Perfect zero knowledge for quantum multiprover interactive proofs
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In this work we consider the interplay between multiprover interactive proofs, quantum entanglement, and zero knowledge proofs - notions that are central pillars of complexity theory, quantum information and cryptography. In particular, we study the relationship between the complexity class MIP^*, the set of languages decidable by multiprover interactive proofs with quantumly entangled provers, and the class PZKMIP^*, which is the set of languages decidable by MIP^* protocols that furthermore possess the perfect zero knowledge property. Our main result is that the two classes are equal, i.e., MIP^* = PZKMIP^*. This result provides a quantum analogue of the celebrated result of Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP = PZKMIP (in other words, all classical multiprover interactive protocols can be made zero knowledge). We prove our result by showing that every MIP^* protocol can be efficiently transformed into an equivalent zero knowledge MIP^* protocol in a manner that preserves the completeness-soundness gap. Combining our transformation with previous results by Slofstra (Forum of Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we obtain the corollary that all co-recursively enumerable languages (which include undecidable problems as well as all decidable problems) have zero knowledge MIP^* protocols with vanishing promise gap.
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