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Physical Review A

Probabilistic cloning of equidistant states

O. Jiménez, Luis Roa, A. Delgado

Published Aug 25, 2010• 23 citations
Publisher

Abstract

We study the probabilistic cloning of equidistant states. These states are such that the inner product between them is a complex constant or its conjugate. Thereby, it is possible to study their cloning in a simple way. In particular, we are interested in the behavior of the cloning probability as a function of the phase of the overlap among the involved states. We show that for certain families of equidistant states Duan and Guo's cloning machine leads to cloning probabilities lower than the optimal unambiguous discrimination probability of equidistant states. We propose an alternative cloning machine whose cloning probability is higher than or equal to the optimal unambiguous discrimination probability for any family of equidistant states. Both machines achieve the same probability for equidistant states whose inner product is a positive real number.

DOI
10.1103/physreva.82.022328

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We study the probabilistic cloning of equidistant states. These states are such that the inner product between them is a complex constant or its conjugate. Thereby, it is possible to study their cloning in a simple way. In particular, we are interested in the behavior of the cloning probability as a function of the phase of the overlap among the involved states. We show that for certain families of equidistant states Duan and Guo's cloning machine leads to cloning probabilities lower than the optimal unambiguous discrimination probability of equidistant states. We propose an alternative cloning machine whose cloning probability is higher than or equal to the optimal unambiguous discrimination probability for any family of equidistant states. Both machines achieve the same probability for equidistant states whose inner product is a positive real number.
We study the probabilistic cloning of three symmetric states. These states are defined by a single complex quantity, the inner product among them. We show that three different probabilistic cloning machines are necessary to optimally clone all possible families of three symmetric states. We also show that the optimal cloning probability of generating MMM copies out of one original can be cast as the quotient between the success probability of unambiguously discriminating one and MMM copies of symmetric states.
This paper investigates the 1 → M probabilistic quantum cloning of three symmetric states, which are defined by a complex number of their inner product. First, a general transformation of the 1 → M probabilistic quantum cloning is designed, and the ancillary success states are introduced. Then, the explicit expression of the probability of successful cloning is derived, which depends on the number M and the inner product of the three symmetric states. Finally, three examples are provided to demonstrate the efficiency of our cloning transformation.
The method of quantum cloning is divided into two main categories: approximate and probabilistic quantum cloning. The former method is used to approximate an unknown quantum state deterministically, and the latter can be used to faithfully copy the state probabilistically. Thus far, many approximate cloning machines have been experimentally demonstrated, but probabilistic cloning remains an experimental challenge, as it requires more complicated networks and a higher level of precision control. In this work, we design an efficient quantum network with a limited amount of resources and perform the first experimental demonstration of probabilistic quantum cloning in a NMR quantum computer. In our experiment, the optimal cloning efficiency proposed by Duan and Guo [Phys. Rev. Lett. 80, 4999 (1998)] is achieved.
Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation . We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial realizations of the state, where N > M . We also discuss the relationship between this bound and that on unambiguous state discrimination.
Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation. We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial realizations of the state, where N > M. We also discuss the relationship between this bound and that on unambiguous state discrimination.
Probabilistic quantum cloning and identifying machines can be constructed via unitary-reduction processes [Duan and Guo, Phys. Rev. Lett. 80, 4999 (1998)]. Given the cloning (identifying) probabilities, we derive an explicit representation of the unitary evolution and corresponding Hamiltonian to realize probabilistic cloning (identification). The logic networks are obtained by decomposing the unitary representation into universal quantum logic operations. The robustness of the networks is also discussed. Our method is suitable for a k-partite system, such as quantum computer, and may be generalized to general state-dependent cloning and identification.