice), and the pair of
photons 2 and 3 shared by Alice and Bob is in the polarization-entangled
state
| χ 1 = α |
H
1 + β |
V
| 23 . This state is one of the four maximally entangled Bell states
| ± ij =
1
| ± ij =
1
, where H and
V denote horizontal and vertical linear polarizations, and i and j index the
spatial modes of the photons. The overall state of photons 1, 2, and 3 can be
2 ( |
HV
ij ±|
VH
ij )
and
2 ( |
HH
ij ±|
VV
ij )
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