量子混合态与量子纠错中的复本拓扑序
Replica topological order in quantum mixed states and quantum error correction
AI总结：
本文提出了混合态中复本拓扑序的定义，将拓扑序分为量子、经典和平凡三类，并基于 toric code 模型揭示了不同相的纠错能力，通过 PEPS 边界态对称性保护拓扑序与体拓扑的关联解释了信息恢复机制。
摘要：
物质的拓扑相为量子计算和量子纠错提供了一个有前景的平台。然而，与纯态中的对应物不同，混合态中拓扑序的描述仍然相对缺乏探索。我们的工作给出了混合态中复本拓扑序的两个定义，这些定义涉及混合态密度矩阵的 $n$ 个副本。我们的框架将混合态中的拓扑序分类为量子、经典或平凡的，这取决于可以编码的信息类型。对于存在退相干的 toric code 模型的情况，我们为每个相关联一个量子信道，并描述了码空间的结构。我们表明，在量子拓扑相中，存在一种基于后选择的纠错协议可以恢复量子信息，而在经典拓扑相中，量子信息已发生退相干且无法完全恢复。我们通过将混合态描述为投影纠缠对态（PEPS）并将其边界态的对称性保护拓扑序与体拓扑相关联来实现这一点。我们讨论了我们的发现可以外推到 $n \ o 1$ 极限的程度。
英文摘要：
Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively under-explored. Our work give two definitions for replica topological order in mixed states, which involve $n$ copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. We discuss the extent that our findings can be extrapolated to $n \to 1$ limit.
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