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arXiv:quant-ph/0208159 · Journal of Optics B Quantum and Semiclassical Optics

A lower bound on the relative error of mixed-state cloning and related operations

A E Rastegin

Published Oct 16, 2003• 35 citations
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Abstract

We extend the concept of the relative error to mixed-state cloning and related physical operations, in which the ancilla contains some information a priori about the input state. The lower bound on the relative error is obtained. It is shown that this result provides further support for a stronger no-cloning theorem.
DOI
10.1088/1464-4266/5/6/017

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The relative error of cloning of quantum states with arbitrary prior probabilities is considered. It is assumed that the ancilla may contain some a priori information about the input state to be cloned. The lower bound on the relative error for general cloning scenario is derived. Both the case of two-state set and case of multistate set are analyzed in detail. The treated figure of merit is compared with other optimality criteria. It is posed that the relative error is more expressive in the case of nearly indistinguishable states. The quantum circuit for optimal cloning of a pair of pure states is built.
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It is believed that the more we have {\it a priori} information on input states, the better we can make the quality of clones in quantum cloning machines. This common sense idea was confirmed several years ago by analyzing a situation, where the input state is either one of two non-orthogonal states. If the {\it a priori} information is measured by the Shannon entropy, common sense predicts that the quality of the clone becomes poorer with increasing NNN, where NNN is the number of possible input states. We show, however, that the {\it a priori} information measured by the Shannon entropy does not affect the quality of the clones. Instead the no-cloning theorem and `denseness' of the possible input states play important roles in determining the quality. Specifically, the factor `denseness' plays a more crucial role than the no-cloning theorem when N≥3N \geq 3N≥3.
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