Classical restrictions of generic matrix product states are quasi-locally Gibbsian
Abstract
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.
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.Attached Files
Published - 093511_1_online.pdf
Submitted - 2010-11643.pdf
Files
Name | Size | Download all |
---|---|---|
md5:ce75b974c2b97dd32b70ecee83a3d4ac
|
6.0 MB | Preview Download |
md5:57d0d136ac3d1f77aeb44b8fa6638c4f
|
424.7 kB | Preview Download |
Additional details
- Eprint ID
- 111068
- Resolver ID
- CaltechAUTHORS:20210927-225705861
- EXC 2004/1-390534769
- Deutsche Forschungsgemeinschaft (DFG)
- Created
-
2021-09-28Created from EPrint's datestamp field
- Updated
-
2023-10-04Created from EPrint's last_modified field