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60
NonEquilibrium Steady States of Finite Quantum Systems Coupled to Thermal Reservoirs
 COMMUN. MATH. PHYS
, 2001
"... We study the nonequilibrium statistical mechanics of a #level quantum system, # , coupled to two independent free Fermi reservoirs # # , # # , which are in thermal equilibrium at inverse temperatures # # ## # # .Weprove that, at small coupling, the combined quantum system ### # ## # has ..."
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Cited by 46 (8 self)
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We study the nonequilibrium statistical mechanics of a #level quantum system, # , coupled to two independent free Fermi reservoirs # # , # # , which are in thermal equilibrium at inverse temperatures # # ## # # .Weprove that, at small coupling, the combined quantum system ### # ## # has a unique nonequilibrium steady state (NESS) and that the approach to this NESS is exponentially fast. We show that the entropy production of the coupled system is strictly positive and relate this entropy production to the heat uxes through the system. A part of our argument is general and deals with spectral theory of NESS. In the abstract setting of algebraic quantum statistical mechanics we introduce the new concept of # Liouvillean, #, and relate the NESS to zero resonance eigenfunctions of # # . In the specific model ### # ## # we study the resonances of # # using the complex deformation technique developed previously by the authors in [JP1].
Inequalities for quantum entropy. A review with conditions with equality
"... This paper presents selfcontained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, whic ..."
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Cited by 33 (7 self)
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This paper presents selfcontained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A → Tr e K+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2) ≤ S(ρ12) + S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123 = log ρ12 − log ρ2 + log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s
Mathematical Theory of NonEquilibrium Quantum Statistical Mechanics
, 2002
"... We review and further develop a mathematical framework for nonequilibrium quantum statistical mechanics recently proposed in [JP4, JP5, JP6, Ru3, Ru4, Ru5, Ru6]. In the algebraic formalism of quantum statistical mechanics we introduce notions of nonequilibrium steady states, entropy production and ..."
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Cited by 33 (3 self)
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We review and further develop a mathematical framework for nonequilibrium quantum statistical mechanics recently proposed in [JP4, JP5, JP6, Ru3, Ru4, Ru5, Ru6]. In the algebraic formalism of quantum statistical mechanics we introduce notions of nonequilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.
On Capacities of Quantum Channels
, 1997
"... Capacities of quantum mechanical channels are dened in terms of mutual information quantities. Geometry of the relative entropy is used to express capacity as a divergence radius. The symmetric quantum spin 1=2 channel and the attenuation channel of Boson elds are discussed as examples. 1. Introduct ..."
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Cited by 26 (4 self)
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Capacities of quantum mechanical channels are dened in terms of mutual information quantities. Geometry of the relative entropy is used to express capacity as a divergence radius. The symmetric quantum spin 1=2 channel and the attenuation channel of Boson elds are discussed as examples. 1. Introduction. A discrete communication system { as modeled by Shannon { is capable of transmitting succesively symbols of a nite input alphabet fx 1 ; x 2 ; : : : ; xm g. In the stochastic approach to the communication model it is assumed that the input symbols show up with certain probability. Let p ji be the probability that a symbol x i is sent over the channel and the output symbol y j appears at the destination. The joint distribution p ji yields marginal distributions (p 1 ; p 2 ; : : : ; p m ) and (q 1 ; q 2 ; : : : ; q k ) on the set of input symbols and output symbols, respectively. Shannon introduced the mutual information I = X i;j p ji log p ji p i q j (1:1) in order to measur...
H.: Equilibrium statistical mechanics of Fermion lattice systems
"... We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the noncommutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle wit ..."
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Cited by 26 (7 self)
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We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the noncommutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C ∗dynamical systems for the Fermion (CAR) algebra A with all or a part of the following assumptions: (I) The interaction is even, namely, the dynamics αt commutes with the evenoddness automorphism Θ. (Automatically satisfied when (IV) is assumed.) (II) The domain of the generator δα of αt contains the set A ◦ of all strictly local elements of A. (III) The set A ◦ is the core of δα.
On Entropy Production in Quantum Statistical Mechanics
, 2000
"... We propose a definition of entropy production in the framework of algebraic quantum statistical mechanics. We relate our definition to heat flows through the system. We also prove that entropy production is nonnegative in natural nonequilibrium steady states. ..."
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Cited by 17 (3 self)
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We propose a definition of entropy production in the framework of algebraic quantum statistical mechanics. We relate our definition to heat flows through the system. We also prove that entropy production is nonnegative in natural nonequilibrium steady states.
Noncommutative pressure and the variational principle in CuntzKriegertype C∗algebras
 J. FUNCT. ANAL
, 2000
"... Let a be a selfadjoint element of an exact C∗–algebra A and θ: A→A a contractive completely positive map. We define a notion of dynamical pressure Pθ(a) which adopts Voiculescu’s approximation approach to noncommutative entropy and extends the Voiculescu–Brown topological entropy and Neshveyev and ..."
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Cited by 13 (1 self)
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Let a be a selfadjoint element of an exact C∗–algebra A and θ: A→A a contractive completely positive map. We define a notion of dynamical pressure Pθ(a) which adopts Voiculescu’s approximation approach to noncommutative entropy and extends the Voiculescu–Brown topological entropy and Neshveyev and Størmer unitalnuclear pressure. A variational inequality bounding Pθ(a) below by the free energies hσ(θ)+σ(a) with respect to the Sauvageot–Thouvenot entropy hσ(θ) is established in two stages via the introduction of a local state approximation entropy, whose associated free energies function as an intermediate term. Pimsner C ∗ –algebras furnish a framework for investigating the variational principle, which asserts the equality of Pθ(a) with the supremum of the free energies over all θ–invariant states. In one direction we extend Brown’s result on the constancy of the Voiculescu–Brown entropy upon passing to the crossed product, and in another we show that the pressure of a selfadjoint element over the Markov subshift underlying the canonical map on the Cuntz–Kreiger algebra OA is equal to its classical pressure. The latter result is extended to a more general setting comprising an expanded class of Cuntz–Kriegertype Pimsner algebras, leading to the variational principle for selfadjoint elements in a diagonal subalgebra. Equilibrium states are constructed from KMS states under certain conditions in the case of Cuntz–Krieger algebras.
A Minkowski type trace inequality and strong subadditivity of quantum entropy
 Advances in the Mathematical Sciences, AMS Transl., 189 Series 2
, 1999
"... We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is ..."
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Cited by 12 (3 self)
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We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is