Matrix product states and projected entangled pair states (PEPS) are powerful analytical and numerical tools to assess quantum many-body systems in one and higher dimensions, respectively. While matrix product states are comprehensively understood, in PEPS fundamental questions, relevant analytically as well as numerically, remain open, such as how to encode symmetries in full generality, or how to stabilize numerical methods using canonical forms. Here, we show that these key problems, as well as a number of related questions, are algorithmically undecidable, that is, they cannot be fully resolved in a systematic way. Our work thereby exposes fundamental limitations to a full and unbiased understanding of quantum many-body systems using PEPS.

中文翻译：

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预测的纠缠对状态：基本的分析和数值限制

矩阵乘积状态和投影纠缠对状态（PEPS）是强大的分析和数值工具，可以分别评估一维和更高维的量子多体系统。虽然对矩阵乘积的状态有全面的了解，但在PEPS的基本问题中，无论是在分析上还是在数值上，仍然是悬而未决的，例如如何完全通用地编码对称性，或如何使用规范形式来稳定数值方法。在这里，我们表明这些关键问题以及许多相关问题在算法上是无法确定的，也就是说，它们无法以系统的方式完全解决。因此，我们的工作暴露了对使用PEPS的量子多体系统的全面而公正的理解的基本限制。