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The superposition of states is one of the most fundamental issues in thequantum world. Generally there do not exist physical operations to superposetwo unknown random states with nonzero probability. We investigate thesuperposition problem of unknown qubit states with respect to a known qubitstate. It is shown that under trace-nonincreasing completely positiveoperations the superposable state sets are located in some circles on the Blochsphere. Meanwhile, we prove that the quantum states in a circle on the Blochsphere are superposable with respect to a known state. Finally, for thehigh-dimensional case, we illustrate that any superposition transformationprotocols will violate the no-cloning principle for almost all the states. Ourresults also promote the understanding and applications of the superpositionprinciple in view of quantum no-go theorems.

Characterizing the superposition of arbitrary random quantum states and a known quantum state