Quantum conditional relative entropy and quasi-factorization of the relative entropy

Angela Capel; Angelo Lucia; David Pérez-García

doi:10.1088/1751-8121/aae4cf

Quantum conditional relative entropy and quasi-factorization of the relative entropy

Angela Capel, Angelo Lucia, David Pérez-García

7 Citationer (Scopus)

Abstract

The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-σ-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.

Originalsprog	Engelsk
Artikelnummer	484001
Tidsskrift	Journal of Physics A: Mathematical and Theoretical
Vol/bind	51
Udgave nummer	48
Antal sider	41
ISSN	1751-8113
DOI	https://doi.org/10.1088/1751-8121/aae4cf
Status	Udgivet - 2018

Adgang til dokumentet

10.1088/1751-8121/aae4cf

http://arxiv.org/pdf/1804.09525v1

Andre filer og links

Link to publication in Scopus

Citationsformater

@article{17b2b02d615d488ebceac9f1a315db56,

title = "Quantum conditional relative entropy and quasi-factorization of the relative entropy",

abstract = "The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-σ-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.",

keywords = "conditional relative entropy, log-Sobolev inequality, mixing time, quantum dissipative evolution, quantum relative entropy, quasi-factorization of the relative entropy",

author = "Angela Capel and Angelo Lucia and David P{\'e}rez-Garc{\'i}a",

year = "2018",

doi = "10.1088/1751-8121/aae4cf",

language = "English",

volume = "51",

journal = "Journal of Physics A: Mathematical and Theoretical",

issn = "1751-8113",

publisher = "Institute of Physics Publishing Ltd",

number = "48",

}

TY - JOUR

T1 - Quantum conditional relative entropy and quasi-factorization of the relative entropy

AU - Capel, Angela

AU - Lucia, Angelo

AU - Pérez-García, David

PY - 2018

Y1 - 2018

N2 - The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-σ-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.

AB - The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-σ-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.

KW - conditional relative entropy

KW - log-Sobolev inequality

KW - mixing time

KW - quantum dissipative evolution

KW - quantum relative entropy

KW - quasi-factorization of the relative entropy

UR - http://www.scopus.com/inward/record.url?scp=85056477037&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aae4cf

DO - 10.1088/1751-8121/aae4cf

M3 - Journal article

SN - 1751-8113

VL - 51

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 48

M1 - 484001

ER -

Quantum conditional relative entropy and quasi-factorization of the relative entropy

Abstract

Adgang til dokumentet

Andre filer og links

Fingeraftryk

Citationsformater