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arXiv 2405.07970quant-phcond-mat.str-el
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拓扑编码与具有反常对称性的混合态需要多少纠缠?

How Much Entanglement Is Needed for Topological Codes and Mixed States with Anomalous Symmetry?

Zhi Li, Dongjin Lee, Beni Yoshida

AI总结:

该研究利用几何纠缠度量(GEM)定量表征拓扑编码与反常对称系统中涌现粒子所需的纠缠,证明二维稳定子编码GEM与码距二次相关、涌现任意子/费米子系统GEM随系统尺寸线性标度,并将结果推广至混合态,建立量子反常与长程纠缠的关联。

摘要:

众所周知,由于长程量子纠缠,具有奇异性质的粒子可以在由量子比特等简单组分构成的系统中涌现。在本文中,我们使用几何纠缠度量(GEM,用于给定量子态与任意短程纠缠态之间的最大重叠进行量化),对涌现任意子和费米子所需的纠缠给出了定量表征。对于存在涌现任意子的系统,基于编织统计,我们证明无论微观细节如何,GEM都随系统尺寸线性标度。涌现任意子现象也可以在量子纠错(QEC)框架下理解。具体而言,我们证明任意二维稳定子编码的GEM至少与编码距离呈二次方关系。我们的证明基于构造弦算符的通用方案,在涌现任意子与量子纠错之间建立了严格且直接的联系。对于存在涌现费米子的系统,尽管基态子空间可能呈指数级庞大,且其编码特性可能相当差,我们证明GEM同样随系统尺寸线性标度。我们的分析建立了量子反常与纠缠之间的有趣关联:满足反常1-形式对称性的量子态,必然是长程纠缠的,且具有较大的GEM。我们的结果还可推广到混合态,例如ZX相位阻尼环面编码,提供了一类可证明非平凡的内禀混合态相。

英文摘要:

It is known that particles with exotic properties can emerge in systems made of simple constituents such as qubits, due to long-range quantum entanglement. In this paper, we provide quantitative characterizations of entanglement necessary for emergent anyons and fermions by using the geometric entanglement measure (GEM), which quantifies the maximal overlap between a given state and any short-range-entangled states. For systems with emergent anyons, based on the braiding statistics, we show that the GEM scales linearly in the system size regardless of microscopic details. The phenomenon of emergent anyons can also be understood within the framework of quantum error correction (QEC). Specifically, we show that the GEM of any 2D stabilizer codes must be at least quadratic in the code distance. Our proof is based on a generic prescription for constructing string operators, establishing a rigorous and direct connection between emergent anyons and QEC. For systems with emergent fermions, despite that the ground state subspaces could be exponentially huge and their coding properties could be rather poor, we show that the GEM also scales linearly in the system size. Our analysis establishes an intriguing link between quantum anomaly and entanglement: A quantum state respecting anomalous 1-form symmetries, must be long-range-entangled and have large GEM. Our results also extend to mixed states, such as the $ZX$-dephased toric code, providing a provably nontrivial class of intrinsically mixed-state phases.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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