The thermal equilibrium properties of physical systems can be described using Gibbs states. It is therefore of great interest to know when such states allow for an easy description. In particular, this is the case if correlations between distant regions are small. In this work, we consider 1D quantum spin systems with local, finite-range, translation-invariant interactions at any temperature. In this setting, we show that Gibbs states satisfy uniform exponential decay of correlations and, moreover, the mutual information between two regions decays exponentially with their distance, irrespective of the temperature. In order to prove the latter, we show that exponential decay of correlations of the infinite-chain thermal states, exponential uniform clustering and exponential decay of the mutual information are equivalent for 1D quantum spin systems with local, finite-range interactions at any temperature. In particular, Araki's seminal results yields that the three conditions hold in the translation-invariant case. The methods we use are based on the Belavkin-Staszewski relative entropy and on techniques developed by Araki. Moreover, we find that the Gibbs states of the systems we consider are superexponentially close to saturating the data-processing inequality for the Belavkin-Staszewski relative entropy.
We prove that the quantum Gibbs states of spin systems above a certain threshold temperature are approximate quantum Markov networks, meaning that the conditional mutual information decays rapidly with distance. We demonstrate the exponential decay for short-ranged interacting systems and power-law decay for long-ranged interacting systems. Consequently, we establish the efficiency of quantum Gibbs sampling algorithms, a strong version of the area law, the quasi-locality of effective Hamiltonians on subsystems, a clustering theorem for mutual information, and a polynomial-time algorithm for cl…
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. …
Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙高等科学研究理事会理论物理研究所）；Fachbereich Mathematik, Universität Tübingen（蒂宾根大学数学系）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid（马德里康普顿斯大学数学分析与应用数学系）
Classical Estimation of the Free Energy and Quantum Gibbs Sampling from the Markov Entropy Decomposition
University of Cambridge（剑桥大学）；Universität Tübingen（蒂宾根大学）；IBM Quantum, IBM T. J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）；Inria, ENS Lyon, UCBL, LIP（法国国家信息与自动化研究所，里昂高等师范学院，里昂第一大学，计算与并行实验室）；UC Davis（加州大学戴维斯分校）；University of Sydney（悉尼大学）
We provide an analysis of the correlation properties of spin and fermionic systems on a lattice evolving according to open system dynamics generated by a local primitive Liouvillian. We show that if the Liouvillian has a spectral gap which is independent of the system size, then the correlations between local observables decay exponentially as a function of the distance between their supports. We prove, furthermore, that if the Log-Sobolev constant is independent of the system size, then the system satisfies clustering of correlations in the mutual information - a much more stringent form of c…
We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel frameworkfor computing determinants of Toeplitz operators with smooth symbols, and for treating Toeplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the one-dimensional lattice satisfies an area law, with a prefactor which we compute explicitly. As examples, we discuss the fermionic XX model in one dimension and free…
Efficiently estimating quantum thermal properties from exponentially fewer samples
University of California, Berkeley（加州大学伯克利分校）；Center for Theoretical Physics, Massachusetts Institute of Technology（麻省理工学院理论物理中心）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）
Exponential Lindbladian fast forwarding and exponential amplification of certain Gibbs state properties
QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（量子信息与计算倡议，计算与数据科学学院，香港大学）；Beijing International Center for Mathematical Research, Peking University（北京国际数学研究中心，北京大学）；SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（数学力学研究所，数学与系统科学研究院，中国科学院）
Quasi-optimal sampling from Gibbs states via non-commutative optimal transport metrics
Fachbereich Mathematik, Universität Tübingen（蒂宾根大学数学系）；Dep. of App. Mathematics and Theo. Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（巴黎理工学院电信学院LTCI实验室法国国家信息与自动化研究所）
The connection between quantum state recovery and quantum conditional mutual information (QCMI) is studied for the class of purely generated finitely correlated states (pgFCS) of one-dimensional quantum spin chains. For a tripartition of the chain into two subsystems separated by a buffer region, it is shown that a pgFCS is an approximate quantum Markov chain, and stronger, may be approximated by a quantum Markov chain in trace distance, with an error exponentially small in the buffer size. This implies that, (1) a locally corrupted state can be approximately recovered by action of a quantum c…
We propose a quantum algorithm for many-body state preparation. It is especially suited for injective PEPS and thermal states of local commuting Hamiltonians on a lattice. We show that for a uniform gap and sufficiently smooth paths, an adiabatic runtime and circuit depth of $O(\operatorname{polylog}N)$ can be achieved for $O(N)$ spins. This is an almost exponential improvement over previous bounds. The total number of elementary gates scales as $O(N\operatorname{polylog}N)$. This is also faster than the best known upper bound of $O(N^2)$ on the mixing times of Monte Carlo Markov chain algorit…
Quantum Gibbs state sampling algorithms generally suffer from either scaling exponentially with system size or requiring specific knowledge of spectral properties \textit{a priori}. Also, these algorithms require a large overhead of bath or scratch/ancilla qubits. We propose a method, termed the minimal effective Gibbs ansatz (MEGA), which uses a quantum computer to determine a minimal ensemble of pure states that accurately reproduce thermal averages of typical observables. This technique employs properties of correlation functions that can be split into a lesser and greater part; here, we pr…
Renormalization group and long-range conditional mutual information in hierarchical models
University of California at San Diego（圣地亚哥加利福尼亚大学）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣巴巴拉分校卡弗里理论物理研究所）
