Quantum Approximate Markov Chains are Thermal

TL;DR

This paper establishes a quantum analogue of the classical Hammersley-Clifford theorem for 1D systems, showing that quantum approximate Markov chains can be approximated by Gibbs states of short-range Hamiltonians, with implications for efficient state preparation.

Contribution

It proves that 1D quantum approximate Markov chains are well-approximated by Gibbs states, extending classical results to quantum systems and enabling efficient quantum state preparation.

Findings

Quantum approximate Markov chains are close to Gibbs states of short-range Hamiltonians.

Conditional mutual information in 1D Gibbs states decays exponentially with the square root of the region length.

Efficient preparation of 1D Gibbs states at finite temperature is possible with constant-depth circuits.

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.