Quantum-to-quantum Bernoulli factory problem
Jiaqing Jiang, Jialin Zhang, Xiaoming Sun · Physical Review A · 2018
Given a coin with unknown bias $p\ensuremath{\in}[0,1]$, can we exactly simulate another coin with bias $f(p)$? The exact set of simulable functions has been well characterized 20 years ago. In this paper, we ask the quantum counterpart of this question: Given the quantum coin $|p\ensuremath{\rangle}=\sqrt{p}|0\ensuremath{\rangle}+\sqrt{1\ensuremath{-}p}|1\ensuremath{\rangle}$, can we exactly simulate another quantum coin $|f(p)\ensuremath{\rangle}=\sqrt{f(p)}|0\ensuremath{\rangle}+\sqrt{1\ensuremath{-}f(p)}|1\ensuremath{\rangle}$? We give the full characterization of simulable quantum state ${k}_{0}(p)|0\ensuremath{\rangle}+{k}_{1}(p)|1\ensuremath{\rangle}$ from quantum coin $|p\ensuremath{\rangle}=\sqrt{p}|0\ensuremath{\rangle}+\sqrt{1\ensuremath{-}p}|1\ensuremath{\rangle}$, and present an algorithm to transform it. Surprisingly, we show that simulable sets in the quantum-to-quantum case and classical-to-classical case have no inclusion relationship with each other.