Quantum-to-Quantum Bernoulli Factory

Given a coin with unknown bias $p\in [0,1]$, 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 $\mathrm{\mid }p⟩=\sqrt{p}\mathrm{\mid }0⟩+\sqrt{1-p}\mathrm{\mid }1⟩$, can we exactly simulate another quantum coin $\mathrm{\mid }f(p)⟩=\sqrt{f(p)}\mathrm{\mid }0⟩+\sqrt{1-f(p)}\mathrm{\mid }1⟩$? We give the full characterization of simulable quantum state $k_0(p)\mathrm{\mid }0⟩+k_1(p)\mathrm{\mid }1⟩$ from quantum coin $\mathrm{\mid }p⟩=\sqrt{p}\mathrm{\mid }0⟩+\sqrt{1-p}\mathrm{\mid }1⟩$, 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.