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arXiv:2608.10365

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

Xiao Zeng, Kaiyan Yang, Lingxia Zhang, Zizhu Wang

quant-phcond-mat.str-elSubmitted Aug 11, 2026• Announced Aug 12, 2026• 0 citations
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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.

Comments
13 pages, 3 figures

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We study local marginals in one-dimensional translation-invariant quantum systems that may hide finite-period structure. Given an nnn-site reduced density matrix, we ask whether it can be obtained by repeating a finite ppp-site block state along the chain and averaging over the ppp 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.
Bell- and contextuality-type inequalities have become practical probes of many-body quantum correlations, often involving only few-body correlators and quantities with a direct Hamiltonian interpretation such as an energy density. Here we investigate the mechanism by which translation-invariant Hamiltonians generated from representative families of contextuality witnesses organize their ground-state structure in infinite one-dimensional systems. For the witness families considered, maximal quantum violation is realized by ground-state sectors with commensurate enlarged unit cells: the Hamiltonians are invariant under one-site translations, while the optimal ground states are ppp-periodic with p>1p>1p>1. At the corresponding classical-bound points, the ground-state sectors are highly degenerate and support many commensurate periods. Along the interpolation paths analyzed in this work, entering the contextual regime is accompanied by the lifting of this classical period degeneracy in favor of a quantum-selected period. We also identify finite periodic-boundary-condition benchmarks at the selected periods: for each model studied, the finite-ring witness reproduces the same classical bound and quantum value as the corresponding infinite-chain witness, and in several cases the resulting finite inequalities are tight. These reductions turn the infinite-chain contextuality certification into compact energy-estimation benchmarks requiring only local correlator measurements. We establish the mechanism analytically in representative two- and three-body witness models and corroborate it more broadly using translation-invariant semidefinite-program relaxations together with variational matrix-product-state calculations.
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Abstract In this paper, we present a method to solve the quantum marginal problem for symmetric d -level systems. The method is built upon an efficient semi-definite program that uses the compatibility conditions of an m -body reduced density with a global n -body density matrix supported on the symmetric space. We illustrate the applicability of the method in central quantum information problems with several exemplary case studies. Namely, (i) a fast variational ansatz to optimize local Hamiltonians over symmetric states, (ii) a method to optimize symmetric, few-body Bell operators over symmetric states and (iii) a set of sufficient conditions to determine which symmetric states cannot be self-tested from few-body observables. As a by-product of our findings, we also provide a generic, analytical correspondence between arbitrary superpositions of n -qubit Dicke states and translationally-invariant diagonal matrix product states of bond dimension n .
Matrix product operators allow efficient descriptions (or realizations) of states on a 1D lattice. We consider the task of learning a realization of minimal dimension from copies of an unknown state, such that the resulting operator is close to the density matrix in trace norm. For finitely correlated translation-invariant states on an infinite chain, a realization of minimal dimension can be exactly reconstructed via linear algebra operations from the marginals of a size depending on the representation dimension. We establish a bound on the trace norm error for an algorithm that estimates a candidate realization from estimates of these marginals and outputs a matrix product operator, estimating the state of a chain of arbitrary length ttt. This bound allows us to establish an O(t2)O(t^2)O(t2) upper bound on the sample complexity of the learning task, with an explicit dependence on the site dimension, realization dimension and spectral properties of a certain map constructed from the state. A refined error bound can be proven for C∗C^*C∗-finitely correlated states, which have an operational interpretation in terms of sequential quantum channels applied to the memory system. We can also obtain an analogous error bound for a class of matrix product density operators on a finite chain reconstructible by local marginals. In this case, a linear number of marginals must be estimated, obtaining a sample complexity of O~(t3)\tilde{O}(t^3)O~(t3). The learning algorithm also works for states that are sufficiently close to a finitely correlated state, with the potential of providing competitive algorithms for other interesting families of states.
We study the properties of the set of marginal distributions of infinite translation-invariant systems in the two-dimensional square lattice. In cases where the local variables can only take a small number d of possible values, we completely solve the marginal or membership problem for nearest-neighbours distributions ( d = 2, 3) and nearest and next-to-nearest neighbours distributions ( d = 2). Remarkably, all these sets form convex polytopes in probability space. This allows us to devise an algorithm to compute the minimum energy per site of any TI Hamiltonian in these scenarios exactly. We also devise a simple algorithm to approximate the minimum energy per site up to arbitrary accuracy for the cases not covered above. For variables of a higher (but finite) dimensionality, we prove two no-go results. To begin, the exact computation of the energy per site of arbitrary TI Hamiltonians with only nearest-neighbour interactions is an undecidable problem. In addition, in scenarios with d ≥2947, the boundary of the set of nearest-neighbour marginal distributions contains both flat and smoothly curved surfaces and the set itself is not semi-algebraic. This implies, in particular, that it cannot be characterized via semidefinite programming, even if we allow the input of the programme to include polynomials of nearest-neighbour probabilities.
Based on the matrix product state (MPS) formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing density-matrix renormalization group or MPS ground-state code. Although this approach is built on open-boundary MPSs, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.