We study quantum cloning machines (QCM) that act on an unknown N-levelquantum state and make M copies. We give a formula for the maximum of thefidelity of cloning and exhibit the unitary transformations that realize thisoptimal fidelity. We also extend the results to treat the case of M copies from$N^\prime$ ($M>N^\prime$) identical N-level quantum systems.
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机译：我们研究了作用于未知N级量子态并产生M拷贝的量子克隆机（QCM）。我们给出了最大克隆保真度的公式，并展示了实现这种最佳保真度的单一变换。我们还扩展了结果以处理来自$ N ^ \ prime $（$ M> N ^ \ prime $）相同的N级量子系统的M个副本的情况。
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