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14
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
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 compact closed ..."
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Cited by 28 (10 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 14 (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.
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 10 (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.
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 6 (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.
Operational distance and fidelity for quantum channels
 J. Math. Phys
, 2005
"... ABSTRACT. We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minim ..."
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Cited by 4 (1 self)
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ABSTRACT. We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is welldefined for channels between finitedimensional algebras, but it also applies to a certain class of channels between infinitedimensional algebras (explicitly, those channels that possess an operatorvalued Radon– Nikodym density with respect to the trace in the sense of Belavkin–Staszewski) and induces a metric on the set of quantum channels which is topologically equivalent to the CBnorm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantummechanical systems, derived from the wellknown fidelity (‘generalized transition probability’) of Uhlmann, is topologically
Lacunary matrices
, 2001
"... We study unconditional subsequences of the canonical basis (erc) of elementary matrices in the Schatten class S p. They form the matrix counterpart to Rudin’s Λ(p) sets of integers in Fourier analysis. In the case of p an even integer, we find a sufficient condition in terms of trails on a bipartite ..."
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Cited by 3 (1 self)
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We study unconditional subsequences of the canonical basis (erc) of elementary matrices in the Schatten class S p. They form the matrix counterpart to Rudin’s Λ(p) sets of integers in Fourier analysis. In the case of p an even integer, we find a sufficient condition in terms of trails on a bipartite graph. We also establish an optimal density condition and present a random construction of bipartite graphs. As a byproduct, we get a new proof for a theorem of Erdős on circuits in graphs. 1
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 3 (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.
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 3 (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