Learning unknown pure quantum states
Sang Min Lee, Jinhyoung Lee, Jeongho Bang · Physical Review A · 2018
We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}$ that transforms a given unknown state $|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}$ to a known fiducial state $|f\ensuremath{\rangle}$. Then, after completion of the learning process, we can estimate and reproduce $|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}$ based on the learned $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}$ and $|f\ensuremath{\rangle}$. 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 finite number, say $N$, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large $N$, namely $\ensuremath{\simeq}O({N}^{\ensuremath{-}1})$ 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.