MIP*=RE
Abstract
We show that the class MIP* of languages that can be decided by a classical verifier interacting with multiple allpowerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages. Our proof builds upon the quantum lowdegree test of (Natarajan and Vidick, FOCS 2018) and the classical lowindividual degree test of (Ji, et al., 2020) by integrating recent developments from (Natarajan and Wright, FOCS 2019) and combining them with the recursive compression framework of (Fitzsimons et al., STOC 2019). An immediate byproduct of our result is that there is an efficient reduction from the Halting Problem to the problem of deciding whether a twoplayer nonlocal game has entangled value $1$ or at most $1/2$. Using a known connection, undecidability of the entangled value implies a negative answer to Tsirelson's problem: we show, by providing an explicit example, that the closure $C_{qa}$ of the set of quantum tensor product correlations is strictly included in the set $C_{qc}$ of quantum commuting correlations. Following work of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our results provide a refutation of Connes' embedding conjecture from the theory of von Neumann algebras.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.04383
 Bibcode:
 2020arXiv200104383J
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity;
 Mathematics  Operator Algebras
 EPrint:
 206 pages. v2: Updated to use arXiv:2009.12982. New appendix