imits on the Power of Quantum Statistical Zero-Knowledge

In this paper we propose a definition for (honest verifier) quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZKHV . We prove several facts regarding this class, including: The following problem is a complete promise problem for QSZKHV : given instructions for preparing two mixed quantum states, are the states close together or far apart in the trace norm metric? This problem is a quantum generalization of the complete promise problem of Sahai and Vadhan [25] for (classical) statistical zeroknowledge. QSZKHV is closed under complement. QSZKHV PSPACE. (At present it is not known if arbitrary quantum interactive proof systems can be simulated in PSPACE, even for one-round proof systems.) Any polynomial-round honest verifier quantum statistical zero-knowledge proof system can be simulated by a two-message (i.e., one-round) honest verifier quantum statistical zero-knowledge proof system. Similarly, any polynomial-round honest verifier quantum statistical zero-knowledge proof system can be simulated by a three-message public-coin honest verifier quantum statistical zero-knowledge proof system. These facts establish close connections between classical statistical zero-knowledge and our definition for quantum statistical zero-knowledge, and give some insight regarding the effect of this zero-knowledge restriction on quantum interactive proof systems. The relationship between our definition and possible definitions of general (i.e., not necessarily honest) quantum statistical zero-knowledge are also discussed. E-mail: [email protected]. Supported by Canada’s NSERC and the Canada Research Chair program.