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21
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 58 (8 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
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The comp ..."
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Cited by 39 (12 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Conditional intensity and Gibbsianness of determinantal point processes
 J. Stat. Phys
, 2005
"... The Papangelou intensities of determinantal (or fermion) point processes are investigated. These exhibit a monotonicity property expressing the repulsive nature of the interaction, and satisfy a bound implying stochastic domination by a Poisson point process. We also show that determinantal point pr ..."
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Cited by 26 (2 self)
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The Papangelou intensities of determinantal (or fermion) point processes are investigated. These exhibit a monotonicity property expressing the repulsive nature of the interaction, and satisfy a bound implying stochastic domination by a Poisson point process. We also show that determinantal point processes satisfy the socalled condition (�λ), which is a general form of Gibbsianness. Under a continuity assumption, the Gibbsian conditional probabilities can be identified explicitly. KEY WORDS: Determinantal point process; fermion point process; Gibbs point process; Papangelou intensity; stochastic domination; percolation.
Entropy of open lattice systems
 J. Stat. Phys
, 2007
"... Abstract: We investigate the behavior of the GibbsShannon entropy of the stationary nonequilibrium measure describing a onedimensional lattice gas, of L sites, with symmetric exclusion dynamics and in contact with particle reservoirs at different densities. In the hydrodynamic scaling limit, L → ∞ ..."
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Cited by 20 (5 self)
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Abstract: We investigate the behavior of the GibbsShannon entropy of the stationary nonequilibrium measure describing a onedimensional lattice gas, of L sites, with symmetric exclusion dynamics and in contact with particle reservoirs at different densities. In the hydrodynamic scaling limit, L → ∞, the leading order (O(L)) behavior of this entropy has been shown by Bahadoran to be that of a product measure corresponding to strict local equilibrium; we compute the first correction, which is O(1). The computation uses a formal expansion of the entropy in terms of truncated correlation functions; for this system the k th such correlation is shown to be O(L −k+1). This entropy correction depends only on the scaled truncated pair correlation, which describes the covariance of the density field. It coincides, in the large L limit, with the corresponding correction obtained from a Gaussian measure with the same covariance. 1
Integral Operators, Pseudodifferential Operators, and Gabor Frames
, 2003
"... This chapter illustrates the use of Gabor frame analysis to derive results on the spectral properties of integral and pseudodifferential operators. In particular, we obtain a sufficient condition on the kernel of an integral operator or the symbol of a pseudodifferential operator which implies that ..."
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Cited by 16 (4 self)
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This chapter illustrates the use of Gabor frame analysis to derive results on the spectral properties of integral and pseudodifferential operators. In particular, we obtain a sufficient condition on the kernel of an integral operator or the symbol of a pseudodifferential operator which implies that the operator is traceclass. This result significantly improves a sufficient condition due to Daubechies and Hörmander.
RadonNikodym derivatives of quantum operations
 JOURNAL OF MATHEMATICAL PHYSICS
, 2003
"... Given a completely positive (CP) map T, there is a theorem of the RadonNikodym type [W.B. Arveson, Acta Math. 123, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. 24, 49 (1986)] that completely characterizes all CP maps S such that T − S is also a CP map. This theorem is reviewed, and ..."
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Cited by 12 (2 self)
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Given a completely positive (CP) map T, there is a theorem of the RadonNikodym type [W.B. Arveson, Acta Math. 123, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. 24, 49 (1986)] that completely characterizes all CP maps S such that T − S is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the RadonNikodym formalism to study the structure of order intervals of quantum operations, as well as a certain onetoone correspondence between CP maps and positive operators, already fruitfully exploited in many quantum informationtheoretic treatments. We also comment on how the RadonNikodym theorem can be used to derive norm estimates for differences of CP maps in general, and of quantum operations in particular.
Darboux Transformations from nKdV to KP
 Acta Applicandae Mathematicae
, 1997
"... Abstract. The iterated Darboux transformations of an ordinary differential operator are constructively parametrized by an infinite dimensional grassmannian of finitely supported distributions. In the case that the operator depends on time parameters so that it is a solution to the nKdV hierarchy, i ..."
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Cited by 7 (4 self)
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Abstract. The iterated Darboux transformations of an ordinary differential operator are constructively parametrized by an infinite dimensional grassmannian of finitely supported distributions. In the case that the operator depends on time parameters so that it is a solution to the nKdV hierarchy, it is shown that the transformation produces a solution of the KP hierarchy. The standard definitions of the theory of τfunctions are applied to this grassmannian and it is shown that these new τfunctions are quotients of KP τfunctions. The application of this procedure for the construction of “higher rank ” KP solutions is discussed. 1.
On the KohnSham equations with periodic background potentials
 Journal of Statistical Physics
"... We study the question of existence and uniqueness for the finite temperature KohnSham equations. For finite volumes, a unique soluion is shown to exists if the effective potential satisfies a set of general conditions and the coupling constant is smaller than a certain value. For periodic backgroun ..."
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Cited by 5 (1 self)
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We study the question of existence and uniqueness for the finite temperature KohnSham equations. For finite volumes, a unique soluion is shown to exists if the effective potential satisfies a set of general conditions and the coupling constant is smaller than a certain value. For periodic background potentials, this value is proven to be volume independent. In this case, the finite volume solutions are shown to converge as the thermodynamic limit is considered. The local density approximation is shown to satisfy the general conditions mentioned above. Key words: density functional theory, KohnSham equations, existence and uniqueness, thermodynamic limit, periodic potentials. 1 1
Resolvent estimates for operators belonging to exponential classes
 Integr. Equ. Oper. Theory
"... Abstract. For a, α> 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that sn(A) = O(exp(−an α)), where sn(A) denotes the nth singular number of A. We provide upper bounds for the norm of the resolvent (zI − A) −1 of A in terms of a quantity describing the de ..."
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Cited by 5 (3 self)
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Abstract. For a, α> 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that sn(A) = O(exp(−an α)), where sn(A) denotes the nth singular number of A. We provide upper bounds for the norm of the resolvent (zI − A) −1 of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α). 1.