Quantum cloning

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. In Dirac notation, the process of quantum cloning is described by:

$U|\psi \rangle _{A}|e\rangle _{B}=|\psi \rangle _{A}|\psi \rangle _{B}$,

where $U$ is the actual cloning operation, $|\psi\rangle_A$ is the state to be cloned, and $|e\rangle _{B}$ is the initial state of the copy.

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 $|\psi\rangle_A$. Though perfect quantum cloning is not possible, it is possible to perform imperfect cloning, where the copies have a non-unit fidelity. A universal cloning machine can have a fidelity as high as 5/6.^[1]

The quantum cloning operation is the best way to make copies of quantum information therefore cloning is an important task in quantum information processing, especially in the context of quantum cryptography. Researchers are seeking ways to build quantum cloning machines, which work at the so-called quantum limit. The first cloning machine relied on stimulated emission to copy quantum information encoded into single photons. Teleportation, nuclear magnetic resonance, quantum amplification and superior phase conjugation have been some other methods utilized to realize a quantum cloning machine.^[2] Ion trapping techniques have been applied to cloning quantum states of ions.^[3]

It may be possible to clone a quantum state to arbitrary accuracy in the presence of closed timelike curves.^[4]

We can also consider quantum cloning in more complicated cases such as, the input states are restricted to a special form such that they are equally distributed in the equator of the Bloch sphere which can represents arbitrary states of qubit; or we can consider to quantum copy approximately but optimally N identical states to M states (M is larger than N); on the other hand, we can quantum clone a quantum state perferctly but with highest probability. Based on different aims, we can construct various quantum cloning machines. There are wide applications of those different quantum cloning machines. The universal and phase-covariant quantum cloning machines can be directly related with BB84 and six-state protocols of quantum cryptography. The probabilistical quantum cloning machine can be related with

B92 protocol of quantum cryptography. Those quantum cloning machines can be implemented in various physical systems for quantum information processing. A complete and update review about various quantum cloning machines, their applications and the implementations can be found in ^[5]

References

1. Bužek V. and Hillery, M. Quantum Copying: Beyond the No-Cloning Theorem. Phys. Rev. A 54, 1844 (1996)
2. Antía Lamas-Linares, Christoph Simon, John C. Howell, Dik Bouwmeester, Experimental Quantum Cloning of Single Photons, Science 296 5568 (2002)
3. YANG, Rong-Can; LI, Hong-Cai†; LIN, Xiu; HUANG, Zhi-Ping; XIE, Hong (Jan 2008). "Implementing a Universal Quantum Cloning Machine via Adiabatic Evolution in Ion-Trap System". Bibcode:2008CoTPh..49...80Y. doi:10.1088/0253-6102/49/1/17.
4. 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
5. H. Fan, Y. N. Wang, L. Jing, J. D. Yue, H. D. Shi, Y. L. Zhang, and L. Z. Mu, Quantum cloning machines and the applications, Physics Reports-Review Section of Physics Letters 544, 241-322 (2014).

Additional References

• V. Buzek and M. Hillery, Quantum cloning, Physics World 14 (11) (2001), pp. 25–29.
• Zhao Zhi, Zhang An-Ning, Zhou Xiao-Qi, Chen Yu-Ao, Lu Chao-Yang, Karlsson Anders, Pan Jian-Wei, Experimental Realization of Optimal Asymmetric Cloning and Telecloning via Partial Teleportation, Phys. Rev. Lett 95 030502 (2005)
• Cummins Holly K., Jones Claire, Furze Alistair, Soffe Nicholas F., Mosca Michele, Peach Josephine M., Jones Jonathan A., Approximate Quantum Cloning with Nuclear Magnetic Resonance, Phys. Rev. Lett 88 187901 (2002)
• Samuel L. Braunstein, Nicolas J. Cerf, Sofyan Iblisdir, Peter van Loock, and Serge Massar Optimal Cloning of Coherent States with a Linear Amplifier and Beam Splitters, Phys. Rev. Lett 86 4938 (2001)
• Metin Sabuncu, Ulrik L. Andersen, and Gerd Leuchs Experimental Demonstration of Continuous Variable Cloning with Phase-Conjugate Inputs, Phys. Rev. Lett 98 170503 (2007)
• Johannes Nokkala Universal Quantum Cloning (2013), (Master's thesis). Retrieved from http://urn.fi/

See also

• No-cloning theorem
• No-broadcast theorem
• Quantum no-deleting theorem