# Background

Earlier this month (Jan 2020) a pre-print was posted to the arXiv claiming to have proved the equivalence of the complexity classes $$\mathrm{MIP}^{*}$$ and $$\mathrm{RE}$$ (see below for definitions). The paper has not yet passed peer-review at the time of this question, but there seems to be reasonable confidence that the result is correct (see e.g. Scott Aaronson's blog post about it), so for the rest of this post I will assume this is the case.

It turns out that this result is equivalent to a negative answer to Tsirelson's problem from quantum information. As I understand it, this is a question about the equivalence of two ways of defining entanglement, one in the Hilbert space formalism of quantum mechanics, the other in the operator algebraic formalism (so we don't necessarily have a natural tensor product structure). Naturally these approaches are equivalent for finite-dimensional systems, but $$\mathrm{MIP}^{*} = \mathrm{RE}$$ implies that, surprisingly, for infinite-dimensional systems this is not always the case. It also implies a negative answer to the Connes embedding problem from the theory of von Neumann algebras, but I do not know enough to say more than this.

### Acronyms

$$\mathrm{MIP}^{*}$$ : the class of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement.

$$\mathrm{RE}$$ : the class of recursively enumerable languages.

# Question

At a discussion group I attended about this paper, we hit upon the question of whether this has any implications for physics "in practice", or whether the systems for which the Tsirelson bound is inequivalent between the two approaches are somehow "edge cases". Certainly infinite-dimensional systems frequently pop up in physics. Prosaically, a bosonic system has an infinite-dimensional Hilbert space. As a more non-trivial example, in quantum field theory one often approximates a continuum theory with a lattice theory, and then eventually takes the lattice spacing to zero. Does $$\mathrm{MIP}^{*} = \mathrm{RE}$$ imply that this approach is not always possible? I am not necessarily interested in an answer to that specific question, but more generally whether the counterexamples to the Tsirelson problem in this paper should be considered "physical" (for a suitable definition of "physical").