Quantum Approximate Markov Chains are Thermal

• DOI: 10.1007/s00220-019-03485-6
• Publication Date: 2019-06-21T14:03:03Z
• Authors: Kohtaro Kato; Fernando G. S. L....
• Keywords: Gibbs state; Hamiltonian (control theory)

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.<br/>v1 22 pages, 3 figures; v2 31 pages, 5 figures. Presentations improved and new result (Theorem 3) is added v3; 32 pages, 5 figures. Accepted for publication in Communications in Mathematical Physics<br/>