An area law for 2d frustration-free spin systems
An area law for 2d frustration-free spin systems
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DOI：
10.1145/3519935.3519962
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Anurag Anshu;I. Arad;David Gosset
中科院分区：
文献类型：
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作者：
Anurag Anshu;I. Arad;David Gosset
我们证明了局部有隙无挫败二维晶格自旋系统的基态纠缠熵满足关于晶格垂直二分到左右区域的面积定律。我们首先确定任何局部有隙无挫败一维自旋系统的基态投影仪都可以在哈密顿量相互作用项中以 O(√nlog(є−1)) 度多元多项式逼近到误差 є 内。这概括了布尔 AND 函数近似度的最佳界限，对应于交换哈密顿项的特殊情况。对于 2D 自旋系统，我们构建一个近似基态投影仪 (AGSP)，它在感兴趣的二分边界附近采用最佳一维近似。该 AGSP 具有足够低的纠缠和误差，可以使用已知技术建立面积定律。
We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error є by a degree O(√nlog(є−1)) multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special case of commuting Hamiltonian terms. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation in the vicinity of the boundary of the bipartition of interest. This AGSP has sufficiently low entanglement and error to establish the area law using a known technique.