A Dichotomy for MIP* in the Presence of Unital Noise

A recent work studied quantum two-prover one-round proof systems ${\mathsf{M}\mathsf{I}\mathsf{P}}^{\psi }[\text{poly},O(1)]$. It showed that when $\psi$ is a noisy EPR state with quantum maximal correlation ${\rho }_{\max }(\psi )<1$, the resulting complexity class is equivalent to $\mathsf{N}\mathsf{E}\mathsf{X}\mathsf{P}=\mathsf{M}\mathsf{I}\mathsf{P}$. In contrast, ${\mathsf{M}\mathsf{I}\mathsf{P}}^{\text{EPR}}[\text{poly},O(1)]=\mathsf{R}\mathsf{E}$. Not all noise, however, decreases quantum maximal correlation. A basic example is dephasing noise: a dephased EPR pair retain quantum maximal correlation $1$. In this work, we remove the constant-answer restriction and establish a complete dichotomy for ${\mathsf{M}\mathsf{I}\mathsf{P}}^{\psi }[\text{poly},\text{poly}]$, where the provers share arbitrarily many copies of EPR pairs subject to unital noise. Equivalently, let $\psi$ be a fixed two-qubit state with maximally mixed marginals. We prove that $${\mathsf{M}\mathsf{I}\mathsf{P}}^{\psi }[\text{poly},\text{poly}]=\mathsf{M}\mathsf{I}\mathsf{P}=\mathsf{N}\mathsf{E}\mathsf{X}\mathsf{P},\text{if }\psi \text{ is separable or }{\rho }_{\max }(\psi )<1;$$$${\mathsf{M}\mathsf{I}\mathsf{P}}^{\psi }[\text{poly},\text{poly}]={\mathsf{M}\mathsf{I}\mathsf{P}}^{\psi }[\text{poly},O(1)]=\mathsf{R}\mathsf{E},\text{if }\psi \text{ is entangled and }{\rho }_{\max }(\psi )=1\text{.}$$ Beyond the complexity classification, our proof develops two techniques that may be useful more broadly in quantum information and quantum complexity. First, we introduce a positivity-preserving low-degree approximation framework for quantum measurements: instead of truncating positive operators directly in the Pauli basis, we approximate suitable square-root factorizations, simultaneously achieving low Pauli degree, positivity, and global control over an entire POVM. Second, we develop a new dimension-reduction method, called Pauli folding, for low-degree positive operators. Its key ingredient is an almost-multiplicativity theorem for randomized Pauli hashing, which allows noncommutative products, positivity, and normalization to survive compression with controlled error.