The ‘operator entanglement’ of a quantum operator O is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global U(1) conservation law, and on operators O with a well-defined U(1) charge, for which it is possible to resolve the operator entanglement of O according to the U(1) symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in Rath et al (2023 PRX Quantum 4 010318) and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix ρβ=e−βH and of charged local operators evolving in Heisenberg picture O=eitHOe−itH . Our main results are: i) the SROE of ρβ obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.

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有关对称性解决算子纠缠的更多信息

量子算子的“算子纠缠”氧是其复杂性的有用指标，并且在一维中，是矩阵乘积运算符的近似性的有用指标。在这里，我们专注于具有全球影响力的旋转链U(1) 守恒定律和算子氧具有明确的U(1)收费，可以解决运营商的纠结氧根据U(1)对称性。我们采用 Rath 中引入的对称解析算子纠缠 (SROE) 的概念等人(2023PRX量子 4010318）并将后一篇论文的结果扩展到几个方向。结合共形场理论以及临界自由费米子链中的精确分析和数值计算，我们研究了热密度矩阵的 SROE ρβ=e-βH 以及海森堡图中不断演变的收费本地算子 氧=e我tH氧e-我tH 。我们的主要结果是： i) 的 SROE ρβ 遵守运营商区域法； ii) 对于自由费米子，海森堡图中的局部算子可以具有随时间对数增长或饱和到恒定值的 SROE； iii) 除了一对费米子产生和湮灭算子之外，所有电荷扇区之间的纠缠是均分的。