发表机构
The University of Texas at Dallas; National Research Council Canada; University of Waterloo; Iowa State University（德克萨斯大学达拉斯分校; 加拿大国家研究委员会; 滑铁卢大学; 爱荷华州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Pauli错误检测中Knill-Laflamme系数的几何结构，证明表示重数足以实现精确标量压缩，并统一了交换Pauli族与子系统稳定子码的系数范围性质。
AI 中文摘要
Knill-Laflamme系数通过将错误算子标量压缩到码空间来表征精确的量子错误检测。对于给定的Pauli可观测量和固定的码维度，可达到的系数向量构成一个联合高阶数值范围，其欧几里得范数图像即为特征谱。我们研究算子表示、码维度和检测约束如何支配这一几何结构。我们的主要定理将表示重数识别为足以将活动期望数据提升为精确标量压缩的充分条件：当每个占据块的重数至少为码维度乘以相应活动态的秩时，提升即存在。当重数足以实现所有活动态时，系数范围即为完整活动期望体，因此是凸且连通的，其特征谱为闭区间。该框架统一了交换Pauli族（其中足够的公共本征空间简并产生多面体）和子系统稳定子码（其中受保护的逻辑子系统为非交换规范可观测量提供重数空间）。具体示例表明，增加码维度可使范围收缩、坍缩或变为空集，而增加重数可将Bloch球变为实心球。在子系统设置中，母哈密顿量构造将精确检测码实现为简并基态空间，在这些范围内选择特定向量和连续路径。最后的三量子比特示例表明，额外的非交换检测约束将四面体范围缩减为其重心和四个顶点，产生不连通的特征谱。这些结果为Knill-Laflamme系数几何提供了结构性框架，并激发在结构化错误模型下对连通系数范围和区间特征谱的更精确判据。
英文摘要
Knill-Laflamme coefficients characterize exact quantum error detection through scalar compressions of error operators to the code space. For prescribed Pauli observables and a fixed code dimension, attainable coefficient vectors form a joint higher-rank numerical range, whose Euclidean norm image is the signature spectrum. We study how operator representation, code dimension, and detection constraints govern this geometry. Our main theorem identifies representation multiplicity as sufficient to lift active expectation data to exact scalar compressions: a lift exists whenever each occupied block has multiplicity at least the code dimension times the rank of the corresponding active state. When multiplicity suffices to realize all active states, the coefficient range is the full active expectation body and is therefore convex and connected, with a closed interval as its signature spectrum. This framework unifies commuting Pauli families, where sufficient common-eigenspace degeneracy yields polytopes, and subsystem stabilizer codes, where protected logical subsystems supply multiplicity spaces for noncommuting gauge observables. Explicit examples show how increasing the code dimension can make a range shrink, collapse, or become empty, while increasing multiplicity can turn a Bloch sphere into a filled ball. In the subsystem setting, parent-Hamiltonian constructions realize exact detecting codes as degenerate ground spaces, selecting distinguished vectors and continuous paths within these ranges. A final three-qubit example shows additional noncommuting detection constraints reduce a tetrahedral range to its barycenter and four vertices, producing a disconnected signature spectrum. The results provide a structural framework for Knill-Laflamme coefficient geometry and motivate sharper criteria for connected coefficient ranges and interval signature spectra under structured error models.
Comments26 pages, 5 figures
