Optimal universal and state-dependent quantum cloning

Abstract

We establish the best possible approximation to a perfect quantum cloning machine that produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two nonorthogonal states and discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive an upper bound on the quantum capacity of the depolarizing quantum channel.

Quantum cloning machines and the applications

“…For qubit case, by mapping several identical pure states into the cloning of output states assisted by ancillary states, the general universal cloning machine is realized (Gisin and Massar, 1997). The optimality of the fidelity is later proved by considering that the case of infinite copies should be equivalent to quantum state estimation (Bruß et al, 1998b). Along this line, the one to many universal cloning machine for higher-dimensional case is also studied in (Albeverio and Fei, 2000).…”

“…In this sense, the corresponding cloning machine is "universal". One may find that the above presented cloning machine achieves higher fidelity than the trivial one, and it is proved to be optimal (Bruß et al, 1998a;Gisin and Massar, 1997)). Gisin and Massar (Gisin and Massar, 1997) then generalize the cloning machine to N → M case, that is M copies are created from N identical qubits.…”

“…, 7. The complete proof of the optimality is finished in (Bruß et al, 1998b) where the connection between optimal quantum cloning and optimal state estimation is established. The upper bound of N to M UQCM is found to be exactly equal to (55), hence Gisin and Massar UQCM is optimal.…”

Separability in asymmetric phase-covariant cloning

“…However, extension to triplicators is also straightforward. The question of how well one can design an approximate duplicator of a qubit (or qudit), provided that the qualities of the two outputs be independent of the input states, has been investigated by Bužek and Hillery [5,6] and the others [7,8,9,10,11,12,13].…”

“…For example in BB84 protocol [4] there is a link between optimal cloning of equatorial states and optimal eavesdropping attack. There have been extensive studies on this subject that illuminate some aspects of quantum cloning [5,6,7,8,9,10,11,12,13].…”