Approximate traces on groups and the quantum complexity class $\operatorname{MIP}^{co,s}$

TL;DR

This paper investigates the relationship between quantum interactive proof classes and recursively enumerable languages, introducing a new concept called qc-modulus to analyze quantum correlations and their approximations.

Contribution

It introduces the concept of a qc-modulus and demonstrates that its computability implies a negative answer to a key open question in quantum complexity theory.

Findings

Existence of a computable qc-modulus implies $ ext{MIP}^{co}$ does not equal $coRE$

Provides a new approach to approximate quantum commuting correlations

Offers insights into the structure of quantum complexity classes

Abstract

An open question in quantum complexity theory is whether or not the class $\operatorname{MIP}^{co}$, consisting of languages that can be efficiently verified using interacting provers sharing quantum resources according to the quantum commuting model, coincides with the class $coRE$ of languages with recursively enumerable complement. We introduce the notion of a qc-modulus, which encodes approximations to quantum commuting correlations, and show that the existence of a computable qc-modulus gives a negative answer to a natural variant of the aforementioned question.