存在单位噪声时 MIP* 的二分类定理
A Dichotomy for MIP* in the Presence of Unital Noise
- Nanjing University（南京大学）
- Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
- Hefei National Laboratory（合肥国家实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究单位噪声下量子双证明者证明系统 MIP* 的复杂度分类，证明当共享态可分离或量子最大相关性小于1时等价于NEXP，当纠缠且相关性为1时等价于RE，并引入保正低度近似与泡利折叠两种新技术。
AI中文摘要：
最近的一项工作研究了量子双证明者单轮证明系统 $\mathsf{MIP}^{\psi}[\text{poly},O(1)]$。该工作表明，当 $\psi$ 是具有量子最大相关性 $\rho_{\max}(\psi)<1$ 的含噪 EPR 态时，所得复杂度类等价于 $\mathsf{NEXP}=\mathsf{MIP}$。相比之下，$\mathsf{MIP}^{\text{EPR}}[\text{poly},O(1)]=\mathsf{RE}$。然而，并非所有噪声都会降低量子最大相关性。一个基本例子是退相噪声：退相后的 EPR 对仍保持量子最大相关性为 1。在本工作中，我们移除了常数答案数限制，并为 $\mathsf{MIP}^{\psi}[\text{poly},\text{poly}]$ 建立了完整的二分类定理，其中证明者共享任意多份受单位噪声影响的 EPR 对。等价地，设 $\psi$ 是一个具有最大混合边缘的固定两量子比特态。我们证明：若 $\psi$ 是可分离的或 $\rho_{\max}(\psi)<1$，则 $\mathsf{MIP}^{\psi}[\text{poly},\text{poly}]=\mathsf{MIP}=\mathsf{NEXP}$；若 $\psi$ 是纠缠的且 $\rho_{\max}(\psi)=1$，则 $\mathsf{MIP}^{\psi}[\text{poly},\text{poly}]=\mathsf{MIP}^{\psi}[\text{poly},O(1)]=\mathsf{RE}$。除了复杂度分类之外，我们的证明还发展了两种可能在量子信息和量子复杂度领域更广泛有用的技术。首先，我们引入了一种用于量子测量的保正低度近似框架：不是直接在泡利基中截断正算子，而是近似合适的平方根分解，同时实现低泡利度、正性以及对整个 POVM 的全局控制。其次，我们为低度正算子开发了一种新的降维方法，称为泡利折叠。其关键成分是随机化泡利哈希的几乎乘法性定理，该定理允许非交换乘积、正性和归一化在受控误差下经受压缩。
英文摘要：
A recent work studied quantum two-prover one-round proof systems $\mathsf{MIP}^ψ[\text{poly},O(1)]$. 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 $\mathsf{NEXP}=\mathsf{MIP}$. In contrast, $\mathsf{MIP}^{\text{EPR}}[\text{poly},O(1)]=\mathsf{RE}$. 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 $\mathsf{MIP}^ψ[\text{poly},\text{poly}]$, 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 \[\mathsf{MIP}^ψ[\text{poly},\text{poly}]=\mathsf{MIP} =\mathsf{NEXP},~\text{if $ψ$ is separable or $ρ_{\max}(ψ)<1$};\]\[\mathsf{MIP}^ψ[\text{poly},\text{poly}]=\mathsf{MIP}^ψ[\text{poly},O(1)]=\mathsf{RE},\text{if $ψ$ is entangled and $ρ_{\max}(ψ)=1$.}\] 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.
