Mip* = Re

Abstract

Note from the Research Highlights Co-Chairs: A Research Highlights paper appearing in Communications is usually peer-reviewed prior to publication. The following paper is unusual in that it is still under review. However, the result has generated enormous excitement in the research community, and came strongly nominated by SIGACT, a nomination seconded by external reviewers. The complexity class NP characterizes the collection of computational problems that have efficiently verifiable solutions. With the goal of classifying computational problems that seem to lie beyond NP, starting in the 1980s complexity theorists have considered extensions of the notion of efficient verification that allow for the use of randomness (the class MA), interaction (the class IP), and the possibility to interact with multiple proofs, or provers (the class MIP). The study of these extensions led to the celebrated PCP theorem and its applications to hardness of approximation and the design of cryptographic protocols. In this work, we study a fourth modification to the notion of efficient verification that originates in the study of quantum entanglement. We prove the surprising result that every problem that is recursively enumerable, including the Halting problem, can be efficiently verified by a classical probabilistic polynomial-time verifier interacting with two all-powerful but noncommunicating provers sharing entanglement. The result resolves long-standing open problems in the foundations of quantum mechanics (Tsirelson's problem) and operator algebras (Connes' embedding problem).

Entanglement-Separability Boundary Within a Quantum State

“…Our results can be generalized to general convex resource theories [30] and the operator theory. Quantum entanglement was shown being at the centre of a new mathematical proof recently [31]. We expected that our findings will stimulate further investigation on the quantum theory and practical applications in other fields.…”

Non-embeddable II$_1$ factors resembling the hyperfinite II$_1$ factor

“…The reference[6] actually only deals with strongly self-absoring C * -algebras, but the proof there implies the result that we mention above 7. By the work in[7, Remark 5.8], the failure of CEP already implies that E, should it exist, could not be isomorphic to E ⊗ E 8. In general, the prime model of a theory is a model which elementarily embeds into any other model of the theory and, if it exists, it is automatically unique.…”

“…In the embeddable situation, R is a finitely generic embeddable factor (see [7,Corollary 3.14]), whence the finite forcing companion is simply the complete theory of R. Since R embeds in any model of its theory (due to the axiomatizability of being McDuff), the fact that R has the generalized Jung property implies that it is the prime model of its theory. 8 It is currently unknown whether or not T f has a prime model. However, if it does, then it is a non-embeddable factor with the Jung property: Theorem 3.…”

“…Recall that the Connes Embedding Problem (CEP) asks whether or not every separable 1 tracial von Neumann algebra is embeddable. As announced in the recent landmark paper [8], the Connes Embedding Problem has a negative answer. It thus makes sense to ask whether or not there are separable nonembeddable II 1 factors that have the Jung property or are self-tracially stable.…”

Extreme Points and Factorizability for New Classes of Unital Quantum Channels

“…In [18] it was shown that for every d ≥ 11, there are UCPT maps with an exact factorization through M d (C) ⊗ N , where N a von Neumann algebra of type II 1 which cannot be replaced by any finite dimensional von Neumann algebra. In view of [6,Theorem 3.7] and the announcement in [12] that the Connes Embedding Problem has a negative answer, there are factorizable maps, for some (presumably very large) d, which cannot even be approximated by maps with exact factorizations through matrix algebras.…”