Quantum cloning

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Quantum cloning is a process that takes an arbitrary, unknown quantum state and makes an exact copy without altering the original state in any way. Quantum cloning is forbidden by the laws of quantum mechanics as shown by the no cloning theorem, which states that there is no operation for cloning any arbitrary state perfectly. In Dirac notation, the process of quantum cloning is described by:

Contents

where is the actual cloning operation, is the state to be cloned, and is the initial state of the copy.

Though perfect quantum cloning is not possible, it is possible to perform imperfect cloning, where the copies have a non-unit (i.e. non-perfect) fidelity. The possibility of approximate quantum copying was first addressed by Buzek and Hillery, [1] and theoretical bounds were derived on the fidelity of cloned quantum states. [2]

One of the applications of quantum cloning is to analyse the security of quantum key distribution protocols. [3] Teleportation, nuclear magnetic resonance, quantum amplification, and superior phase conjugation are examples of some methods utilized to realize a quantum cloning machine. [4] [3] Ion trapping techniques have been applied to cloning quantum states of ions. [5]

Types of quantum cloning machines

It may be possible to clone a quantum state to arbitrary accuracy in the presence of closed timelike curves. [6]

Universal quantum cloning

Universal quantum cloning (UQC) implies that the quality of the output (cloned state) is not dependent on the input, thus the process is "universal" to any input state. [7] [8] The output state produced is governed by the Hamiltonian of the system. [9]

One of the first cloning machines, a 1 to 2 UQC machine, was proposed in 1996 by Buzek and Hillery. [10] As the name implies, the machine produces two identical copies of a single input qubit with a fidelity of 5/6 when comparing only one output qubit, and global fidelity of 2/3 when comparing both qubits. This idea was expanded to more general cases such as an arbitrary number of inputs and copies, [11] as well as d-dimensional systems. [12]

Multiple experiments have been conducted to realize this type of cloning machine physically by using photon stimulated emission. [13] The concept relies on the property of certain three-level atoms to emit photons of any polarization with equally likely probability. This symmetry ensures the universality of the machine. [13]

Phase covariant cloning

When input states are restricted to Bloch vectors corresponding to points on the equator of the Bloch Sphere, more information is known about them. [7] [14] The resulting clones are thus state-dependent, having an optimal fidelity of . Although only having a fidelity slightly greater than the UQCM (≈0.83), phase covariant cloning has the added benefit of being easily implemented through quantum logic gates consisting of the rotational operator and the controlled-NOT (CNOT). Output states are also separable according to Peres-Horodecki criterion. [14]

The process has been generalized to the 1 → M case and proven optimal. [11] This has also been extended to the qutrit [15] and qudit [16] cases. The first experimental asymmetric quantum cloning machine was realized in 2004 using nuclear magnetic resonance. [17]

Asymmetric quantum cloning

The first family of asymmetric quantum cloning machines was proposed by Nicolas Cerf in 1998. [18] A cloning operation is said to be asymmetric if its clones have different qualities and are all independent of the input state. This is a more general case of the symmetric cloning operations discussed above which produce identical clones with the same fidelity. Take the case of a simple 1 → 2 asymmetric cloning machine. There is a natural trade-off in the cloning process in that if one clone's fidelity is fixed to a higher value, the other must decrease in quality and vice versa. [19] The optimal trade-off is bounded by the following inequality: [20]

where Fd and Fe are the state-independent fidelities of the two copies. This type of cloning procedure was proven mathematically to be optimal as derived from the Choi-Jamiolkowski channel state duality. However, even with this cloning machine perfect quantum cloning is proved to be unattainable. [19]

The trade-off of optimal accuracy between the resulting copies has been studied in quantum circuits, [21] and with regards to theoretical bounds. [22]

Optimal asymmetric cloning machines are extended to in dimensions.[ clarification needed ] [23]

Probabilistic quantum cloning

In 1998, Duan and Guo proposed a different approach to quantum cloning machines that relies on probability. [7] [24] [25] This machine allows for the perfect copying of quantum states without violation of the No-Cloning and No-Broadcasting Theorems, but at the cost of not being 100% reproducible. The cloning machine is termed "probabilistic" because it performs measurements in addition to a unitary evolution. These measurements are then sorted through to obtain the perfect copies with a certain quantum efficiency (probability). [25] As only orthogonal states can be cloned perfectly, this technique can be used to identify non-orthogonal states. The process is optimal when where η is the probability of success for the states Ψ0 and Ψ1. [8] [26]

The process was proven mathematically to clone two pure, non-orthogonal input states using a unitary-reduction process. [27] One implementation of this machine was realized through the use of a "noiseless optical amplifier" with a success rate of about 5% . [28]

Applications of approximate quantum cloning

Cloning in discrete quantum systems

The simple basis for approximate quantum cloning exists in the first and second trivial cloning strategies. In first trivial cloning, a measurement of a qubit in a certain basis is made at random and yields two copies of the qubit. This method has a universal fidelity of 2/3. [29]

The second trivial cloning strategy, also called "trivial amplification", is a method in which an original qubit is left unaltered, and another qubit is prepared in a different orthogonal state. When measured, both qubits have the same probability, 1/2, (check) and an overall single copy fidelity of 3/4. [29]

Quantum cloning attacks

Quantum information is useful in the field of cryptography due to its intrinsic encrypted nature. One such mechanism is quantum key distribution. In this process, Bob receives a quantum state sent by Alice, in which some type of classical information is stored. [29] He then performs a random measurement, and using minimal information provided by Alice, can determine whether or not his measurement was "good". This measurement is then transformed into a key in which private data can be stored and sent without fear of the information being stolen.

One reason this method of cryptography is so secure is because it is impossible to eavesdrop due to the no-cloning theorem. A third party, Eve, can use incoherent attacks in an attempt to observe the information being transferred from Bob to Alice. Due to the no-cloning theorem, Eve is unable to gain any information. However, through quantum cloning, this is no longer entirely true.

Incoherent attacks involve a third party gaining some information into the information being transmitted between Bob and Alice. These attacks follow two guidelines: 1) third party Eve must act individually and match the states that are being observed, and 2) Eve's measurement of the traveling states occurs after the sifting phase (removing states that are in non-matched bases [30] ) but before reconciliation (putting Alice and Bob's strings back together [31] ). Due to the secure nature of quantum key distribution, Eve would be unable to decipher the secret key even with as much information as Bob and Alice. These are known as an incoherent attacks because a random, repeated attack yields the highest chance of Eve finding the key. [32]

Nuclear magnetic resonance

While classical nuclear magnetic resonance is the phenomenon of nuclei emitting electromagnetic radiation at resonant frequencies when exposed to a strong magnetic field and is used heavily in imaging technology, [33] quantum nuclear magnetic resonance is a type of quantum information processing (QIP). The interactions between the nuclei allow for the application of quantum logic gates, such as the CNOT.

One quantum NMR experiment involved passing three qubits through a circuit, after which they are all entangled; the second and third qubit are transformed into clones of the first with a fidelity of 5/6. [34]

Another application allowed for the alteration of the signal-noise ratio, a process that increased the signal frequency while decreasing the noise frequency, allowing for a clearer information transfer. [35] This is done through polarization transfer, which allows for a portion of the signal's highly polarized electric spin to be transferred to the target nuclear spin.

The NMR system allows for the application of quantum algorithms such as Shor factorization and the Deutsch-Joza algorithm.

Stimulated emission

Stimulated emission is a type of Universal Quantum Cloning Machine that functions on a three-level system: one ground and two degenerates that are connected by an orthogonal electromagnetic field.[ clarification needed ] The system is able to emit photons by exciting electrons between the levels. The photons are emitted in varying polarizations due to the random nature of the system, but the probability of emission type is equal for all – this is what makes this a universal cloning machine. [36] By integrating quantum logic gates into the stimulated emission system, the system is able to produce cloned states. [36]

Telecloning

Telecloning is the combination of quantum teleportation and quantum cloning. [37] This process uses positive operator-valued measurements, maximally entangled states, and quantum teleportation to create identical copies, locally and in a remote location. Quantum teleportation alone follows a "one-to-one" or "many-to-many" method in which either one or many states are transported from Alice, to Bob in a remote location. The teleclone works by first creating local quantum clones of a state, then sending these to a remote location by quantum teleportation. [38]

The benefit of this technology is that it removes errors in transmission that usually result from quantum channel decoherence. [38]

See also

Related Research Articles

In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist. The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem.

<span class="mw-page-title-main">Quantum teleportation</span> Physical phenomenon

Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. The sender does not have to know the particular quantum state being transferred. Moreover, the location of the recipient can be unknown, but to complete the quantum teleportation, classical information needs to be sent from sender to receiver. Because classical information needs to be sent, quantum teleportation cannot occur faster than the speed of light.

<span class="mw-page-title-main">Quantum entanglement</span> Correlation between quantum systems

Quantum entanglement is the phenomenon of a group of particles being generated, interacting, or sharing spatial proximity in such a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.

In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.

Quantum error correction (QEC) is a set of techniques used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum state preparation, and faulty measurements. Effective quantum error correction would allow quantum computers with low qubit fidelity to execute algorithms of higher complexity or greater circuit depth.

In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the principle of locality. These models attempt to account for the probabilistic features of quantum mechanics via the mechanism of underlying, but inaccessible variables, with the additional requirement that distant events be statistically independent.

<span class="mw-page-title-main">Greenberger–Horne–Zeilinger state</span> "Highly entangled" quantum state of 3 or more qubits

In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state that involves at least three subsystems. The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989, and the three-particle version was introduced by N. David Mermin in 1990. Extremely non-classical properties of the state have been observed, contradicting intuitive notions of locality and causality. GHZ states for large numbers of qubits are theorized to give enhanced performance for metrology compared to other qubit superposition states.

The W state is an entangled quantum state of three qubits which in the bra-ket notation has the following shape

In quantum computing, a graph state is a special type of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. In particular, they are a convenient way of representing certain types of entangled states.

BB84 is a quantum key distribution scheme developed by Charles Bennett and Gilles Brassard in 1984. It is the first quantum cryptography protocol. The protocol is provably secure assuming a perfect implementation, relying on two conditions: (1) the quantum property that information gain is only possible at the expense of disturbing the signal if the two states one is trying to distinguish are not orthogonal ; and (2) the existence of an authenticated public classical channel. It is usually explained as a method of securely communicating a private key from one party to another for use in one-time pad encryption. The proof of BB84 depends on a perfect implementation. Side channel attacks exist, taking advantage of non-quantum sources of information. Since this information is non-quantum, it can be intercepted without measuring or cloning quantum particles.

Time-bin encoding is a technique used in quantum information science to encode a qubit of information on a photon. Quantum information science makes use of qubits as a basic resource similar to bits in classical computing. Qubits are any two-level quantum mechanical system; there are many different physical implementations of qubits, one of which is time-bin encoding.

<span class="mw-page-title-main">One-way quantum computer</span> Method of quantum computing

The one-way quantum computer, also known as measurement-based quantum computer (MBQC), is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.

In quantum information and quantum computing, a cluster state is a type of highly entangled state of multiple qubits. Cluster states are generated in lattices of qubits with Ising type interactions. A cluster C is a connected subset of a d-dimensional lattice, and a cluster state is a pure state of the qubits located on C. They are different from other types of entangled states such as GHZ states or W states in that it is more difficult to eliminate quantum entanglement in the case of cluster states. Another way of thinking of cluster states is as a particular instance of graph states, where the underlying graph is a connected subset of a d-dimensional lattice. Cluster states are especially useful in the context of the one-way quantum computer. For a comprehensible introduction to the topic see.

SARG04 is a 2004 quantum cryptography protocol derived from the first protocol of that kind, BB84.

In quantum mechanics, the cat state, named after Schrödinger's cat, refers to a quantum state composed of a superposition of two other states of flagrantly contradictory aspects. Generalizing Schrödinger's thought experiment, any other quantum superposition of two macroscopically distinct states is also referred to as a cat state. A cat state could be of one or more modes or particles, therefore it is not necessarily an entangled state. Such cat states have been experimentally realized in various ways and at various scales.

Magic state distillation is a method for creating more accurate quantum states from multiple noisy ones, which is important for building fault tolerant quantum computers. It has also been linked to quantum contextuality, a concept thought to contribute to quantum computers' power.

The Eastin–Knill theorem is a no-go theorem that states: "No quantum error correcting code can have a continuous symmetry which acts transversely on physical qubits". In other words, no quantum error correcting code can transversely implement a universal gate set, where a transversal logical gate is one that can be implemented on a logical qubit by the independent action of separate physical gates on corresponding physical qubits.

In quantum physics, the "monogamy" of quantum entanglement refers to the fundamental property that it cannot be freely shared between arbitrarily many parties.

This glossary of quantum computing is a list of definitions of terms and concepts used in quantum computing, its sub-disciplines, and related fields.

References

  1. ↑ Bužek, V.; Hillery, M. (1996-09-01). "Quantum copying: Beyond the no-cloning theorem". Physical Review A. 54 (3): 1844–1852. arXiv: quant-ph/9607018 . Bibcode:1996PhRvA..54.1844B. doi:10.1103/physreva.54.1844. ISSN   1050-2947. PMID   9913670. S2CID   1446565.
  2. ↑ Bruss, Dagmar; Ekert, Artur; Macchiavello, Chiara (1998-09-21). "Optimal Universal Quantum Cloning and State Estimation". Physical Review Letters. 81 (12): 2598–2601. arXiv: quant-ph/9712019 . Bibcode:1998PhRvL..81.2598B. doi:10.1103/PhysRevLett.81.2598. S2CID   119535353.
  3. 1 2 Fan, Heng; Wang, Yi-Nan; Jing, Li; Yue, Jie-Dong; Shi, Han-Duo; Zhang, Yong-Liang; Mu, Liang-Zhu (2014-11-20). "Quantum cloning machines and the applications". Physics Reports. 544 (3): 241–322. arXiv: 1301.2956 . Bibcode:2014PhR...544..241F. doi:10.1016/j.physrep.2014.06.004. ISSN   0370-1573. S2CID   118509764.
  4. ↑ Lamas-Linares, A. (2002-03-28). "Experimental Quantum Cloning of Single Photons". Science. 296 (5568): 712–714. arXiv: quant-ph/0205149 . Bibcode:2002Sci...296..712L. doi:10.1126/science.1068972. ISSN   0036-8075. PMID   11923493. S2CID   17723881.
  5. ↑ Rong-Can, Yang; Hong-Cai, Li; Xiu, Lin; Zhi-Ping, Huang; Hong, Xie (January 15, 2008). "Implementing a Universal Quantum Cloning Machine via Adiabatic Evolution in Ion-Trap System". Communications in Theoretical Physics. 49 (1): 80–82. Bibcode:2008CoTPh..49...80Y. doi:10.1088/0253-6102/49/1/17. ISSN   0253-6102. S2CID   250903268.
  6. ↑ Todd A. Brun, Mark M. Wilde, Andreas Winter, Quantum state cloning using Deutschian closed timelike curve. Physical Review Letters 111, 190401 (2013); arXiv:1306.1795
  7. 1 2 3 Imai, H.; Hayashi, Masahito, eds. (2006). Quantum computation and information: from theory to experiment. Berlin: Springer. ISBN   978-3-540-33133-9. OCLC   262693032.
  8. 1 2 Fan, H. (2014). "Quantum Cloning Machines and the Applications". Physics Reports. 544 (3): 241–322. arXiv: 1301.2956 . Bibcode:2014PhR...544..241F. doi:10.1016/j.physrep.2014.06.004. S2CID   118509764.
  9. ↑ Fan, H. (2002). "Cloning of d-level photonic states in physical systems". Phys. Rev. A. 66 (2): 024307. arXiv: quant-ph/0112094 . doi:10.1103/PhysRevA.66.024307. S2CID   40081413.
  10. ↑ Bužek, V.; Hillery, M. (1996-09-01). "Quantum copying: Beyond the no-cloning theorem". Physical Review A. 54 (3): 1844–1852. arXiv: quant-ph/9607018 . Bibcode:1996PhRvA..54.1844B. doi:10.1103/PhysRevA.54.1844. PMID   9913670. S2CID   1446565.
  11. 1 2 Gisin, N.; Massar, S. (1997-09-15). "Optimal Quantum Cloning Machines". Physical Review Letters. 79 (11): 2153–2156. arXiv: quant-ph/9705046 . Bibcode:1997PhRvL..79.2153G. doi:10.1103/PhysRevLett.79.2153. ISSN   0031-9007. S2CID   40919047.
  12. ↑ Werner, R. F. (1998-09-01). "Optimal cloning of pure states". Physical Review A. 58 (3): 1827–1832. arXiv: quant-ph/9804001 . Bibcode:1998PhRvA..58.1827W. CiteSeerX   10.1.1.251.4724 . doi:10.1103/physreva.58.1827. ISSN   1050-2947. S2CID   1918105.
  13. 1 2 Lamas-Linares, Antía; Simon, Christoph; Howell, John C.; Bouwmeester, Dik (2002-04-26). "Experimental Quantum Cloning of Single Photons". Science. 296 (5568): 712–714. arXiv: quant-ph/0205149 . Bibcode:2002Sci...296..712L. doi:10.1126/science.1068972. ISSN   0036-8075. PMID   11923493. S2CID   17723881.
  14. 1 2 Fan, Heng; Matsumoto, Keiji; Wang, Xiang-Bin; Wadati, Miki (2001-12-10). "Quantum cloning machines for equatorial qubits". Physical Review A. 65 (1): 012304. arXiv: quant-ph/0101101 . Bibcode:2001PhRvA..65a2304F. CiteSeerX   10.1.1.251.3934 . doi:10.1103/PhysRevA.65.012304. S2CID   14987216.
  15. ↑ Cerf, Nicolas; Durt, Thomas; Gisin, Nicolas (December 3, 2010). "Cloning a qutrit". Journal of Modern Optics. 49 (8): 1355–1373. arXiv: quant-ph/0110092 . doi:10.1080/09500340110109043. ISSN   0950-0340. S2CID   15872282.
  16. ↑ Fan, Heng (2003-11-03). "Quantum Cloning of Mixed States in Symmetric Subspace". Physical Review A. 68 (5): 054301. arXiv: quant-ph/0308058 . Bibcode:2003PhRvA..68e4301F. doi:10.1103/PhysRevA.68.054301. ISSN   1050-2947. S2CID   119423569.
  17. ↑ Du, Jiangfeng; Durt, Thomas; Zou, Ping; Li, Hui; Kwek, L. C.; Lai, C. H.; Oh, C. H.; Ekert, Artur (2005-02-02). "Experimental Quantum Cloning with Prior Partial Information". Physical Review Letters. 94 (4): 040505. arXiv: quant-ph/0405094 . Bibcode:2005PhRvL..94d0505D. doi:10.1103/PhysRevLett.94.040505. PMID   15783542. S2CID   10764176.
  18. ↑ Cerf, Nicolas J. (2000-05-08). "Pauli Cloning of a Quantum Bit". Physical Review Letters. 84 (19): 4497–4500. arXiv: quant-ph/9803058 . Bibcode:2000PhRvL..84.4497C. doi:10.1103/PhysRevLett.84.4497. PMID   10990720. S2CID   119417153.
  19. 1 2 K. Hashagen, A. "Universal Asymmetric Quantum Cloning Revisited". ResearchGate. Retrieved 2018-11-13.
  20. ↑ Zhao, Zhi; Zhang, An-Ning; Zhou, Xiao-Qi; Chen, Yu-Ao; Lu, Chao-Yang; Karlsson, Anders; Pan, Jian-Wei (2005-07-15). "Experimental realization of optimal asymmetric cloning and telecloning via partial teleportation". Physical Review Letters. 95 (3): 030502. arXiv: quant-ph/0412017 . Bibcode:2005PhRvL..95c0502Z. doi:10.1103/PhysRevLett.95.030502. ISSN   0031-9007. PMID   16090727. S2CID   15815406.
  21. ↑ A. T. Rezakhani, S. Siadatnejad, and A. H. Ghaderi. Separability in Asymmetric Phase-Covariant Cloning (First submitted on 2 December 2003). Physics Letters A 336 (4), 278.
  22. ↑ L.-P. Lamoureux, N. J. Cerf Asymmetric phase-covariant d-dimensional cloning. Physics Letters A 336 (4), 278 (First submitted 7 October 2004).
  23. ↑ A. Kay, R. Ramanathan, D. Kaszlikowski Optimal Asymmetric Quantum Cloning
  24. ↑ Duan & Guo (1997). "Two non-orthogonal states can be cloned by a unitary-reduction process". arXiv: quant-ph/9704020 .
  25. 1 2 Duan, Lu-Ming; Guo, Guang-Can (1998). "Probabilistic cloning and identification of linearly independent states". Physical Review Letters. 80 (22): 4999–5002. arXiv: quant-ph/9804064 . doi:10.1103/PhysRevLett.80.4999. S2CID   14154472 – via APS Physics.
  26. ↑ Chen, Hongwei; Lu, Dawei; Chong, Bo; Qin, Gan; Zhou, Xianyi; Peng, Xinhua; Du, Jiangfeng (2011-05-06). "Experimental Demonstration of Probabilistic Quantum Cloning". Physical Review Letters. 106 (18): 180404. arXiv: 1104.3643 . Bibcode:2011PhRvL.106r0404C. doi:10.1103/physrevlett.106.180404. ISSN   0031-9007. PMID   21635072. S2CID   39554511.
  27. ↑ Duan, Lu-Ming; Guo, Guang-Can (1998-07-06). "A probabilistic cloning machine for replicating two non-orthogonal states". Physics Letters A. 243 (5–6): 261–264. Bibcode:1998PhLA..243..261D. doi:10.1016/S0375-9601(98)00287-4. ISSN   0375-9601.
  28. ↑ Lam, Ping Koy; Ralph, Timothy C.; Symul, Thomas; Blandino, Rémi; Bradshaw, Mark; Assad, Syed M.; Dias, Josephine; Zhao, Jie; Haw, Jing Yan (2016-10-26). "Surpassing the no-cloning limit with a heralded hybrid linear amplifier for coherent states". Nature Communications. 7: 13222. arXiv: 1610.08604 . Bibcode:2016NatCo...713222H. doi:10.1038/ncomms13222. ISSN   2041-1723. PMC   5095179 . PMID   27782135.
  29. 1 2 3 "Quantum Cloning"; Valerio Scarani, Sofyan Iblisdir, Nicolas Gisin; Group of Applied Physics, University of Geneva, 20, rue de l'Ecole-de-M´edecine, CH-1211 Geneva 4, Switzerland
  30. ↑ "Sifting attacks in finite-size quantum key distribution"; Corsin Pfister, Norbert Lütkenhaus, Stephanie Wehner, and Patrick J. Coles; QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, Netherlands
  31. ↑ "Information Reconciliation for Quantum Key Distribution"; David Elkouss, Jesus Martinez-Mateo, Vicente Martina; Research group on Quantum Information and Computationb Facultad de Inform´atica, Universidad Polit´ecnica de Madrid Campus de Montegancedo, 28660 Boadilla del Monte, Madrid, Spain
  32. ↑ "Quantum Cloning"; Valerio Scarani, Sofyan Iblisdir, Nicolas Gisin; Group of Applied Physics, University of Geneva, 20, rue de l'Ecole-de-M´edecine, CH-1211 Geneva 4, Switzerland
  33. ↑ "Nuclear Magnetic Resonance"; Andrew, E.R.; Cambridge University Press, Cambridge, UK, 2009
  34. ↑ "Approximate Quantum Cloning with Nuclear Magnetic Resonance"; Holly K. Cummins, Claire Jones, Alistair Furze, Nicholas F. Soffe, Michele Mosca, Josephine M. Peach, and Jonathan A. Jones; Centre for Quantum Computation, Clarendon Laboratory, University of Oxford, Parks Road, OX1 3PU, United Kingdom
  35. ↑ "Nuclear magnetic resonance for quantum computing: Techniques and recent achievements"; Tao Xin et al 2018 Chinese Phys. B 27 020308
  36. 1 2 "Optimal Quantum Cloning via Stimulated Emission"; Christoph Simon, Gregor Weihs, and Anton Zeilinger; Institut für Experimentalphysik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria
  37. ↑ Murao, Mio; Jonathan, Daniel; Plenio, Martin B; Vedral, Vlatko (1999). "Quantum telecloning and multiparticle entanglement". Phys. Rev. A. 59 (1): 156–161. arXiv: quant-ph/9806082 . Bibcode:1999PhRvA..59..156M. doi:10.1103/PhysRevA.59.156. hdl: 10044/1/246 . S2CID   119348617.
  38. 1 2 "Quantum Cloning Machines and the Applications"; Heng Fan, Yi-Nan Wang, Li Jing, Jie-Dong Yue, Han-Duo Shi, Yong-Liang Zhang, and Liang-Zhu Mu; Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
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