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Quantum Approximate Markov Chains are Thermal

Kato, Kohtaro and Brandão, Fernando G. S. L. (2019) Quantum Approximate Markov Chains are Thermal. Communications in Mathematical Physics, 370 (1). pp. 117-149. ISSN 0010-3616. http://resolver.caltech.edu/CaltechAUTHORS:20170726-094724141

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Abstract

We prove that any one-dimensional (1D) quantum state with small quantum conditional mutual information in all certain tripartite splits of the system, which we call a quantum approximate Markov chain, can be well-approximated by a Gibbs state of a short-range quantum Hamiltonian. Conversely, we also derive an upper bound on the (quantum) conditional mutual information of Gibbs states of 1D short-range quantum Hamiltonians. We show that the conditional mutual information between two regions A and C conditioned on the middle region B decays exponentially with the square root of the length of B. These two results constitute a variant of the Hammersley–Clifford theorem (which characterizes Markov networks, i.e. probability distributions which have vanishing conditional mutual information, as Gibbs states of classical short-range Hamiltonians) for 1D quantum systems. The result can be seen as a strengthening—for 1D systems—of the mutual information area law for thermal states. It directly implies an efficient preparation of any 1D Gibbs state at finite temperature by a constant-depth quantum circuit.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00220-019-03485-6DOIArticle
https://rdcu.be/bOa5BPublisherFree ReadCube access
https://arxiv.org/abs/1609.06636arXivDiscussion Paper
ORCID:
AuthorORCID
Kato, Kohtaro0000-0003-3317-2004
Brandão, Fernando G. S. L.0000-0003-3866-9378
Additional Information:© 2019 Springer-Verlag GmbH Germany, part of Springer Nature. Received: 16 October 2017; Accepted: 22 April 2019; First Online: 21 June 2019. Part of this work was done when both of us were working in the QuArC group of Microsoft Research. KK thanks Advanced Leading Graduate Course for Photon Science (ALPS) and JSPS KAKENHI Grant Number JP16J05374 for financial support. We thank Matthew Hastings and Michael Kastoryano for useful discussions.
Funders:
Funding AgencyGrant Number
Advanced Leading Graduate Course for Photon ScienceUNSPECIFIED
Japan Society for the Promotion of Science (JSPS)JP16J05374
Issue or Number:1
Record Number:CaltechAUTHORS:20170726-094724141
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170726-094724141
Official Citation:Kato, K. & Brandão, F.G.S.L. Commun. Math. Phys. (2019) 370: 117. https://doi.org/10.1007/s00220-019-03485-6
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79404
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:26 Jul 2017 22:36
Last Modified:09 Aug 2019 17:35

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