Quantum teleportation

Quantum teleportation is the name of any method of quantum communication that transfers the quantum properties, i.e. the state of a system (source), to another, possibly distant system (target). In the source, the state is inevitably deleted because the principles of the quantum world do not allow states to be duplicated ( no-cloning theorem ). Because particles of the same type are fundamentally indistinguishable , the transfer of the quantum properties can be sufficient to fully realize an existing quantum object at the target location. The known method also uses a 'sender object' for transmission and uses the fact that the measured values are correlated on entangled quantum objects.

The use of the term teleportation repeatedly leads to misleading newspaper reports. Quantum teleportation is not a teleportation in the common sense, nor is it a preliminary stage. Neither matter nor energy is transferred. It differs fundamentally from teleportation in science fiction in that it requires a target object prepared at the target location, onto which the state of the source object is transplanted, so to speak.

In the non-classical part of quantum teleportation, there is no transmission path, no transmission time, no speed. In particular, nothing moves faster than light. However, the completion of quantum teleportation requires the transmission of measurement results, for which a 'classic' information channel, such as a radio link, is needed, whose signal crosses space and which works at the speed of light at the most.

Unlike in the macroscopic, where the current state of a source system can be determined and thus reproduced on the target system, the quantum mechanical state of a source system cannot generally be determined at all. Each of the infinitely many polarization states of a photon, for example, provides only one bit of information during a measurement and is destroyed in the process. It is therefore remarkable in itself that a state can be transferred from one object to another at all.

Quantum teleportation is an important component of quantum communication, cryptography and computing protocols. An essential property of the quantum teleportation protocol is that it works even if the state to be sent is not known to the sender or is linked to another system. In addition, it does not matter in which physical system the initial and target state exist (the four systems involved (input system, the two interlinked systems and the carrier of the classical information) can be implemented by four different physical systems): only the state of one becomes Quantum system, transmitted, not the system itself transported. Therefore, sometimes the "disembodied" (Engl .: disembodied speech) Transport.

invention

The idea of ​​quantum teleportation was published by Asher Peres , William Wootters , Gilles Brassard , Charles H. Bennett , Richard Jozsa, and Claude Crépeau in the 1993 Physical Review Letters . Quantum teleportation was first demonstrated in 1997 by Anton Zeilinger , almost simultaneously with Sandu Popescu , Francesco De Martini and others, through quantum optical experiments with photons . Teleportation of the states of individual atoms is now also possible.

Process in the overview

States of particles in quantum theory are not properties in the sense of intuition, but a promise that this or that will come out if one measures. There is therefore no contradiction in the statement that two particles can be 'entangled' in such a way that neither has a state but both have the same: It is only promised that both produce the same result; what that will be remains open. - Alice and Bob have particles and that are entangled in this way. Chris has a particle in the state that Alice is supposed to teleport onto Bob's . To do this, she interlocks with hers , thereby releasing the entanglement of with . The entanglement states used here do not leave the components with any individual state information. The status information from , therefore , is no longer available in the subsystem . However, the overall system preserves them. A learns what state the subsystem is located now, but only he can to reconstruct. He can then query the bit of information that was originally promised for . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle z}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle z}$ ${\displaystyle \{a,c\}}$ ${\displaystyle \{a,b,c\}}$ ${\displaystyle \{a,c\}}$ ${\displaystyle z}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle b}$

Process in detail

Quantum teleportation requires two kinds of connections between transmitter A (Alice) and receiver B (Bob):

• A system consisting of two qubits and , which are entangled in such a way that the measured values ​​found in them are maximally correlated (always the same or always opposite). This connection is sometimes misunderstood as an information channel. ${\displaystyle a}$ ${\displaystyle b}$
• A conventional communication channel, for example a radio link.

1. First of all, the system must be suitably prepared and the components transferred to A and B. Let us assume as an example that the qubits are photons, and we measure whether the polarization is (horizontal) or (vertical). Each of the four bell states is suitable as an entanglement state. We take , therefore, a superposition of the two states in which and are both horizontally or both vertically polarized, both with amplitude . The symbol for the tensor product of the states may be omitted to relieve the formula: instead of etc. ${\displaystyle \{a,b\}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle h}$ ${\displaystyle v}$ ${\displaystyle B\{a,b\}=1/{\sqrt {2}}\,(h_{a}h_{b}+v_{a}v_{b})}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle 1/{\sqrt {2}}}$ ${\displaystyle \otimes }$ ${\displaystyle h_{a}h_{b}}$ ${\displaystyle h_{a}\otimes h_{b}}$
2. C (Chris) prepares another qubit into a state and hands it over to A, who should teleport its state onto the qubit . A does not need to experience the state . ${\displaystyle c}$ ${\displaystyle z}$ ${\displaystyle b}$ ${\displaystyle z}$
3. A carries out a 'bell measurement' on the qubits and , a (technically difficult) measurement that brings the system into one of the four possible bell states and thereby entangles the two qubits. It communicates the result to B via the conventional channel in a 2-bit message. This is the only information that is transmitted during teleportation. ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle \{a,c\}}$ ${\displaystyle B_{1}\{a,c\},\,\dots ,B_{4}\{a,c\}}$
4. B now knows which of four possible states is in: or  ( he does not know himself). This stands for a rotation of the polarization direction by 90 ° and for a mirroring of this direction on the axis. With a corresponding rotation and / or mirroring, he can reverse these transformations and thus bring them into the state that the qubit originally had . On the other hand, in is deleted , which is now entangled with . ${\displaystyle b}$ ${\displaystyle z,\,Dz,\,Sz}$ ${\displaystyle SDz}$ ${\displaystyle z}$ ${\displaystyle D}$ ${\displaystyle S}$ ${\displaystyle h}$ ${\displaystyle b}$ ${\displaystyle z}$ ${\displaystyle c}$ ${\displaystyle z}$ ${\displaystyle c}$ ${\displaystyle b}$
5. The fact that the state was transferred from to means the following in detail: If B now tests for the polarization state , this test will definitely pass. If it tests for any condition , this test passes with the same probability as it would have passed before the Bell measurement. In contrast, any polarization test now passes with a probability of 1/2. Since the state of a qubit cannot be determined (here ), B does not find out about it, unless it is communicated in the traditional way. Only then he can also monitor the success of teleportation by on tests. ${\displaystyle z}$ ${\displaystyle c}$ ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle z}$ ${\displaystyle b}$ ${\displaystyle u}$ ${\displaystyle b}$ ${\displaystyle |\langle z,u\rangle |^{2},}$ ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle z_{b}}$ ${\displaystyle b}$ ${\displaystyle z}$

Mathematical consideration

Indices denote the qubit, for which the state is specified with etc. The sign (tensor product) can be set or omitted. ${\displaystyle a,b,c}$ ${\displaystyle h_{a},v_{b}}$ ${\displaystyle \otimes }$

The readiness for teleportation is given when the system is , as assumed, in the entangled state . ${\displaystyle \{a,b\}}$ ${\displaystyle \textstyle B\{a,b\}={\frac {1}{\sqrt {2}}}(h_{a}h_{b}+v_{a}v_{b})}$

If Chris' qubit is included in the consideration, we have to calculate in the Hilbert space of the system . The state of sei . Since there is no interaction with or , the overall system has the product status ${\displaystyle c}$ ${\displaystyle {\mathcal {H}}_{a}\otimes {\mathcal {H}}_{b}\otimes {\mathcal {H}}_{c}}$ ${\displaystyle \{a,b,c\}}$ ${\displaystyle c}$ ${\displaystyle z=\alpha h+\beta v}$ ${\displaystyle c}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle \textstyle Z\{a,b,c\}\,=\,B\{a,b\}\otimes z_{c}\,=\,~{\frac {1}{\sqrt {2}}}(h_{a}h_{b}+v_{a}v_{b})(\alpha h_{c}+\beta v_{c}).}$

With regard to the bell measurement on the qubits and at the transmitter, the orthonormal system of the four bell states should be used instead of the standard basis : ${\displaystyle a}$ ${\displaystyle c}$ ${\displaystyle {\mathcal {H}}_{a}\otimes {\mathcal {H}}_{c}}$ ${\displaystyle \{h_{a}h_{c},\,h_{a}v_{c},\,v_{a}h_{c},\,v_{a}v_{c}\}}$ ${\displaystyle B_{i}\{a,c\},\,i=1,2,3,4}$

${\displaystyle \textstyle {\frac {1}{\sqrt {2}}}(h_{a}h_{c}+v_{a}v_{c}),\,~~{\frac {1}{\sqrt {2}}}(h_{a}h_{c}-v_{a}v_{c}),\,~~{\frac {1}{\sqrt {2}}}(h_{a}v_{c}+v_{a}h_{c}),\,~~{\frac {1}{\sqrt {2}}}(h_{a}v_{c}-v_{a}h_{c})}$

be used. Are to be used

${\displaystyle \textstyle h_{a}h_{c}={\frac {1}{\sqrt {2}}}{\big (}B_{1}\{a,c\}+B_{2}\{a,c\}{\big )},\,~~v_{a}v_{c}={\frac {1}{\sqrt {2}}}{\big (}B_{1}\{a,c\}-B_{2}\{a,c\}{\big )},}$

${\displaystyle \textstyle h_{a}v_{c}={\frac {1}{\sqrt {2}}}{\big (}B_{3}\{a,c\}+B_{4}\{a,c\}{\big )},\,~~v_{a}h_{c}={\frac {1}{\sqrt {2}}}{\big (}B_{3}\{a,c\}-B_{4}\{a,c\}{\big )}.}$

After arranging the terms you have

${\displaystyle Z\{a,b,c\}\,=\,\textstyle {\frac {1}{2}}\sum _{i}B_{i}\{a,c\}\otimes z_{ib}}$   With    ${\displaystyle z_{1}=\alpha h+\beta v,}$ ${\displaystyle z_{2}=\alpha h-\beta v,}$ ${\displaystyle z_{3}=\beta h+\alpha v,}$ ${\displaystyle z_{4}=\beta h-\alpha v.}$

Designates the reflection in the plane of polarization on the -axis and the clockwise rotation, so and we get the equation on which the teleportation is based: ${\displaystyle S}$ ${\displaystyle h}$ ${\displaystyle D}$ ${\displaystyle 90^{o}}$ ${\displaystyle z_{1}=z,\,z_{2}=Sz,\,z_{3}=SDz,\,z_{4}=Dz,}$

${\displaystyle (h_{a}h_{b}+v_{a}v_{b})\otimes z_{c}\,=\,{\frac {1}{2}}{\big (}B_{1}\{a,c\}\otimes z_{b}+B_{2}\{a,c\}\otimes Sz_{b}+B_{3}\{a,c\}\otimes SDz_{b}+B_{4}\{a,c\}\otimes Dz_{b}{\big )}\,.}$

The left-hand side describes the structure of : Sender A and receiver B have qubits and are entangled , and are therefore ready for transmission. Qubit , the carrier of the state to be sent, is almost uninvolved. The same state is shown on the right-hand side as a superposition of four states, which express the possible reactions of the system to the planned measurements. The products are orthogonal to one another in pairs and each occur with an amplitude of 1/2. The chance therefore determines with the same probability 1/4 in which of the four states the system will show itself during the planned measurements: A decision for both measurements, for the on and the on . This leads to a strict correlation: Makes A's measurement , a test falls on to certainly positive. The spatial / temporal distance of the measurement events is irrelevant, in principle also their chronological order. It is only the protocol of teleportation that prescribes that A measure first, in order to be able to give instructions to B after the result of your measurement, how he can manipulate, transform, in order not to but measure. That he doesn't know doesn't matter. He just needs to know what transformation to perform. The protocol ends with the determination that the recipient qubit is now in the state . A measurement on can be made later at an indefinite point in time. ${\displaystyle Z\{a,b,c\}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle B\{a,b\}}$ ${\displaystyle c}$ ${\displaystyle B_{i}\{a,c\}z_{ib}}$ ${\displaystyle \{a,c\}}$ ${\displaystyle b}$ ${\displaystyle B_{i}}$ ${\displaystyle b}$ ${\displaystyle z_{i}}$ ${\displaystyle b}$ ${\displaystyle z}$ ${\displaystyle z_{i}}$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle b}$ ${\displaystyle z}$ ${\displaystyle b}$

Experimental successes

Experiments 2003 and 2004

In 2003 Nicolas Gisin and his team demonstrated quantum teleportation with photons over long distances (2 km glass fiber at 55 m distance) at the University of Geneva, and in 2007 also in commercial glass fiber communication networks (Swisscom).

In 2004 two working groups ( University of Innsbruck and NIST in Boulder Colorado ) succeeded for the first time in carrying out quantum teleportation with atoms (more precisely: with ions ).

Also in 2004, Viennese researchers working with Rupert Ursin and Anton Zeilinger succeeded for the first time in teleporting a quantum state of a photon outside the laboratory. They bridged a distance of 600 m under the Danube . For this purpose, an optical fiber was laid in a sewer under the Danube in order to transfer the quantum state (the polarization ) of the photon to be teleported from the Danube Island (Alice) to the southern Danube side (Bob) to another photon. At Alice, the source of the entangled photons was set up and one of the couple's entangled photons was transmitted to Bob via a fiber optic cable . Alice superimposed the other photon of the couple with the photon to be teleported and carried out a Bell state measurement - the original polarization state of Alice's photon to be transmitted was destroyed. The results of Alice's Bell state measurement, which can distinguish two of the possible four Bell states from one another, were transmitted via a classic information channel to Bob, who then - if necessary - applied a corresponding unitary transformation (a rotation of the polarization direction) to his entangled photon in order to achieve the transmission of the quantum state (i.e. the original polarization direction of Alice's photon) to complete this.

Progress since 2004

In July 2009, researchers from the universities of Auckland (New Zealand), Griffith University in Queensland (Australia) and Doha ( Qatar ) proposed a method for teleporting a beam of light or a complete quantum field, including the fluctuations over time. This “strong” teleportation (including fluctuations) is seen as a prerequisite for some quantum information applications and could lead to the teleportation of quantum images.

In May 2010, the science magazine Nature reported on the successful quantum teleportation over a distance of 16 kilometers, carried out in the field by a Chinese team led by Xian-Min Jin . An average accuracy of 89 percent was achieved, which is well above the classically expected limit of 2/3.

In May 2012, researchers at the Chinese University of Science and Technology said they used a laser to cover a distance of 97 kilometers, setting a new record.

In September 2012 the science magazine Nature published a report on a quantum teleportation over a distance of 143 km from La Palma to Tenerife.

In August 2014, Nature reported on an experimental setup for quantum teleportation with photons of different energies. It makes it possible to illuminate an object with low-frequency infrared light, the interaction of which with the object being examined affects the entangled photons in visible light, which can be captured with simple digital cameras.

An equivalent description of quantum teleportation in the context of quantum gravity was found by Ping Gao, Daniel Louis Jafferis and Aron C. Wall in 2016 when they introduced a new type of wormhole .

Practical meaning

The practical importance of quantum teleportation does not lie in the fact that information or even objects can be transported faster than light, as would be the case with (fictitious) classic teleportation . On the other hand, quantum teleportation is of practical importance because it allows quantum states to be transferred without simultaneously changing them through a measurement process (compare: quantum mechanical measurement ) and without having to transport a quantum system (the transport of a quantum system before the Teleportation and the sending of classic information is sufficient). For quantum computers, this opens up technically promising possibilities for the transmission, storage and processing of qubits , especially for a quantum Internet .

literature

Essays

• Dik Bouwmeester, Jian-Wei Pan , Klaus Mattle, Manfred Eibl, Harald Weinfurter, Anton Zeilinger: Experimental quantum teleportation. In: Nature 390, 575-579 (1997), doi: 10.1038 / 37539 , pdf
• A. Zeilinger: Experimental quantum teleportation. In: Scientific American , April 2000, pp. 32-41
• D. Bouwmeester, Pan, Mattle, Eibl, Weinfurter, Zeilinger: Experimental quantum teleportation. In: Phil. Trans. R. Soc. Lond. A 356, 1733 (1998), doi: 10.1098 / rsta.1998.0245
• Rupert Ursin, Thomas Jennewein, Markus Aspelmeyer, Rainer Kaltenbaek, Michael Lindenthal, Philip Walther, Anton Zeilinger: Quantum teleportation across the Danube. In: Nature 430, 849 (2004), doi: 10.1038 / 430849a
• M. Riebe, H. Häffner, CF Roos, W. Hänsel, J. Benhelm, GPT Lancaster, TW Körber, C. Becher, Ferdinand Schmidt-Kaler , DFV James, Rainer Blatt : Deterministic quantum teleportation with atoms. In: Nature 429, 734 (2004), doi: 10.1038 / nature02570
• MD Barrett, J. Chiaverini, T. Schaetz, J. Britton, WM Itano, JD Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, David Wineland : Deterministic quantum teleportation of atomic qubits. In: Nature 429, 737 (2004), doi: 10.1038 / nature02608

Books

• M. Homeister: Understanding Quantum Computing Springer Vieweg, Wiesbaden 2018, fifth edition, ISBN 978-3-6582-2883-5 , pp. 125ff.
• B. Lenze: Mathematik und Quantum Computing Logos Verlag, Berlin 2020, second edition, ISBN 978-3-8325-4716-5 , p. 41ff.
• MA Nielsen, IL Chuang: Quantum Computation and Quantum Information Cambridge University Press, Cambridge MA 2010, ISBN 978-1-1070-0217-3 , pp. 26ff.
• W. Scherer: Mathematics of Quanteninformatik Springer-Verlag, Berlin-Heidelberg 2016, ISBN 978-3-6624-9079-2 , p. 191ff.
• CP Williams: Explorations in Quantum Computing Springer-Verlag, London 2011, second edition, ISBN 978-1-8462-8886-9 , pp. 483ff.

Web links

Wiktionary: Quantum teleportation  - explanations of meanings, word origins, synonyms, translations

• Quantum teleportation
• Quantum communication in space
• Not even God knows how it will end  - Interview with Anton Zeilinger in the Weltwoche. Edition 48/05
• Nobel laureate in physics Theodor W. Hänsch speaks about aspects of quantum teleportation Extensive interview on quantum mechanics, July 22, 2008

Individual evidence

1. ↑ Stuttgarter Nachrichten, Stuttgart Germany: Quantum Computers - Is Beaming Soon Possible ?: Google: Successful breakthrough in quantum computing. Retrieved June 11, 2020 . 
2. ↑ Interview with Nobel Laureate in Physics Prof. Dr. Theodor W. Hänsch, part 5: Quantum teleportation and quantum cryptography ( Memento from October 12, 2008 in the Internet Archive )
3. ^ Braunstein, Sam L .: Quantum Teleportation . In: Fortschr. Phys. tape 50 , no. 5-7 . Wiley, May 17, 2002, pp. 608–613 , doi : 10.1002 / 1521-3978 (200205) 50: 5/7 (English).
4. ↑ CH Bennett et al. : Teleporting to unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. In: Phys. Rev. Lett. 70, 1895 (1993), doi: 10.1103 / PhysRevLett.70.1895
5. ↑ D. Bouwmeester, JW Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger Experimental Quantum Teleportation , Nature 390, 575-579 (1997)
6. ↑ Boschi, Branca, De Martini, Hardy, Popescu Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels , Phys. Rev. Lett. 80, 1121 (1998), Arxiv
7. ↑ S. Olmschenk, DN Matsukevich, P. Maunz, D. Hayes, L.-M. Duan, C. Monroe, Quantum Teleportation Between Distant Matter Qubits , Science 323 (5913), 486–489 (2009) ( online , arXiv )
8. ↑ 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) ( online , arXiv )
9. ↑ I. Marcikic, H. de Ried mats, W. Tittel, H. Zbinden, N. Gisin, Long-distance teleportation of qubits at telecommunication Wavelengths, Nature, Vol 421, 2003, pp 509-513, Abstract
10. ↑ Quantum teleportation of Light (English) - Summary in the library of Cornell University , October 3, 2008 (Accessed on May 20, 2012)
11. ↑ Experimental free-space quantum teleportation (English) - summary in Nature , May 16, 2010 (accessed on May 20, 2012)
12. ↑ Teleporting independent qubits through a 97 km free-space channel (English) - Summary at the library at Cornell University , May 9, 2012 (accessed on May 20, 2012)
13. ↑ Chinese Physicists Smash Distance Record For Teleportation - Article in Technology Review , May 11, 2012 (accessed May 20, 2012)
14. ↑ Quantum teleportation Scientists send particles over 97 kilometers - Article at Golem.de , May 12, 2012 (accessed on May 20, 2012)
15. ↑ Quantum teleportation over 143 kilometers using active feed-forward - Publication by Nature , September 5, 2012 (accessed September 7, 2012)
16. ↑ Quantum imaging with undetected photons - Publication by Nature , August 28, 2014 (accessed August 27, 2014)
17. ↑ Physics: Camera photographs with teleported light - Publication by Spiegel Online , August 28, 2014 (accessed August 28, 2014)
18. ↑ Ping Dao, Daniel Jafferis, Aron Wall, Traversable Wormholes via a Double Trace Deformation , Arxiv 2016
19. ↑ Natalie Wolchover: Newfound Wormhole Allows Information to Escape Black Holes , Quanta Magazine, October 23, 2017