Symmetrized Block-Product Periodic Marginals in Infinite Translation-Invariant Quantum Chains

• URL: http://arxiv.org/abs/2608.10365v1
• Date: Tue, 11 Aug 2026 01:45:25 GMT
• Title: Symmetrized Block-Product Periodic Marginals in Infinite Translation-Invariant Quantum Chains
• Abstract summary: We study local marginals in one-dimensional translation-invariant quantum systems that may hide finite-period structure.<n>This defines a symmetrized block-product periodic marginal problem.<n>We benchmark the framework on the Majumdar-Ghosh model, transverse-field Ising, XX, XXZ, and contextuality-related spin models.
• Score: 1.1960668185846597
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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