Large deviations for quantum spin probabilities at temperature zero

Mengue

et al. 2019

Commun. Contemp. Math.

Self Cite

We consider the KMS state associated to the Hamiltonian H = σ x ⊗ σ x over the quantum spin lattice C 2 ⊗ C 2 ⊗ C 2 ⊗ .... For a fixed observable of the form L ⊗ L ⊗ L ⊗ ..., where L : C 2 → C 2 is self adjoint, and for positive temperature T one can get a naturally defined stationary probability µ T on the Bernoulli space {1, 2} N . The Jacobian of µ T can be expressed via a certain continued fraction expansion. We will show that this probability is a Gibbs probability for a Hölder potential. Therefore, this probability is mixing for the shift map. For such probability µ T we will show the explicit deviation function for a certain class of functions. When decreasing temperature we will be able to exhibit the explicit transition value T c where the set of values of the Jacobian of the Gibbs probability µ T changes from being a Cantor set to being an interval.We also present some properties for quantum spin probabilities at zero temperature (for instance, the explicit value of the entropy). arXiv:1805.01784v2 [math.DS] 16 May 2018Given a selfadjoint operator H acting on a finite dimensional complex Hilbert space H and a temperature T > 0, the density operatorwhere Z(T ) = Trace e − 1 T H , is called the KMS operator associated to the Hamiltonian H. It is usual to denote β = 1/T . A general reference on KMS operators and KMS states is [5].The set of linear operators acting on C 2 will be denoted by M 2 . We will call ω = ω n : M 2

Section: Recurrence Formulas and Construction Of The Quantum Spin Promentioning

confidence: 92%

Section: Recurrence Formulas and Construction Of The Quantum Spin Promentioning

confidence: 97%

Section: Recurrence Formulas and Construction Of The Quantum Spin Promentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quantum spin probabilities at positive temperature are Hölder Gibbs probabilities

Brasil

Mengue

et al. 2019

Commun. Contemp. Math.

Self Cite

Thermodynamic formalism for quantum channels: Entropy, pressure, Gibbs channels and generic properties

Brasil

Knorst

2021

Commun. Contemp. Math.

Denote [Formula: see text] the set of complex [Formula: see text] by [Formula: see text] matrices. We will analyze here quantum channels [Formula: see text] of the following kind: given a measurable function [Formula: see text] and the measure [Formula: see text] on [Formula: see text] we define the linear operator [Formula: see text], via the expression [Formula: see text]. A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where [Formula: see text] was the identity. Under some mild assumptions on the quantum channel [Formula: see text] we analyze the eigenvalue property for [Formula: see text] and we define entropy for such channel. For a fixed [Formula: see text] (the a priori measure) and for a given a Hamiltonian [Formula: see text] we present a version of the Ruelle Theorem: a variational principle of pressure (associated to such [Formula: see text]) related to an eigenvalue problem for the Ruelle operator. We introduce the concept of Gibbs channel. We also show that for a fixed [Formula: see text] (with more than one point in the support) the set of [Formula: see text] such that it is [Formula: see text]-Erg (also irreducible) for [Formula: see text] is a generic set. We describe a related process [Formula: see text], [Formula: see text], taking values on the projective space [Formula: see text] and analyze the question of the existence of invariant probabilities. We also consider an associated process [Formula: see text], [Formula: see text], with values on [Formula: see text] ([Formula: see text] is the set of density operators). Via the barycenter, we associate the invariant probability mentioned above with the density operator fixed for [Formula: see text].

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

Mengue

2022

DCDS

Self Cite

<p style='text-indent:20px;'>It is well known that in Information Theory and Machine Learning the Kullback-Leibler divergence, which extends the concept of Shannon entropy, plays a fundamental role. Given an <i>a priori</i> probability kernel <inline-formula><tex-math id="M1">\begin{document}$ \hat{\nu} $\end{document}</tex-math></inline-formula> and a probability <inline-formula><tex-math id="M2">\begin{document}$ \pi $\end{document}</tex-math></inline-formula> on the measurable space <inline-formula><tex-math id="M3">\begin{document}$ X\times Y $\end{document}</tex-math></inline-formula> we consider an appropriate definition of entropy of <inline-formula><tex-math id="M4">\begin{document}$ \pi $\end{document}</tex-math></inline-formula> relative to <inline-formula><tex-math id="M5">\begin{document}$ \hat{\nu} $\end{document}</tex-math></inline-formula>, which is based on previous works. Using this concept of entropy we obtain a natural definition of information gain for general measurable spaces which coincides with the mutual information given from the K-L divergence in the case <inline-formula><tex-math id="M6">\begin{document}$ \hat{\nu} $\end{document}</tex-math></inline-formula> is identified with a probability <inline-formula><tex-math id="M7">\begin{document}$ \nu $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M8">\begin{document}$ X $\end{document}</tex-math></inline-formula>. This will be used to extend the meaning of specific information gain and dynamical entropy production to the model of thermodynamic formalism for symbolic dynamics over a compact alphabet (TFCA model). Via the concepts of involution kernel and dual potential, one can ask if a given potential is symmetric - the relevant information is available in the potential. In the affirmative case, its corresponding equilibrium state has zero entropy production.</p>