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20
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 3 (2 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.
Contravariant densities, operational distances and quantum channel fidelities, presented at the
 Sixth International Conference on Quantum Communication, Measurement, and Computing (QCM&C’02
"... Abstract. Introducing contravariant tracedensities for quantum states on semifinite algebras, we restore one to one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian positive operatorvalued contravariant kernels. The CBnorm distance betwe ..."
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Cited by 2 (2 self)
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Abstract. Introducing contravariant tracedensities for quantum states on semifinite algebras, we restore one to one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian positive operatorvalued contravariant kernels. The CBnorm distance between two quantum operations with type one input algebras is explicitly expressed in terms of these densities, and this formula is also extended to a generalized CBdistances between quantum operations with type two inputs. A larger Cdistance is given as the natural normdistance for the channel densities, and another, Helinger type distance, related to minimax mean square optimization problem for purification of quantum channels, is also introduced and evaluated in terms of their contravariant tracedensities. It is proved that the Helinger type complete fidelity distance between two channels is equivalent to the CB distance at least for type one inputs, and this equivalence is also extended to type two for a generalized CB distance. An operational meaning for these distances and relative complete fidelity for quantum channels is given in terms of quantum encodings as generalized entanglements of quantum states opposite to the inputs and the output states. 1.
CONTENTS
, 2004
"... 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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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 equivalent to the tracenorm distance. 2000 Mathematics Subject Classification. 46L07, 46L55, 46L60, 47L07.
Trace estimates and invariance of the essential spectrum
, 2006
"... We provide sufficient conditions under which the difference of the resolvents of two higherorder operators acting in R N belongs to trace classes C p. We provide explicit estimates on the norm of the resolvent difference in terms of L p norms of the difference of the coefficients. Such inequalities ..."
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We provide sufficient conditions under which the difference of the resolvents of two higherorder operators acting in R N belongs to trace classes C p. We provide explicit estimates on the norm of the resolvent difference in terms of L p norms of the difference of the coefficients. Such inequalities are useful in estimating the effect of localized perturbations of the coefficients.
CONTRAVARIANT DENSITIES, COMPLETE DISTANCES AND RELATIVE FIDELITIES FOR QUANTUM CHANNELS
, 2005
"... In celebration of the 100th anniversary of the birth of John von Neumann Abstract. Introducing contravariant tracedensities for quantum states, we restore onetoone correspondence between quantum operations described by normal CP maps and their trace densities as Hermitianpositive operatorvalued ..."
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In celebration of the 100th anniversary of the birth of John von Neumann Abstract. Introducing contravariant tracedensities for quantum states, we restore onetoone correspondence between quantum operations described by normal CP maps and their trace densities as Hermitianpositive operatorvalued contravariant kernels. The CBnorm distance between two quantum operations is explicitly expressed in terms of these densities as the supremum over the input states. A larger Cdistance is given as the natural normdistance for the channel densities, and another, Helinger type complete distance (CHdistance), related to the minimax mean square fidelity optimization problem by purification of quantum channels, is also introduced and evaluated in terms of their contravariant tracedensities. It is proved that the CH distance between two channels is equivalent to the CB distance. An operational meaning for these distances and relative complete fidelity for quantum channels is given in terms of quantum encodings producing optimal entanglements of quantum states for an opposite and output systems. 1.
Edinburgh, Scotland
, 1998
"... Most tensored \Lambdacategories 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 ..."
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Most tensored \Lambdacategories 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 \Lambdacategories, all morphisms are nuclear, and in the tensored \Lambdacategory of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored \Lambdacategories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations.
DOI: 10.1007/s1095500691605 Entropy of Open Lattice Systems
, 2005
"... 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 lea ..."
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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 kth 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. KEY WORDS: Entropy, exclusion process, hydrodynamic scaling 1.