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| '''Quantum teleportation''' is a process by which quantum information (e.g. the exact state of an atom or photon) can be transmitted (exactly, in principle) from one location to another, with the help of [[Classical_information#Classical_versus_quantum_information|classical communication]] and previously shared [[quantum entanglement]] between the sending and receiving location. Because it depends on classical communication, which can proceed no faster than the speed of light, it cannot be used for [[superluminal]] transport or communication of classical bits. It also cannot be used to make copies of a system, as this violates the [[no-cloning theorem]]. Although the name is inspired by the [[teleportation in fiction|teleportation]] commonly used in [[fiction]], fiction far outpaces current technology: although single atoms have been teleported,<ref name="nyt"/><ref name="barrett"/><ref name="riebe"/> molecules or anything larger, such as living things, have not. One may think of teleportation as either a kind of transportation, or as a kind of communication; it provides a way of transporting a [[qubit]] from one location to another, without having to actually move a physical particle along with it. | | I'm Bridgett and was born on 13 September 1980. My hobbies are Vehicle restoration and Table tennis.<br><br>Also visit my homepage ... facebook app [[http://facebookvouchers.com facebookvouchers.com]] |
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| The seminal paper first expounding the idea was published by [[Charles H. Bennett (computer scientist)|C. H. Bennett]], [[Gilles Brassard|G. Brassard]], [[Claude Crépeau|C. Crépeau]], [[Richard Jozsa|R. Jozsa]], [[Asher Peres|A. Peres]] and [[William Wootters|W. K. Wootters]] in 1993.<ref>[[Charles H. Bennett (computer scientist)|C. H. Bennett]], [[Gilles Brassard|G. Brassard]], [[Claude Crépeau|C. Crépeau]], [[Richard Jozsa|R. Jozsa]], [[Asher Peres|A. Peres]], [[William Wootters|W. K. Wootters]], ''Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels'', [[Phys. Rev. Lett.]] '''70''', 1895-1899 (1993) ([http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.9405 online]).</ref> Since then, quantum teleportation has been realized in various physical systems. Presently, the record distance for quantum teleportation is {{convert|143|km|mi|abbr=on}} with photons,<ref>{{Cite journal
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| | last1 = Ma | first1 = X. S.
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| | last2 = Herbst | first2 = T.
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| | last3 = Scheidl | first3 = T.
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| | last4 = Wang | first4 = D.
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| | last5 = Kropatschek | first5 = S.
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| | last6 = Naylor | first6 = W.
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| | last7 = Wittmann | first7 = B.
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| | last8 = Mech | first8 = A.
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| | last9 = Kofler | first9 = J.
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| | displayauthors=8
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| | doi = 10.1038/nature11472
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| | title = Quantum teleportation over 143 kilometres using active feed-forward
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| | journal = Nature
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| | volume = 489
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| | issue = 7415
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| | pages = 269–273
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| | year = 2012
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| | pmid = 22951967
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| |bibcode = 2012Natur.489..269M }}</ref>
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| and 21m with material systems.<ref>C. Nölleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, S. Ritter}, ''Efficient Teleportation Between Remote Single-Atom Quantum Memories'', [[Phys. Rev. Lett.]] '''110''', 140403 (2013) ([http://prl.aps.org/abstract/PRL/v110/i14/e140403 online], [http://arxiv.org/abs/1212.3127 arXiv])</ref> On September 11, 2013, the "Furusawa group at the University of Tokyo has succeeded in demonstrating complete quantum teleportation of photonic quantum bits by a hybrid technique for the first time worldwide." <ref>http://akihabaranews.com/2013/09/11/article-en/world-first-success-complete-quantum-teleportation-750245129</ref>
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| ==Non-technical summary==
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| It is known, from [[axiom]]atizations of [[quantum mechanics]] (such as [[categorical quantum mechanics]]), that the universe is fundamentally composed of two things: [[bit]]s and [[qubit]]s.<ref>Lluís Masanes, Markus P. Müller, Remigiusz Augusiak, and David Pérez-García, "Existence of an information unit as a postulate of quantum theory", (2012) [http://arxiv.org/abs/1208.0493 aRxIV 1208.0493]</ref><ref name=coecke>Bob Coecke, "Quantum Picturalism", (2009) ''Contemporary Physics'' vol '''51''', pp59-83. ([http://arxiv.org/abs/0908.1787 ArXiv 0908.1787])</ref> Bits are units of information, and are commonly represented using zero or one, true or false. These bits are sometimes called "classical" bits, to distinguish them from quantum bits, or [[qubit]]s. Qubits also encode a type of information, called [[quantum information]], which differs sharply from "classical" information. For example, a qubit cannot be used to encode a classical bit (this is the content of the [[no-communication theorem]]). Conversely, classical bits cannot be used to encode qubits: the two are quite distinct, and not inter-convertible. Qubits differ from classical bits in dramatic ways: they cannot be copied (the [[no-cloning theorem]]) and they cannot be destroyed (the [[no-deleting theorem]]).
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| Quantum teleportation provides a mechanism of moving a qubit from one location to another, without having to physically transport the underlying particle that a qubit is normally attached to. This is important, as heretofore, the only way of moving qubits was to move the actual physical particles that qubits are attached to (e.g. by foot, car or airplane). Much like the invention of the [[telegraph]] allowed classical bits to be transported at high speed across continents, so also quantum teleportation holds the promise that one day, qubits could be moved likewise. However, as of 2013, only photons and single atoms have been teleported; molecules have not, nor does this even seem likely in the upcoming years, as the technology remains daunting. Specific distance and quantity records are stated below.
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| The movement of qubits does require the movement of "things"; in particular, the actual teleportation protocol requires that an [[entangled state|entangled quantum state]] or [[Bell state]] be created, and its two parts shared between two locations (the source and destination, or [[Alice and Bob]]). In essence, a certain kind of "[[quantum channel]]" between two sites must be established first, before a qubit can be moved. Teleportation also requires a [[communication channel|classical information link]] to be established, as two classical bits must be transmitted to accompany each qubit. The need for such links may, at first, seem disappointing; however, this is not unlike ordinary communications, which requires wires, radios or lasers. What's more, Bell states are most easily shared using [[photon]]s from [[laser]]s, and so teleportation could be done, in principle, through open space.
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| Single atoms have been teleported,<ref name="nyt">New York Times, [http://www.nytimes.com/2004/06/17/us/scientists-teleport-not-kirk-but-an-atom.html Scientists Teleport Not Kirk, but an Atom] (2004)</ref><ref name="barrett">M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri & D. J. Wineland, [http://www.nature.com/nature/journal/v429/n6993/abs/nature02608.html "Deterministic quantum teleportation of atomic qubits"] ''Nature'' '''429''', 737-739 (17 June 2004) doi:10.1038/nature02608</ref><ref name="riebe">M. Riebe, H. Häffner, C. F. Roos, W. Hänsel, J. Benhelm, G. P. T. Lancaster, T. W. Körber, C. Becher, F. Schmidt-Kaler, D. F. V. James & R. Blatt, [http://www.nature.com/nature/journal/v429/n6993/abs/nature02570.html "Deterministic quantum teleportation with atoms"], ''Nature'' '''429'', 734-737 (17 June 2004) doi:10.1038/nature02570</ref> although not in the science-fiction sense. An atom consists of several parts: the qubits in the [[electronic state]] or [[electron shell]]s surrounding the [[atomic nucleus]], the qubits in the nucleus itself, and, finally, the [[electron]]s, [[proton]]s and [[neutron]]s making up the atom. Physicists have teleported the qubits encoded in the electronic state of atoms; they have not teleported the nuclear state, nor the nucleus itself. Thus, performing this kind of teleportation requires a feedstock of atoms at the receiving site, that are readily available for having qubits imprinted on them. In an abstract sense, the feedstock is not strictly necessary: the atoms and their nuclei could be created, for example, with [[particle accelerator|atom-smashers]] or via [[nuclear fusion]]; this, however, would be economically absurd, and so a feedstock is used instead. The importance of teleporting nuclear state is unclear: nuclear state does affect the atom, e.g. in [[hyperfine splitting]], but whether such state would need to be teleported in some futuristic "practical" application is debatable.
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| The quantum world is strange and unusual; so, aside from no-cloning and no-deleting, there are other oddities. For example, quantum correlations arising from Bell states seem to be instantaneous (the [[Bell test experiments|Alain Aspect experiments]]), whereas classical bits can only be transmitted slower than the speed of light (quantum correlations cannot be used to transmit classical bits; again, this is the [[no-communication theorem]]). Thus, teleportation, as a whole, can never be [[superluminal]], as a qubit cannot be reconstructed until the accompanying classical bits arrive.
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| The proper description of quantum teleportation requires a basic mathematical toolset, which, although complex, is not out of reach of advanced high-school students, and indeed becomes accessible to college students with a good grounding in finite-dimensional [[linear algebra]]. In particular, the theory of [[Hilbert space]]s and [[projection matrix]]es is heavily used. A qubit is described using a two-dimensional [[complex number]]-valued [[vector space]] (a Hilbert space); the formal manipulations given below do not make use of anything much more than that. Strictly speaking, a working knowledge of quantum mechanics is not required to understand the mathematics of quantum teleportation, although without such acquaintance, the deeper meaning of the equations may remain quite mysterious.
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| ==Protocol==
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| [[Image:Quantum teleportation diagram.PNG|300px|thumb|right| Diagram for quantum teleportation of a photon]]
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| The prerequisites for quantum teleportation are a [[qubit]] that is to be teleported, a conventional [[communication channel]] capable of transmitting two classical bits (i.e., one of four states), and means of generating an [[Quantum entanglement|entangled]] [[Bell state|EPR pair]] of qubits, transporting each of these to two different locations,A and B, performing a [[Bell measurement]] on one of the EPR pair qubits, and manipulating the quantum state of the other of the pair. The [[protocol (computing)|protocol]] is then as follows:
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| # An EPR pair is generated, one qubit sent to location A, the other to B.
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| # At location A, a Bell measurement of the EPR pair qubit and the qubit to be teleported (the quantum state <math>|\phi \rangle</math>) is performed, yielding one of four possibilities, which can be encoded in two classical bits of information. Both qubits at location A are then discarded.
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| # Using the classical channel, the two bits are sent from A to B. (This is the only potentially time-consuming step, due to speed-of-light considerations.)
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| # As a result of the measurement performed at location A, the EPR pair qubit at location B is in one of four possible states. Of these four possible states, one is identical to the original quantum state <math>|\phi \rangle</math>, and the other three are closely related. Which of these four possibilities it actually is, is encoded in the two classical bits. Knowing this, the qubit at location B is modified in one of three ways, or not at all, to result in a qubit identical to <math>|\phi \rangle</math>, the qubit that was chosen for teleportation.
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| == Experimental results and records ==
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| Work in 1998 verified the initial predictions,<ref name="Rome1998">{{cite journal
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| |journal=[[Physical Review Letters]]
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| |volume=80
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| |issue=6
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| |page=1121
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| |doi= 10.1103/PhysRevLett.80.1121
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| |title=Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels
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| |author=D. Boschi
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| |coauthors=S. Branca1, F. De Martini1, L. Hardy, and S. Popescu
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| |year=1998
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| |arxiv = quant-ph/9710013 |bibcode = 1998PhRvL..80.1121B }}</ref> and the distance of teleportation was increased in August 2004 to 600 meters, using [[optical fiber]].<ref name="Danube2004">{{cite web
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| |url=http://www.nature.com/nature/journal/v430/n7002/full/430849a.html
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| |title=Quantum teleportation across the Danube
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| |author=Rupert Ursin
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| |date=August 2004
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| |publisher=Nature
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| |accessdate=2010-05-22
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| }}</ref> The longest distance yet claimed to be achieved for quantum teleportation is {{convert|143|km|mi|abbr=on}}, performed in May 2012, between the two Canary Islands of La Palma and Tenerife off the Atlantic coast of north Africa.<ref name="Canary2012">{{cite web
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| |url=http://arxiv.org/abs/1205.3909
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| |title=Quantum teleportation using active feed-forward between two Canary Islands
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| |author=Xiao-song Ma, Thomas Herbst, Thomas Scheidl, Daqing Wang, Sebastian Kropatschek, William Naylor, Alexandra Mech, Bernhard Wittmann, Johannes Kofler, Elena Anisimova, Vadim Makarov, Thomas Jennewein, Rupert Ursin, Anton Zeilinger
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| |date=17 May 2012
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| |accessdate=2012-05-24
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| }}</ref> In April 2011, experimenters reported that they had demonstrated teleportation of wave packets of light up to a bandwidth of 10 MHz while preserving strongly nonclassical superposition states.<ref name=Lee2011>{{cite journal |last=Lee |first=Noriyuki |authorlink= |coauthors=Hugo Benichi, Yuishi Takeno, Shuntaro Takeda, James Webb, Elanor Huntington, and Akira Furusawa |date=April 2011 |title=Teleportation of Nonclassical Wave Packets of Light |journal=Science |volume=332 |issue=6027 |pages=330–333 |doi=10.1126/science.1201034 |url=http://www.sciencemag.org/content/332/6027/330.abstract |accessdate=2011-04-26 |quote= |bibcode = 2011Sci...332..330L |arxiv = 1205.6253 }}</ref><ref name=NewQuantTelep>{{cite web|last=Trute|first=Peter|title=Quantum teleporter breakthrough|url=http://www.unsw.edu.au/news/pad/articles/2011/apr/Quantum_teleport_paper.html|publisher=The University Of New South Wales|accessdate=17 April 2011}}</ref>
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| Researchers at the Niels Bohr Institute successfully used quantum teleportation to transmit information between clouds of gas atoms, notable because the clouds of gas are macroscopic atomic ensembles.<ref>{{cite web|title=Quantum teleportation between atomic systems over long distances|url=http://phys.org/news/2013-06-quantum-teleportation-atomic-distances.html|publisher=Phys.Org}}</ref><ref>http://www.nature.com/nphys/journal/vaop/ncurrent/full/nphys2631.html#auth-1</ref>
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| ==Formal presentation==
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| There are a variety of ways in which the teleportation protocol can be written mathematically. Some are very compact but abstract, and some are verbose but straightforward and concrete. The presentation below is of the latter form: verbose, but has the benefit of showing each quantum state simply and directly. Later sections review more compact notations.
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| The teleportation protocol begins with a quantum state or qubit <math>|\psi\rangle</math>, in Alice's possession, that she wants to convey to Bob. This qubit can be written generally, in [[bra-ket notation]], as:
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| :<math>|\psi\rangle_C = \alpha |0\rangle_C + \beta|1\rangle_C.</math>
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| The subscript ''C'' above is used only to distinguish this state from ''A'' and ''B'', below. The protocol requires that Alice and Bob share a maximally [[quantum entanglement|entangled]] state beforehand. This state is chosen beforehand, by mutual agreement between Alice and Bob, and will be one of the four [[Bell state]]s
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| :<math>|\Phi^+\rangle_{AB} = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_{B} + |1\rangle_A \otimes |1\rangle_{B})</math>,
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| :<math>|\Phi^-\rangle_{AB} = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_{B} - |1\rangle_A \otimes |1\rangle_{B})</math>,
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| :<math>|\Psi^+\rangle_{AB} = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_{B} + |1\rangle_A \otimes |0\rangle_{B})</math>,
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| :<math>|\Psi^-\rangle_{AB} = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_{B} - |1\rangle_A \otimes |0\rangle_{B})</math>.
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| Alice obtains one of the qubits in the pair, with the other going to Bob. The subscripts ''A'' and ''B'' in the entangled state refer to Alice's or Bob's particle. In the following, assume that Alice and Bob shared the entangled state <math>|\Phi^+\rangle_{AB}.</math>
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| At this point, Alice has two particles (''C'', the one she wants to teleport, and ''A'', one of the entangled pair), and Bob has one particle, ''B''. In the total system, the state of these three particles is given by
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| :<math> |\Phi^+\rangle_{AB} \otimes |\psi\rangle_C = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B)\otimes (\alpha |0\rangle_C + \beta|1\rangle_C). </math>
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| Alice will then make a partial measurement in the Bell basis on the two qubits in her possession. To make the result of her measurement clear, it is best to write the state of Alice's two qubits as superpositions of the Bell basis. This is done by using the following general identities, which are easily verified:
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| :<math>|0\rangle \otimes |0\rangle = \frac{1}{\sqrt{2}} (|\Phi^+\rangle + |\Phi^-\rangle),</math>
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| :<math>|0\rangle \otimes |1\rangle = \frac{1}{\sqrt{2}} (|\Psi^+\rangle + |\Psi^-\rangle),</math>
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| :<math>|1\rangle \otimes |0\rangle = \frac{1}{\sqrt{2}} (|\Psi^+\rangle - |\Psi^-\rangle),</math>
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| and
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| :<math>|1\rangle \otimes |1\rangle = \frac{1}{\sqrt{2}} (|\Phi^+\rangle - |\Phi^-\rangle).</math>
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| The total three particle state, of ''A'', ''B'' and ''C'' together, thus becomes the following four-term superposition:
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| :<math>
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| \begin{align}
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| |\Phi^+\rangle_{AB} \ \otimes\ | & \psi\rangle_C = \\
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| \frac{1}{2} \Big \lbrack
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| \ & |\Phi^+\rangle_{AC} \otimes (\alpha |0\rangle_B + \beta|1\rangle_B)
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| \ + \ |\Phi^-\rangle_{AC} \otimes (\alpha |0\rangle_B - \beta|1\rangle_B) \\
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| \ + \ & |\Psi^+\rangle_{AC} \otimes (\beta |0\rangle_B + \alpha|1\rangle_B)
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| \ + \ |\Psi^-\rangle_{AC} \otimes (\beta |0\rangle_B - \alpha|1\rangle_B) \Big \rbrack . \\
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| \end{align}
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| </math>
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| The above is just a change of basis on Alice's part of the system. No operation has been performed and the three particles are still in the same total state. The actual teleportation occurs when Alice measures her two qubits in the Bell basis. Given the above expression, evidently the result of her (local) measurement is that the three-particle state would [[Wave function collapse|collapse]] to one of the following four states (with equal probability of obtaining each):
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| *<math>|\Phi^+\rangle_{AC} \otimes (\alpha |0\rangle_B + \beta|1\rangle_B)</math>
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| *<math>|\Phi^-\rangle_{AC} \otimes (\alpha |0\rangle_B - \beta|1\rangle_B)</math>
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| *<math>|\Psi^+\rangle_{AC} \otimes (\beta |0\rangle_B + \alpha|1\rangle_B)</math>
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| *<math>|\Psi^-\rangle_{AC} \otimes (\beta |0\rangle_B - \alpha|1\rangle_B)</math>
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| Alice's two particles are now entangled to each other, in one of the four [[Bell states]]. The entanglement originally shared between Alice's and Bob's is now broken. Bob's particle takes on one of the four superposition states shown above. Note how Bob's qubit is now in a state that resembles the state to be teleported. The four possible states for Bob's [[qubit]] are unitary images of the state to be teleported.
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| The crucial step, the local measurement done by Alice on the Bell basis, is done. It is clear how to proceed further. Alice now has complete knowledge of the state of the three particles; the result of her Bell measurement tells her which of the four states the system is in. She simply has to send her results to Bob through a classical channel. Two classical bits can communicate which of the four results she obtained.
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| After Bob receives the message from Alice, he will know which of the four states his particle is in. Using this information, he performs a unitary operation on his particle to transform it to the desired state <math>\alpha |0\rangle_B + \beta|1\rangle_B</math>:
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| *If Alice indicates her result is <math>|\Phi^+\rangle_{AC}</math>, Bob knows his qubit is already in the desired state and does nothing. This amounts to the trivial unitary operation, the identity operator.
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| *If the message indicates <math>|\Phi^-\rangle_{AC}</math>, Bob would send his qubit through the unitary gate given by the [[Pauli matrix]]
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| :<math>\sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}</math>
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| to recover the state.
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| *If Alice's message corresponds to <math>|\Psi^+\rangle_{AC}</math>, Bob applies the gate
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| :<math>\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}</math>
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| to his qubit.
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| *Finally, for the remaining case, the appropriate gate is given by
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| :<math> -\sigma_3 \sigma_1 = i \sigma_2 = \begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix}.</math>
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| Teleportation is therefore achieved.
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| Experimentally, the projective measurement done by Alice may be achieved via a series of laser pulses directed at the two particles.
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| === Remarks ===
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| * After this operation, Bob's qubit will take on the state <math>|\psi\rangle_B= \alpha |0\rangle_B + \beta|1\rangle_B</math>, and Alice's qubit becomes an (undefined) part of an entangled state. Teleportation does not result in the copying of qubits, and hence is consistent with the [[no cloning theorem]].
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| * There is no transfer of matter or energy involved. Alice's particle has not been physically moved to Bob; only its state has been transferred. The term "teleportation", coined by Bennett, Brassard, Crépeau, Jozsa, Peres and Wootters, reflects the indistinguishability of quantum mechanical particles.
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| * For every qubit teleported, Alice needs to send Bob two classical bits of information. These two classical bits do not carry complete information about the qubit being teleported. If an eavesdropper intercepts the two bits, she may know exactly what Bob needs to do in order to recover the desired state. However, this information is useless if she cannot interact with the entangled particle in Bob's possession.
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| == Alternative notations ==
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| {{wide image|Teleport.png|780px| Quantum teleportation, as computed in a [[dagger compact category]].<ref name=coecke/> Such diagrams are employed in [[categorical quantum mechanics]], and trace back to [[Penrose graphical notation]], developed in the early 1970s.<ref name=Penrose>R. Penrose, Applications of negative dimensional tensors, In: Combinatorial Mathematics and its Applications, D.~Welsh (Ed), pages 221–244. Academic Press (1971).</ref>}}
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| [[Image:Quantum teleportation circuit.svg|300px|thumb|right| [[Quantum circuit]] representation of quantum teleportation]]
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| There are a variety of different notations in use that describe the teleportation protocol. One common one is by using the notation of [[quantum gate]]s. In the above derivation, the unitary transformation that is the change of basis (from the standard product basis into the Bell basis) can be written using quantum gates. Direct calculation shows that this gate is given by
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| :<math>G = (H \otimes I) \; C_N</math>
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| where ''H'' is the one qubit [[Hadamard gate|Walsh-Hadamard gate]] and <math>C_N</math> is the [[Controlled NOT gate]].
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| == Entanglement swapping ==
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| Teleportation can be applied not just to pure states, but also [[density matrix|mixed states]], that can be regarded as the state of a single subsystem of an entangled pair. The so-called entanglement swapping is a simple and illustrative example.
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| If Alice has a particle which is entangled with a particle owned by Bob, and Bob teleports it to Carol, then afterwards, Alice's particle is entangled with Carol's.
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| A more symmetric way to describe the situation is the following: Alice has one particle, Bob two, and Carol one. Alice's particle and Bob's first particle are entangled, and so are Bob's second and Carol's particle:
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| ___
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| / \
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| Alice-:-:-:-:-:-Bob1 -:- Bob2-:-:-:-:-:-Carol
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| \___/
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| Now, if Bob performs a projective measurement on his two particles in the Bell state basis and communicates the results to Carol, as per the teleportation scheme described above, the state of Bob's first particle can be teleported to Carol's. Although Alice and Carol never interacted with each other, their particles are now entangled.
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| A detailed diagrammatic derivaction of entanglement swapping has been given by [[Bob Coecke]],<ref>Bob Coecke, "The logic of entanglement". ''Research Report PRG-RR-03-12'', 2003. [http://arXiv.org/abs/quant-ph/0402014 arXiv:quant-ph/0402014] (8 page shortversion) [http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-03-12.html (full 160 page version)]</ref> presented in terms of [[categorical quantum mechanics]].
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| == N-state particles ==
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| One can imagine how the teleportation scheme given above might be extended to ''N''-state particles, i.e. particles whose states lie in the ''N'' dimensional Hilbert space. The combined system of the three particles now has an <math>N^3</math> dimensional state space. To teleport, Alice makes a partial measurement on the two particles in her possession in some entangled basis on the <math>N^2</math> dimensional subsystem. This measurement has <math>N^2</math> equally probable outcomes, which are then communicated to Bob classically. Bob recovers the desired state by sending his particle through an appropriate unitary gate.
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| == Logic gate teleportation ==
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| In general, [[mixed state (physics)|mixed states]] ρ may be transported, and a linear transformation ω applied during teleportation, thus allowing data processing of [[quantum information]]. This is one of the foundational building blocks of quantum information processing. This is demonstrated below.
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| === General description ===
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| A general teleportation scheme can be described as follows. Three quantum systems are involved. System 1 is the (unknown) state ''ρ'' to be teleported by Alice. Systems 2 and 3 are in a maximally entangled state ''ω'' that are distributed to Alice and Bob, respectively. The total system is then in the state
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| :<math>\rho \otimes \omega.</math>
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| A successful teleportation process is a [[LOCC]] [[quantum channel]] Φ that satisfies
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| :<math>(\operatorname{Tr}_{12} \circ \Phi ) (\rho \otimes \omega) = \rho\,,</math>
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| where Tr<sub>12</sub> is the [[partial trace]] operation with respect systems 1 and 2, and <math>\circ</math> denotes the composition of maps. This describes the channel in the Schrödinger picture.
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| Taking adjoint maps in the Heisenberg picture, the success condition becomes
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| :<math>\langle \Phi(\rho \otimes \omega)| I \otimes O \rangle = \langle \rho | O \rangle</math>
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| for all observable ''O'' on Bob's system. The tensor factor in <math>I \otimes O</math> is <math>12 \otimes 3</math> while that of <math>\rho \otimes \omega</math> is <math>1 \otimes 23</math>.
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| === Further details ===
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| The proposed channel Φ can be described more explicitly. To begin teleportation, Alice performs a local measurement on the two subsystems (1 and 2) in her possession. Assume the local measurement have ''effects''
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| :<math>{F_i} = {M_i ^2}.</math>
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| If the measurement registers the ''i''-th outcome, the overall state collapses to
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| :<math>(M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I).</math>
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| The tensor factor in <math>(M_i \otimes I)</math> is <math>12 \otimes 3</math> while that of <math>\rho \otimes \omega</math> is <math>1 \otimes 23</math>. Bob then applies a corresponding local operation Ψ''<sub>i</sub>'' on system 3. On the combined system, this is described by
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| :<math>(Id \otimes \Psi_i)(M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I).</math>
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| where ''Id'' is the identity map on the composite system <math>1 \otimes 2</math>.
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| Therefore the channel Φ is defined by
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| :<math>\Phi (\rho \otimes \omega) = \sum_i (Id \otimes \Psi_i)(M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I)</math>
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| Notice Φ satisfies the definition of [[LOCC]]. As stated above, the teleportation is said to be successful if, for all observable ''O'' on Bob's system, the equality
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| :<math>\langle \Phi(\rho \otimes \omega), I \otimes O \rangle = \langle \rho, O \rangle</math>
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| holds. The left hand side of the equation is:
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| :<math>
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| \sum_i \langle (Id \otimes \Psi_i)(M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I), \; I \otimes O \rangle
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| </math>
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| :<math>
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| = \sum_i \langle (M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I), \; I \otimes \Psi_i ^*(O)\rangle
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| </math>
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| where Ψ''<sub>i</sub>*'' is the adjoint of Ψ''<sub>i</sub>'' in the Heisenberg picture. Assuming all objects are finite dimensional, this becomes
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| :<math>\sum_i \operatorname{Tr} \; (\rho \otimes \omega)(F_i \otimes \Psi_i^*(O)).</math>
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| The success criterion for teleportation has the expression
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| :<math>\sum_i \operatorname{Tr} \; (\rho \otimes \omega)(F_i \otimes \Psi_i ^*(O)) = \operatorname{Tr} \; \rho \cdot O.</math>
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| ==See also==
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| * [[Teleportation]]
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| * [[Quantum mechanics]]
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| ** [[Introduction to quantum mechanics]]
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| ** [[Quantum computer]]
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| ** [[Quantum energy teleportation]]
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| ** [[Quantum entanglement]]
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| ** [[Quantum nonlocality]]
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| ** [[Uncertainty principle|Heisenberg uncertainty principle]]
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| ==References==
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| {{More footnotes|date=June 2012}}
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| ;Specific
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| {{Reflist}}
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| ;General
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| *Theoretical proposal:
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| ** [[Charles H. Bennett (computer scientist)|C. H. Bennett]], [[Gilles Brassard|G. Brassard]], [[Claude Crépeau|C. Crépeau]], [[Richard Jozsa|R. Jozsa]], [[Asher Peres|A. Peres]], [[William Wootters|W. K. Wootters]], ''Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels'', [[Phys. Rev. Lett.]] '''70''', 1895-1899 (1993) ([http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.9405 pdf]). This is the seminal paper that laid out the entanglement protocol.
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| ** [[Lev Vaidman|L. Vaidman]], ''Teleportation of Quantum States'', [[Phys. Rev. A]] '''49''', 1473-1476 (1994)
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| ** [[Gilles Brassard|G Brassard]], [[Samuel L. Braunstein|S Braunstein]], [[Richard Cleve|R Cleve]], ''Teleportation as a Quantum Computation'', Physica D '''120''' 43-47 (1998)
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| ** [[Asher Peres|A. Peres]], "What is actually teleported?", IBM Journal of Research and Development, Vol. 48, Issue 1, (2004)([http://arxiv.org/abs/quant-ph/0304158 this document online])
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| ** G. Rigolin, ''Quantum Teleportation of an Arbitrary Two Qubit State and its Relation to Multipartite Entanglement'', [[Phys. Rev. A]] '''71''' 032303 (2005)([http://arxiv.org/abs/quant-ph/0407219 this document online])
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| ** Shi-Biao Zheng, "Scheme for approximate conditional teleportation of an unknown atomic state without the Bell-state measurement", [[Phys. Rev. A]] '''69''', 064302 (2004).
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| ** W. B. Cardoso, A. T. Avelar, B. Baseia, and N. G. de Almeida, "Teleportation of entangled states without Bell-state measurement", [[Phys. Rev. A]] '''72''', [http://pra.aps.org/abstract/PRA/v72/i4/e045802 045802 (2005)].
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| ** Michael N. Leuenberger, Michael E. Flatte, David D. Awschalom, "Teleportation of Electronic Many-Qubit States Encoded in the Electron Spin of Quantum Dots via Single Photons", [[Phys. Rev. Lett.]] '''94''', [http://prl.aps.org/abstract/PRL/v94/i10/e107401 107401 (2005)].
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| *First experiments with photons:
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| **D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, [[Anton Zeilinger|A. Zeilinger]], ''Experimental Quantum Teleportation,'' [[Nature (journal)|Nature]] '''390''', 6660, 575-579 (1997).
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| **D. Boschi, S. Branca, F. De Martini, L. Hardy, & S. Popescu, ''Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual classical and Einstein-Podolsky-Rosen channels,'' [[Phys. Rev. Lett.]] '''80''', 6, 1121-1125 (1998)
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| ** Y.-H. Kim, S.P. Kulik, and Y. Shih, ''Quantum teleportation of a polarization state with a complete bell state measurement,'' [[Phys. Rev. Lett.]] '''86''', 1370 (2001).
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| **I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, N. Gisin, ''Long-Distance Teleportation of Qubits at Telecommunication Wavelengths,'' Nature, '''421''', 509 (2003)
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| **R. Ursin et al., ''Quantum Teleportation Link across the Danube,'' Nature '''430''', 849 (2004)
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| *First experiments with atoms:
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| **M. Riebe, H. Häffner, C. F. Roos, W. Hänsel, M. Ruth, J. Benhelm, G. P. T. Lancaster, T. W. Körber, C. Becher, F. Schmidt-Kaler, D. F. V. James, R. Blatt, ''Deterministic Quantum Teleportation with Atoms,'' Nature '''429''', 734-737 (2004)
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| **M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, D. J. Wineland, ''Deterministic Quantum Teleportation of Atomic Qubits,'' Nature '''429''', 737 (2004).
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| **S. Olmschenk, D. N. Matsukevich, P. Maunz, D. Hayes, L.-M. Duan, and C. Monroe, ''Quantum Teleportation between Distant Matter Qubits,'' Science 323, 486 (2009).
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| ==External links==
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| *''signandsight.com:''[http://www.signandsight.com/features/614.html "Spooky action and beyond"] - Interview with Prof. Dr. [[Anton Zeilinger]] about quantum teleportation. Date: 2006-02-16
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| * [http://www.research.ibm.com/quantuminfo/teleportation/ Quantum Teleportation at IBM]
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| * [http://www.spacedaily.com/news/physics-04zi.html Physicists Succeed In Transferring Information Between Matter And Light]
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| * [http://www.physorg.com/news10924.html Quantum telecloning: Captain Kirk's clone and the eavesdropper]
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| *[http://www.math.uwaterloo.ca/~amchilds/talks/pi03.ppt Teleportation-based approaches to universal quantum computation]
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| *[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6TVK-3TXC7TP-4-1&_cdi=5537&_user=4799849&_orig=search&_coverDate=09%2F01%2F1998&_sk=998799998&view=c&wchp=dGLbVtz-zSkWb&md5=4f615b57a53a67a4268535a9ba311407&ie=/sdarticle.pdf Teleportation as a quantum computation]
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| *[http://www.iop.org/EJ/article/1367-2630/9/7/211/njp7_7_211.html#nj248372s4 Quantum teleportation with atoms: quantum process tomography]
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| *[http://www.imit.kth.se/QEO/qucomm/DelD19QuComm.pdf Entangled State Teleportation]
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| *[http://koasas.kaist.ac.kr/bitstream/10203/1029/1/e022316.pdf Fidelity of quantum teleportation through noisy channels by]
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| *[http://www.cs.technion.ac.il/~talmo/CV/my-new-papers/BHM04-IBM.pdf TelePOVM— A generalized quantum teleportation scheme]
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| *[http://web.am.qub.ac.uk/users/m.s.kim/PRL04236.pdf Entanglement Teleportation via Werner States]
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| *[http://www.physics.ohio-state.edu/~wilkins/writing/Assign/topics/Q-trans-prl.pdf Quantum Teleportation of a Polarization State]
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| *[http://books.google.ca/books?id=zENRkvFO0SEC&pg=PA224&lpg=PA224&dq=teleport+schematic&source=web&ots=qxbRSq4WA6&sig=WXcfKLgM8siRiiqJi-I5YXvl2VI&hl=en&sa=X&oi=book_result&resnum=10&ct=result#PPA224,M1 The Time Travel Handbook: A Manual of Practical Teleportation & Time Travel]
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| *[http://heart-c704.uibk.ac.at/publications/papers/nature04_riebe.pdf letters to nature: Deterministic quantum teleportation with atoms]
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| *[http://qi.phys.msu.ru/kulik/Papers/p221.pdf Quantum teleportation with a complete Bell state measurement]
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| *[http://sciencenews.org/view/feature/id/34762/title/Welcome_to_the_Quantum_Internet Welcome to the quantum Internet.] ''Science News,'' Aug. 16 2008.
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| *[http://www.QuantumLab.de Quantum experiments - interactive.]
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| *“[http://lightlike.com/teleport/ A (mostly serious) introduction to quantum teleportation for non-physicists]”
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| {{Quantum computing}}
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| {{Emerging technologies}}
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| {{DEFAULTSORT:Quantum Teleportation}}
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| [[Category:Quantum information science]]
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| [[Category:Emerging technologies]]
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