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 | ψ ⟩ A {\displaystyle {\displaystyle |\psi \rangle _{A}}} perfectly. In Dirac notation, the process of quantum cloning is described by:
U | ψ ⟩ A | e ⟩ B = | ψ ⟩ A | ψ ⟩ B , {\displaystyle {\displaystyle U|\psi \rangle _{A}|e\rangle _{B}=|\psi \rangle _{A}|\psi \rangle _{B}},}
where U {\displaystyle {\displaystyle U}} is the actual cloning operation, | ψ ⟩ A {\displaystyle {\displaystyle |\psi \rangle _{A}}} is the state to be cloned, and | e ⟩ B {\displaystyle {\displaystyle |e\rangle _{B}}} 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, and theoretical bounds were derived on the fidelity of cloned quantum states. One of the applications of quantum cloning is to analyse the security of quantum key distribution protocols. Teleportation, nuclear magnetic resonance, quantum amplification, and superior phase conjugation are examples of some methods utilized to realize a quantum cloning machine. Ion trapping techniques have been applied to cloning quantum states of ions.
Types of quantum cloning machines It may be possible to clone a quantum state to arbitrary accuracy in the presence of closed timelike curves.
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. The output state produced is governed by the Hamiltonian of the system. One of the first cloning machines, a 1 to 2 UQC machine, was proposed in 1996 by Buzek and Hillery. 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, as well as d-dimensional systems. Multiple experiments have been conducted to realize this type of cloning machine physically by using photon stimulated emission. 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.
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. The resulting clones are thus state-dependent, having an optimal fidelity of 1 / 2 + 1 / 8 ≈ 0.8536 {\textstyle 1/2+{\sqrt {1/8}}\approx 0.8536} . 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 R ^ ( ϑ ) {\textstyle {\hat {R}}(\vartheta )} and the controlled-NOT (CNOT). Output states are also separable according to Peres–Horodecki criterion. The process has been generalized to the 1 → M case and proven optimal. This has also been extended to the qutrit and qudit cases. The first experimental asymmetric quantum cloning machine was realized in 2004 using nuclear magnetic resonance.
Asymmetric quantum cloning The first family of asymmetric quantum cloning machines was proposed by Nicolas Cerf in 1998. 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. The optimal trade-off is bounded by the following inequality:
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