不一致线性约束系统有限维量子解存在性的完整分类
A complete classification of the existence of finite-dimensional quantum solutions to inconsistent linear constraint systems
浏览论文内容
中文总结 AI 辅助
该论文完整分类了不一致线性约束系统的有限维量子解存在性，证明其存在当且仅当维度至少为3且模数与维度不互素，并构造了显式例子。
中文摘要 AI 辅助
线性约束系统（LCS）为非局域博弈和与态无关的上下文性提供了一种紧凑的代数语言。在二元情形下，Mermin-Peres 方格表明，不一致的 LCS 可以通过将变量提升为局域泡利算符来求解，而一系列否定性结果排除了在奇素数模数下基于泡利、克利福德和单项式酉的类似构造。我们首先隔离了这种差异的原因：对于张量积量子解，LCS 在 $\mathbb{Z}_n$ 上经典解的障碍分解为由各个因子携带的障碍类。因此，真正的全局障碍不能仅通过张量局部无阻碍系统而产生。该论证特定于奇模数；在偶模数下，重排序相位不必消失，如基于泡利的例子所示。鉴于这一区别，我们转而将 LCS 与有限维向量空间中的秩一投影算符和恒等分解的有限排列相关联。此类 LCS 的量子可解性是自动的，而寻找量子-经典间隙完全依赖于证明该排列的底层模关联问题的经典不可满足性。通过将此类 LCS 与群值框架函数相关联，我们在 [Harding, Jager, and Smith, Int. J. Theor. Phys. 44, 539 (2005)] 的工作中识别出一族例子。为了确定在所有维度 $d$ 和所有 $n\in\mathbb{N}$ 上，是否存在具有量子解的 $\mathbb{Z}_n$ 上经典不可满足的 LCS，我们进一步为 $d=n$ 为素数的情况构造了显式例子。因此，我们的主要结果肯定地解决了二元情形之外此类 LCS 的存在性问题：此类系统存在当且仅当 $d\geq 3$ 且 $\gcd(n,d)>1$。此外，我们证明了在有限域上存在具有量子-经典间隙的 LCS 当且仅当 $d\geq 3$ 且 $p\mid d$。
英文摘要
Linear constraint systems (LCS) provide a compact algebraic language for nonlocal games and state-independent contextuality. In the binary case the Mermin--Peres square shows that inconsistent LCS can be solved by promoting variables to local Pauli operators, whereas a series of no-go results excludes analogous Pauli-, Clifford- and monomial unitary-based constructions at odd prime modulus. We first isolate the reason for this difference: for tensor-product quantum solutions, the obstruction to a classical solution of a LCS over $\mathbb{Z}_n$ decomposes into obstruction classes carried by the individual factors. As a consequence, a genuinely global obstruction cannot arise solely by tensoring locally unobstructed systems. The argument is specific to odd modulus; at even modulus the reordering phase need not vanish, as in Pauli-based examples. Given this distinction, we instead associate LCS to finite arrangements of rank-one projectors and resolutions of the identity in finite-dimensional vector spaces. Quantum solvability for such LCS is automatic and the search for a quantum-classical gap rests entirely on proving classical unsatisfiability of the underlying modular incidence problem of the arrangement. By relating such LCS with group-valued frame functions, we identify a family of examples in the work of [Harding, Jager, and Smith, Int. J. Theor. Phys. 44, 539 (2005)]. To establish the (non)existence of classically unsatisfiable LCS over $\mathbb{Z}_n$ with quantum solutions in all dimensions $d$ and for all $n\in\mathbb{N}$, we further construct explicit examples for the cases $d=n$ prime. Our main result thus positively resolves the existence problem of such LCS beyond the binary case: such systems exist if and only if $d\geq 3$ and $\gcd(n,d)>1$. Moreover, we establish the existence of LCS with a quantum-classical gap over finite fields if and only if $d\geq 3$ and $p\mid d$.
发表机构
- Leibniz Universität Hannover（汉诺威莱布尼茨大学）
机构由 AI 辅助整理，请以论文原文为准。
