Connes embedding problem

Connes' embedding problem, formulated by <a href="/facts/Alain_Connes/tjFJ9Iou">Alain Connes</a> in the 1970s, is a major problem in <a href="/facts/Von_Neumann_algebra/8NyWUIRH">von Neumann algebra</a> theory. During that time, the problem was reformulated in several different areas of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>. <a href="/facts/Dan-Virgil_Voiculescu/os6zf1pI">Dan Voiculescu</a> developing his free entropy theory found that Connes' embedding problem is related to the existence of microstates. Some results of von Neumann algebra theory can be obtained assuming positive solution to the problem. The problem is connected to some basic questions in quantum theory, which led to the realization that it also has important implications in <a href="/facts/Computer_science/lN3pEyPz">computer science</a>.
The problem admits a number of equivalent formulations. Notably, it is equivalent to the following long standing problems:

<ul><li>Kirchberg's QWEP conjecture in <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a> theory</li>
<li><a href="/facts/Tsirelson%2527s_bound/qlnD5GDC">Tsirelson's problem</a> in quantum information theory</li>
<li>The predual of any (separable) von Neumann algebra is finitely representable in the trace class.</li></ul>
In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen announced a result in <a href="/facts/Quantum_complexity_theory/Gnd4G8Mt">quantum complexity theory</a> that implies a negative answer to Connes' embedding problem. However, an error was discovered in September 2020 in an earlier result they used; a new proof avoiding the earlier result was published as a preprint in September. A broad outline was published in <a href="/facts/Communications_of_the_ACM/Ix1DvEAk">Communications of the ACM</a> in November 2021, and an article explaining the connection between MIP*=RE and the Connes Embedding Problem appeared in October 2022.

Connes embedding problem open-in-new

Connes embedding problem