Probabilistic cloning of three nonorthogonal states

We study the probabilistic cloning of three nonorthogonal states with equal success probabilities. For simplicity, we assume that the three states belong to a special set. Analytical form of the maximal success probability for $$M\rightarrow N$$M→N probabilistic cloning is calculated. With the maximal success probability, we deduce the explicit form of $$M\rightarrow N$$M→N probabilistic quantum cloning machine. In the case of $$1\rightarrow 2$$1→2 cloning, we get the unambiguous form of the unitary operation. It is demonstrated that the upper bound for probabilistic quantum cloning machine in (Qiu in J Phys A 35:6931, 2002) can be reached only if the three states are equidistant.

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