Quantum many-body systems in thermal equilibrium

Abstract

The thermal or equilibrium ensemble is one of the most ubiquitous states of matter. For models comprised of many locally interacting quantum particles, it describes a wide range of physical situations, relevant to condensed matter physics, high energy physics, quantum chemistry and quantum computing, among others. We give a pedagogical overview of some of the most important universal features about the physics and complexity of these states, which have the locality of the Hamiltonian at its core. We focus on mathematically rigorous statements, many of them inspired by ideas and tools from quantum information theory. These include bounds on their correlations, the form of the subsystems, various statistical properties, and the performance of classical and quantum algorithms. We also include a summary of a few of the most important technical tools, as well as some self-contained proofs.Parts of these notes were the basis for a lecture series within the "Quantum Thermo-

Classical and Quantum Iterative Optimization Algorithms Based on Matrix Legendre-Bregman Projections

“…It is an iterative algorithm that minimizes the partition function of the linear family of Hamiltonians. The matrix GIS algorithm is an algorithm for maximum entropy inference and can be applied in understanding many-body quantum systems [16,1]. The convergence of these algorithms follows as they are special cases of the information projection algorithm.…”

Thermal Area Law for the Bose-Hubbard Model

“…Further results of thermal area laws were established for free fermions [37], for the entanglement negativity (instead of the mutual information) [38], as a result of rapid mixing for dissipative quantum lattice systems [39,40] (with a logarithmic correction) and numerically for some spin chains showing a log β-dependence [41]. A current account of thermal area laws and related phenomena is given in [42].…”

“…The applications of thermal areas laws mirror those of their ground state counterparts. In particular, a thermal area law can be used for the important problem of approximating the Gibbs state by matrix product operators (MPO)-mixed-state cousins of the standard matrix product states-and their higher-dimensional analogs [34,[42][43][44]. It is generally believed that the scaling of the mutual information with β in the low temperature regime is related to the computational complexity of the ground space of the models [42].…”

“…In particular, a thermal area law can be used for the important problem of approximating the Gibbs state by matrix product operators (MPO)-mixed-state cousins of the standard matrix product states-and their higher-dimensional analogs [34,[42][43][44]. It is generally believed that the scaling of the mutual information with β in the low temperature regime is related to the computational complexity of the ground space of the models [42]. Another way to use a thermal area law is to note that controlling the mutual information automatically controls all standard correlation functions.…”

“…Corollary 3. Further physical applications, e.g., to the approximability of bosonic Gibbs states in the spirit of [34,42] are postponed to future work.…”