Bond dimension witnesses and the structure of homogeneous matrix product states

Miguel Navascues1,2 and Tamas Vertesi3

1Department of Physics, Bilkent University, Ankara 06800, Turkey
2Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmangasse 3, 1090 Vienna, Austria
3Institute for Nuclear Research, Hungarian Academy of Sciences, H-4001 Debrecen, P.O. Box 51, Hungary

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Abstract

For the past twenty years, Matrix Product States (MPS) have been widely used in solid state physics to approximate the ground state of one-dimensional spin chains. In this paper, we study homogeneous MPS (hMPS), or MPS constructed via site-independent tensors and a boundary condition. Exploiting a connection with the theory of matrix algebras, we derive two structural properties shared by all hMPS, namely: a) there exist local operators which annihilate all hMPS of a given bond dimension; and b) there exist local operators which, when applied over any hMPS of a given bond dimension, decouple (cut) the particles where they act from the spin chain while at the same time join (glue) the two loose ends back again into a hMPS. Armed with these tools, we show how to systematically derive `bond dimension witnesses', or 2-local operators whose expectation value allows us to lower bound the bond dimension of the underlying hMPS. We extend some of these results to the ansatz of Projected Entangled Pairs States (PEPS). As a bonus, we use our insight on the structure of hMPS to: a) derive some theoretical limitations on the use of hMPS and hPEPS for ground state energy computations; b) show how to decrease the complexity and boost the speed of convergence of the semidefinite programming hierarchies described in [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of finite-dimensional quantum correlations.

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[3] Felix Huber, "Positive maps and trace polynomials from the symmetric group", Journal of Mathematical Physics 62 2, 022203 (2021).

[4] Nick G. Jones, Julian Bibo, Bernhard Jobst, Frank Pollmann, Adam Smith, and Ruben Verresen, "Skeleton of matrix-product-state-solvable models connecting topological phases of matter", Physical Review Research 3 3, 033265 (2021).

[5] Raffaele Salvia and Vittorio Giovannetti, "Extracting work from correlated many-body quantum systems", Physical Review A 105 1, 012414 (2022).

[6] Miguel Navascués, Adrien Feix, Mateus Araújo, and Tamás Vértesi, "Characterizing finite-dimensional quantum behavior", Physical Review A 92 4, 042117 (2015).

[7] Claudia De Lazzari, Harshit J Motwani, and Tim Seynnaeve, "The Linear Span of Uniform Matrix Product States", arXiv:2204.10363, (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-28 15:38:10) and SAO/NASA ADS (last updated successfully 2024-03-28 15:38:11). The list may be incomplete as not all publishers provide suitable and complete citation data.