Creating n perfect clones from m copies of one of N prior known quantum states with minimum failure probability is a long-standing problem. We present a general geometrical form of the sufficient and necessary condition of probabilistic cloning of N known quantum states. With only one equality, we delineate in a parameter space the feasible set for probabilistic cloning of N known quantum states with prior probabilities and the surface for the optimal probabilistic cloning. The results are also applicable to the identification of those N quantum states. We provide a complete process of optimal identification of N given states and illustrate it with examples of three states. We further discover the relationship between probabilistic identification and the convexity of the feasible set. In addition, we discuss the differences in parameter selection when N = 2 and N ⩾ 3, plus provide a rigorous proof of some specific co...

中文翻译：

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概率量子克隆的几何分析和N状态的识别

一个长期存在的问题是，从N个已知的量子态之一的m个副本中创建n个完美的克隆，其失败概率最小。我们提出了N个已知量子态的概率克隆的充分必要条件的一般几何形式。仅用一个等式，我们就在参数空间中描绘了具有先验概率的N个已知量子态的概率克隆的可行集和最优概率克隆的表面。该结果也可用于鉴定那些N个量子态。我们提供了一个最佳识别N个给定状态的完整过程，并以三个状态为例进行说明。我们进一步发现概率识别与可行集凸性之间的关系。此外，