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

In one-dimensional quantum systems, local reduced density matrices (RDMs) often hide complex, finite-period structures that are not immediately apparent from one-site translation invariance. The authors investigate whether a given local RDM can be explained by a state formed by repeating a finite $p$-site block and averaging over all possible lattice translations. This problem is central to understanding symmetry-broken phases and calculating ground-state energy densities in infinite systems.

The paper develops two complementary methods to address this marginal compatibility problem:

1. SDP Relaxation: The authors construct a semidefinite programming hierarchy based on block permutation symmetry and positive partial transpose (PPT) constraints. This provides a rigorous way to test if a local RDM is incompatible with a specific periodic structure. If the SDP is infeasible, the periodic explanation is ruled out.
2. Symmetrized MPS Ansatz: For larger systems, the authors use a variational matrix product state (MPS) approach. By representing the $p$-site unit cell as an MPS and optimizing it to match the target local data, they generate explicit variational upper bounds for ground-state energy densities. This method is more scalable than the SDP approach and allows for systematic improvement by increasing the bond dimension and period.

The framework successfully recovers expected periodic structures in exactly solvable models, such as the Majumdar-Ghosh chain, where it identifies the dimerized ground state. In more general spin models, the symmetrized MPS method provides systematically improving energy bounds. Furthermore, the authors adapt their approach into a periodic-NPA (Navascués-Pironio-Acín) relaxation to analyze translation-invariant contextuality witnesses, successfully reproducing known quantum limits. This work bridges the gap between local diagnostic tools and global variational methods, providing a unified language for studying symmetry breaking and correlation bounds in infinite quantum chains.