Zero-knowledge convincing protocol on quantum bit is impossible

Pawel Horodecki,Michal Horodecki,Ryszard Horodecki

doi:10.22331/q-2017-12-23-41

Pawel Horodecki, Michal Horodecki + Show 1 more

Open Access

https://doi.org/10.22331/q-2017-12-23-41

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Abstract

Consider two parties: Alice and Bob and suppose that Bob is given a qubit system in a quantum state ϕ, unknown to him. Alice knows ϕ and she is supposed to convince Bob that she knows ϕ sending some test message. Is it possible for her to convince Bob providing him "zero knowledge" i. e. no information about ϕ 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 ϕ with nonzero probability. This "no-go" result puts general constrains on information processing where information about quantum state is involved.

Highlights

Consider first the most general test message from Alice. It can involve some classical information as well as purely quantum information represented by quantum register or, in other words, quantum system in some state
The general form of test message within quantum formalism .- All the above can taken into account in the test message sent form Alice to Bob consisting of three elements (i) classical prescription of some quantum operation, (ii) possibly - some ancilla in quantum state Alice prepared together with (iii) result of the operation
The quantum operation is supposed to act in general on both ancilla and Bob’s state, but this action could be in particular trivial i. e. not affecting some of them at all

Summary

Introduction

Consider first the most general test message from Alice. It can involve some classical information (some data, function encoded in classical bits) as well as purely quantum information represented by quantum register or, in other words, quantum system in some state. It is obvious that if Alice knows the state and she wants to convince Bob about it the result of the operation she predicts must occur with probability one i.

Results

Conclusion