AI 中文总结
本研究针对矩阵乘积局域可纯化密度算子（LPDOs）的基本定理展开研究，聚焦序贯生成的LPDOs，证明特定条件下其表示的等价关系，同时给出周期性边界条件下LPDOs一般基本定理的阻碍反例，并探讨对混合态对称性保护拓扑相的启示。
AI 中文摘要
张量网络方法是表征量子物质相的强大分析与数值工具。尽管矩阵乘积态（MPS）的数学结构已通过MPS基本定理得到充分理解，但混合态张量网络的类似理解仍基本缺失：若两个纯化张量生成相同的密度矩阵，它们之间存在何种关联？本研究启动了矩阵乘积局域可纯化密度算子（LPDOs）基本定理的研究，聚焦于序贯生成的LPDOs（sLPDOs）——这是一类可通过初始态上量子通道的连续应用来解释的广泛子类。我们证明，在合适的可逆性或循环条件下，对于任意系统尺寸，两个sLPDO表示生成相同密度矩阵当且仅当它们通过作用于纯化键上的矩阵乘积等距变换相关联。超出sLPDO框架，我们提供了一个反例，表明具有周期性边界条件的LPDOs的一般基本定理存在阻碍。最后，我们讨论了其对混合态对称性保护拓扑相的启示，包括仅由弱对称条件保护的非平凡相的可能性。
英文摘要
Tensor network methods provide powerful analytical and numerical tools for characterizing quantum phases of matter. While the mathematical structure of matrix product states (MPS) is well understood through the MPS fundamental theorem, an analogous understanding for mixed-state tensor networks remains largely absent: if two purification tensors generate the same density matrix, how are they related? In this work, we initiate the study of a fundamental theorem for matrix product locally purifiable density operators (LPDOs) and focus on sequentially generated LPDOs (sLPDOs), a broad subclass admitting an interpretation in terms of successive applications of quantum channels on an initial state. We prove that, under suitable invertibility or cyclic conditions, two sLPDO representations generate the same density matrix for arbitrary system sizes if and only if they are related by a matrix product isometry acting on the purification bonds. Beyond the sLPDO setting, we provide a counterexample that suggests an obstruction to a general fundamental theorem for LPDOs with periodic boundary conditions. Finally, we discuss implications for mixed-state symmetry-protected topological phases, including the possibility of nontrivial phases protected only by weak symmetry conditions.
Comments12+5 pages
