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 $\hat{U}$ that transforms a given unknown state $|\psi_\tau\rangle$ to a known fiducial state $|f\rangle$. Then, after completion of the learning process, we can estimate and reproduce $|\psi_\tau\rangle$ based on the learned $\hat{U}$ and $|f\rangle$. 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 {\em finite} number, say $N$, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large $N$, namely $\simeq O(N^{-1})$ scales of average infidelity. This result is comparable to those yielded from the standard quantum tomographic method in the case where additional information is available. It highlights a non-trivial message, that is, a random-based adaptive strategy can potentially be as accurate as other standard statistical approaches.