Abstract not found

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...

We study the non-equilibrium 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 non-equilibrium 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].

This paper presents self-contained 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

We review and further develop a mathematical framework for non-equilibrium 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 non-equilibrium 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.

Abstract not found

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 non-negative in natural nonequilibrium steady states.

Let a be a self-adjoint 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 unital-nuclear 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 self-adjoint 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–Krieger-type Pimsner algebras, leading to the variational principle for self-adjoint elements in a diagonal subalgebra. Equilibrium states are constructed from KMS states under certain conditions in the case of Cuntz–Krieger algebras. Supported by NATO–CNR.

Entropy-like functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most well-known are the von Neumann entropy I(ρ): = −trace(ρ log ρ) and a generalization of the Kullback-Leibler distance S(ρ||σ): = trace(ρ log ρ − ρ log σ), refered to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore I and S as regularizing functionals in seeking solutions to multi-variable and multi-dimensional moment problems. It will be shown that extrema can be effectively constructed via a suitable homotopy. The homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems. This is accomplished by a renormalization of a Riemannian metric induced by entropy functionals. As an application we discuss the inverse problem of describing power spectra which are consistent with second-order statistics, which has been the main motivation behind the present work.

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is