AI 中文总结
该研究针对无限平移不变量子链提出对称化块积周期边际方法，结合半定规划松弛与矩阵乘积态近似，可诊断周期序、给出基态能量上界，在多自旋模型测试中验证了其有效性。
AI 中文摘要
我们研究一维平移不变量子系统中的局域边际，这类系统可能隐藏有限周期结构。给定一个n位点约化密度矩阵，我们询问它是否可通过沿链重复有限p位点块态并对p个晶格平移取平均得到，这定义了一个对称化块积周期边际问题，它既为从局域数据诊断隐藏周期序提供了途径，也为无限平移不变局域哈密顿量的基态能量密度提供了上界。我们提出两种互补方法：第一种是基于块置换对称性和部分转置正性约束的半定规划松弛，它外近似这类边际的凸包并产生可验证的不可行性测试；第二种是对称化矩阵乘积态（MPS）近似，它构造显式块积周期态并给出变分上界。我们在Majumdar-Ghosh模型、横场伊辛模型、XX模型、XXZ模型及与上下文相关性相关的自旋模型上对该框架进行基准测试。结果表明，该方法在可精确求解的案例中捕获了预期的有限周期结构，且随着周期和键维度的增加，变分能量系统地改善。我们还为平移不变上下文相关性见证者构建了周期-NPA松弛，并在测试案例中恢复了已知的量子极限。
英文摘要
We study local marginals in one-dimensional translation-invariant quantum systems that may hide finite-period structure. Given an $n$-site reduced density matrix, we ask whether it can be obtained by repeating a finite $p$-site block state along the chain and averaging over the $p$ lattice translations. This defines a symmetrized block-product periodic marginal problem, which provides a route both to diagnosing hidden periodic order from local data and to upper bounding ground-state energy densities of infinite translation-invariant local Hamiltonians. We develop two complementary methods. The first is a semidefinite-programming relaxation based on block permutation symmetry and positive partial transpose constraints, which outer-approximates the convex hull of such marginals and yields certified infeasibility tests. The second is a symmetrized matrix product state ansatz, which constructs explicit block-product periodic states and gives variational upper bounds. We benchmark the framework on the Majumdar-Ghosh model, transverse-field Ising, XX, XXZ, and contextuality-related spin models. The results show that the method captures the expected finite-period structure in exactly solvable cases and gives systematically improving variational energies as the period and bond dimension increase. We also formulate a periodic-NPA relaxation for translation-invariant contextuality witnesses and recover the known quantum limits in the tested examples.
Comments13 pages, 3 figures
