Zero knowledge convincing protocol on quantum bit is impossible

Abstract

Consider two parties: Alice and Bob and suppose that Bob is given a qubit system in a quantum state $\phi$, unknown to him. Alice knows $\phi$ and she is supposed to convince Bob that she knows $\phi$ sending some test message. Is it possible for her to convince Bob providing him "zero knowledge" i. e. no information about $\phi$ he has? We prove that there is no "zero knowledge" protocol of that kind. In fact it turns out that basing on Alice message, Bob (or third party - Eve - who can intercept the message) can synthetize a copy of the unknown qubit state $\phi$ with nonzero probability. This "no-go" result puts general constrains on information processing where information {\it about} quantum state is involved.