A Dichotomy for MIP* in the Presence of Unital Noise

Abstract

A recent work studied quantum two-prover one-round proof systems MIP^ψ[poly,O(1)]. It showed that when ψ is a noisy EPR state with quantum maximal correlation ρ_max(ψ)<1, the resulting complexity class is equivalent to NEXP=MIP. In contrast, MIP^EPR[poly,O(1)]=RE. 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 MIP^ψ[poly,poly], where the provers share arbitrarily many copies of EPR pairs subject to unital noise. Equivalently, let ψ be a fixed two-qubit state with maximally mixed marginals. We prove that MIP^ψ[poly,poly]=MIP =NEXP, if $ψ$ is separable or $ρ_max(ψ)<1$;MIP^ψ[poly,poly]=MIP^ψ[poly,O(1)]=RE,if $ψ$ is entangled and $ρ_max(ψ)=1$. 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.

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BibTeX RIS

BibTeX

@misc{dong2026dichotomy,
  title = {{A Dichotomy for MIP* in the Presence of Unital Noise}},
  author = {Yangjing Dong and Zhenyu Jiang and Minglong Qin and Penghui Yao and Haigang Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39318},
  eprint = {2609.39318},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39318},
}

RIS

TY  - UNPB
TI  - A Dichotomy for MIP* in the Presence of Unital Noise
AU  - Yangjing Dong
AU  - Zhenyu Jiang
AU  - Minglong Qin
AU  - Penghui Yao
AU  - Haigang Zhou
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39318
UR  - https://arxiv.org/abs/2609.39318
AB  - A recent work studied quantum two-prover one-round proof systems MIP^ψ[poly,O(1)]. It showed that when ψ is a noisy EPR state with quantum maximal correlation ρ_max(ψ)<1, the resulting complexity class is equivalent to NEXP=MIP. In contrast, MIP^EPR[poly,O(1)]=RE. 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 MIP^ψ[poly,poly], where the provers share arbitrarily many copies of EPR pairs subject to unital noise. Equivalently, let ψ be a fixed two-qubit state with maximally mixed marginals. We prove that MIP^ψ[poly,poly]=MIP =NEXP, if $ψ$ is separable or $ρ_max(ψ)<1$;MIP^ψ[poly,poly]=MIP^ψ[poly,O(1)]=RE,if $ψ$ is entangled and $ρ_max(ψ)=1$. 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.
ER  -

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