Zero-knowledge convincing protocol on quantum bit is impossible

Pawel Horodecki1, Michal Horodecki2, and Ryszard Horodecki1

1Faculty of Applied Physics and Mathematics National Quantum Information Centre Gdánsk University of Technology, 80–233 Gdánsk, Poland
2Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics University of Gdánsk, 80-308 Gdánsk, Poland

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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 about quantum state is involved.

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