Dissipative dynamics for quantum spin systems on a lattice

Majewski, Adam; Olkiewicz, Robert; Zegarliński, Bogusław

doi:10.1088/0305-4470/31/8/015

Cited by 25 publications

(21 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Refs. [33][34][35] one variant of conditional expectations was studied in very much the same context as we do here, where ergodic properties of Gibbs samplers were the main focus. Also, Petz considered a similar set of maps is the context of corse graining operations [36].…”

Section: Conditional Expectationsmentioning

confidence: 99%

Quantum Gibbs Samplers: The Commuting Case

Kastoryano

Brandão

2016

Commun. Math. Phys.

109

View full text Add to dashboard Cite

We analyze the problem of preparing quantum Gibbs states of lattice spin Hamiltonians with local and commuting terms on a quantum computer and in nature. Our central result is an equivalence between the behavior of correlations in the Gibbs state and the mixing time of the semigroup which drives the system to thermal equilibrium (the Gibbs sampler). We introduce a framework for analyzing the correlation and mixing properties of quantum Gibbs states and quantum Gibbs samplers, which is rooted in the theory of non-commutative Lp spaces. We consider two distinct classes of Gibbs samplers, one of them being the well-studied Davies generator modelling the dynamics of a system due to weak-coupling with a large Markovian environment. We show that their spectral gap is independent of system size if, and only if, a certain strong form of clustering of correlations holds in the Gibbs state. Therefore every Gibbs state of a commuting Hamiltonian that satisfies clustering of correlations in this strong sense can be prepared efficiently on a quantum computer. As concrete applications of our formalism, we show that for every one-dimensional lattice system, or for systems in lattices of any dimension at temperatures above a certain threshold, the Gibbs samplers of commuting Hamiltonians are always gapped, giving an efficient way of preparing the associated Gibbs states on a quantum computer. Contents

show abstract

Section: Conditional Expectationsmentioning

confidence: 99%

Quantum Gibbs Samplers: The Commuting Case

Kastoryano

Brandão

2016

Commun. Math. Phys.

109

View full text Add to dashboard Cite

show abstract

“…As demonstrated in [13 16] such family provides a useful framework for construction and study of Markov semigroups, (that is positivity and unit preserving semigroups P t , t 0, which are contractive in L p spaces), for which a given state | is invariant or even which are symmetric in the corresponding L 2 (|) space. In particular in [16] we have constructed an extension of commutative Glauber dynamics with generator containing an additional nonsymmetric term (non available on the commutative level). We have shown that such dynamics possesses a Feller property in the sense that it maps A into itself.…”

Section: Introductionmentioning

confidence: 99%

Hypercontractivity in Noncommutative LpSpaces

Olkiewicz

Zegarliński

1999

Journal of Functional Analysis

Self Cite

172

View full text Add to dashboard Cite

“…It is worth pointing out that both jump-type and diffusion-type processes were analyzed. (3) Concrete illustrative models of quantum dynamical systems were given, cf [36].…”

Section: 2mentioning

confidence: 99%

On Quantum Statistical Mechanics: A Study Guide

Majewski

2017

Advances in Mathematical Physics

View full text Add to dashboard Cite

We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed. Basic IdeasIn this paper, we will try to give an overview and road map to the area of quantum statistical mechanics without becoming too diverted by details. In contrast, we put a strong emphasis on evolution of calculus which is used in the description of statistical mechanics. To make our exposition abundantly clear, we begin with a historical remark. Newton has given his principles for classical mechanics at the end of the 17th century. However, classical mechanics blossomed into a rich mathematical theory only in the second half of the 19th century. After a moment of reverie, we realize that although Newton and Leibniz introduced the basic principles of (classical) calculus, it was Cauchy (around 1830' , albeit the "epsilon-delta definition of limit" was first given by Bolzano in 1817) who finally clarified the concept of limit and then Riemann (around 1860') clarified the concept of integral. Consequently, in the second half of the 19th century, the principles of (classical) calculus were fully established. This gave the opportunity to transform classical mechanics into a well-developed theory (Lagrange, Hamilton, Liouville, etc.). So with a mature theory of calculus available, it took a few more decades to obtain a fully fledged theory of classical mechanics. Subsequently, (classical) statistical mechanics has appeared as a combined development of classical mechanics and probability theory.We will show that similar situation occurred also in the 20th century but in the context of quantum theory. The starting point was Heisenberg's equation of motion in quantum theory. He for the first time wrote a noncommutative derivation, a commutator. (We recall that a derivation is a unary function satisfying the Leibniz product law.) To see this, it is enough to note that a commutator satisfies the Leibniz rule! This can be considered as an analogy of Newton's introduction of (classical) differentiation to write the equations of motion for a classical system. Then Heisenberg, Born, Jordan, and Dirac realized that noncommutativity is the raison d'être of quantum mechanics and they have introduced the so-called canonical quantization. It means that the basic relations of classical mechanics, { 푖 , 푗 } ∝ 푖 푗 1, , = 1, 2, 3, . . . ,should be replaced by where {⋅, ⋅} stands for the Poisson bracket, while [ , ] = − denotes the commutator.

show abstract

Dissipative dynamics for quantum spin systems on a lattice

Cited by 25 publications

References 7 publications

Quantum Gibbs Samplers: The Commuting Case

Quantum Gibbs Samplers: The Commuting Case

Hypercontractivity in Noncommutative LpSpaces

On Quantum Statistical Mechanics: A Study Guide

Contact Info

Product

Resources

About