Honghao Fu

In the recent breakthrough result (Mastel and Slofstra, STOC24), the authors show that there is a two-player one-round perfect zero-knowledge MIP* protocol for RE. We build on their result to show that there exists a succinct two-player one-round perfect zero-knowledge MIP* protocol for RE with polylog question size and O(1) answer size, or with O(1) question size and polylog answer size. To prove our result, we analyze the four central compression techniques underlying the MIP*=RE proof (Ji et al., arXiv:2001.04383) --- question reduction, oracularization, answer reduction, and parallel repetition --- and show that they all preserve the perfect (as well as statistical and computational) zero-knowledge properties of the original protocol. Furthermore, we complete the study of the conversion between constraint-constraint and constraint-variable binary constraint system (BCS) nonlocal games, which provide a quantum information characterization of MIP* protocols. While Paddock (arXiv:2203.02525) established that any near perfect strategy for a constraint-variable game can be mapped to a constraint-constraint version, we prove the converse, fully establishing their equivalence.

Abstract When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is quantum? This question is central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations --- that is, correlations for which the number of measurements and number of measurement outcomes are fixed --- such that solving the quantum membership problem for this family is computationally impossible. Intuitively, our result means that the undecidability that arises in understanding Bell experiments is innate, and is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and from undecidability results of the third author for linear system nonlocal games.