Learning unknown pure quantum states

We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation (U) over cap that transforms a given unknown state vertical bar psi(tau)> to a known fiducial state vertical bar f >. Then, after completion of the learning process, we can estimate and reproduce vertical bar psi(tau)> based on the learned (U) over cap (a) under bar nd vertical bar f >. To realize this idea, we cast a random-based learning algorithm, called "single-shot measurement learning," in which the learning rule is based on an intuitive and reasonable criterion: the greater the number of success (or failure), the less (or more) changes are imposed. Remarkably, the learning process occurs by means of a single-shot measurement outcome. We demonstrate that our method works effectively, i.e., the learning is completed with a finite number, say N, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large N, namely similar or equal to O (N-1) scales of average infidelity. It highlights a nontrivial message, that is, a random-based strategy can potentially be as accurate as other standard statistical approaches.