# 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:

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

where ${\displaystyle U}$ is the actual cloning operation, ${\displaystyle |\psi \rangle _{A}}$ is the state to be cloned, and ${\displaystyle |e\rangle _{B}}$ 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 ${\displaystyle |\psi \rangle _{A}}$ perfectly. 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. As a result of cloning, the copies resulted from a pure state will be entangled. A cloner is often increases the entanglement between the states. However, outputs of an optimal phase-covariant cloner (where equatorial states) can be unentangled (see refs.).

The accuracy of copies is measured by fidelity and other measures. The universal property of cloning machine means the input state is equally likely to be any pure state.[1] In universal cloning, the qualities of the two outputs have to be independent of the input states. Other alternative prior distributions include phase-covariant cloners and real cloners. A universal ${\displaystyle 1\rightarrow 2}$ cloning machine can have a fidelity as high as 5/6.[2]

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[who?] 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.[3] Ion trapping techniques have been applied to cloning quantum states of ions.[4]

A complete and updated review about various quantum cloning machines, their applications and the implementations can be found in [5]

## Special cases

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

### Phase-Covariant Cloning

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 represent 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 perfectly but with the 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 probabilistic 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.

### Asymmetric Quantum Cloning

The Uncertainty principle puts a limit on the fidelity of cloning. However, a higher fidelity can be achieved in one of the copies if the other copy(ies) require less fidelity. If the fidelities of the clones is designed to be not equal, the optimal quantum cloning machine is asymmetric. This can be used to customise the accuracy by choosing any arbitrary point in the trade-off curve between the qualities of the copies. The trade-off of optimal accuracy between the resulting copies has been studied independently by [7] (with quantum circuit) and [8] (theoretical bounds).

The optimal asymmetric cloning machines are extended to ${\displaystyle M\rightarrow N}$ in ${\displaystyle d}$ dimensions.[9]

## References

1. ^ Alastair Kay Optimal Universal Quantum Cloning: Asymmetries and Fidelity Measures First submitted on July 2014
2. ^ Bužek V. and Hillery, M. Quantum Copying: Beyond the No-Cloning Theorem. Phys. Rev. A 54, 1844 (1996)
3. ^ Antía Lamas-Linares, Christoph Simon, John C. Howell, Dik Bouwmeester, Experimental Quantum Cloning of Single Photons, Science 296 5568 (2002)
4. ^ 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.
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).
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. ^ A. T. Rezakhani, S. Siadatnejad, and A. H. Ghaderi. Separability in Asymmetric Phase-Covariant Cloning (First submitted on 2 Dec 2003). Physics Letters A 336 (4), 278.
8. ^ L.-P. Lamoureux, N. J. Cerf Asymmetric phase-covariant d-dimensional cloning. Physics Letters A 336 (4), 278 (First submitted 7 Oct 2004).
9. ^ A. Kay, R. Ramanathan, D. Kaszlikowski Optimal Asymmetric Quantum Cloning