A Caltech Library Service

Classical restrictions of generic matrix product states are quasi-locally Gibbsian

Aragonés-Soria, Y. and Åberg, J. and Park, C.-Y. and Kastoryano, M. J. (2021) Classical restrictions of generic matrix product states are quasi-locally Gibbsian. Journal of Mathematical Physics, 62 (9). Art. No. 093511. ISSN 0022-2488. doi:10.1063/5.0040256.

[img] PDF - Accepted Version
See Usage Policy.


Use this Persistent URL to link to this item:


We show that norm squared amplitudes with respect to a local orthonormal basis (the classical restriction) of finite quantum systems on one-dimensional lattices can be exponentially well approximated by Gibbs states of local Hamiltonians (i.e., they are quasi-locally Gibbsian) if the classical conditional mutual information (CMI) of any connected tripartition of the lattice is rapidly decaying in the width of the middle region. For injective matrix product states, we, moreover, show that the classical CMI decays exponentially whenever the collection of matrix product operators satisfies a “purity condition,” a notion previously established in the theory of random matrix products. We, furthermore, show that violation of the purity condition enables a generalized notion of error correction on the virtual space, thus indicating the non-generic nature of such violations. We make this intuition more concrete by constructing a probabilistic model where purity is a typical property. The proof of our main result makes extensive use of the theory of random matrix products and may find applications elsewhere.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Aragonés-Soria, Y.0000-0002-8880-8957
Kastoryano, M. J.0000-0001-5233-7957
Additional Information:© 2021 Author(s). Published under an exclusive license by AIP Publishing. Submitted: 11 December 2020 • Accepted: 19 August 2021 • Published Online: 15 September 2021. We thank T. Benoist for clarifying some details in Ref. 20. We thank David Gross for helpful discussions. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1-390534769. This work was completed while M.J.K. was at the University of Cologne. We also thank an anonymous referee for pointing out improvements of Proposition 1 and Lemma 2, which strengthened the results and simplified the proofs. DATA AVAILABILITY. Data sharing is not applicable to this article as no new data were created or analyzed in this study.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)EXC 2004/1-390534769
Issue or Number:9
Record Number:CaltechAUTHORS:20210927-225705861
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111068
Deposited By: George Porter
Deposited On:28 Sep 2021 16:24
Last Modified:28 Sep 2021 16:24

Repository Staff Only: item control page