Zhou Peng-Fei, Lu Ying, Wang Jia-Hao, Ran Shi-Ju
Center for Quantum Physics and Intelligent Sciences, Department of Physics, Capital Normal University, Beijing 10048, China.
Phys Rev Lett. 2023 Jul 14;131(2):020403. doi: 10.1103/PhysRevLett.131.020403.
Efficient methods to access the entanglement of a quantum many-body state, where the complexity generally scales exponentially with the system size N, have long been a concern. Here we propose the Schmidt tensor network state (Schmidt TNS) that efficiently represents the Schmidt decomposition of finite- and even infinite-size quantum states with nontrivial bipartition boundary. The key idea is to represent the Schmidt coefficients (i.e., entanglement spectrum) and transformations in the decomposition to tensor networks (TNs) with linearly scaled complexity versus N. Specifically, the transformations are written as the TNs formed by local unitary tensors, and the Schmidt coefficients are encoded in a positive-definite matrix product state (MPS). Translational invariance can be imposed on the TNs and MPS for the infinite-size cases. The validity of Schmidt TNS is demonstrated by simulating the ground state of the quasi-one-dimensional spin model with geometrical frustration. Our results show that the MPS encoding the Schmidt coefficients is weakly entangled even when the entanglement entropy of the decomposed state is strong. This justifies the efficiency of using MPS to encode the Schmidt coefficients, and promises an exponential speedup on the full-state sampling tasks.
长期以来，如何高效地获取量子多体状态的纠缠一直备受关注，因为其复杂度通常随系统规模N呈指数增长。在此，我们提出了施密特张量网络态（Schmidt TNS），它能有效地表示具有非平凡二分边界的有限甚至无限规模量子态的施密特分解。关键思想是将施密特系数（即纠缠谱）及其在分解中的变换表示为复杂度随N呈线性增长的张量网络（TNs）。具体而言，变换被写成由局部酉张量构成的TNs，施密特系数则编码在正定矩阵乘积态（MPS）中。对于无限规模的情况，可对TNs和MPS施加平移不变性。通过模拟具有几何阻挫的准一维自旋模型的基态，验证了施密特TNS的有效性。我们的结果表明，即使分解态的纠缠熵很强，编码施密特系数的MPS的纠缠程度也很弱。这证明了使用MPS编码施密特系数的效率，并有望在全态采样任务上实现指数级加速。
