Results

**11 - 14**of**14**### Trace estimates and invariance of the essential spectrum

, 2006

"... We provide sufficient conditions under which the difference of the resolvents of two higher-order 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 higher-order 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.

### 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 well-defined for channels between finitedimensional algebras, but it also applies to a certain class of channels between infinitedimensional algebras (explicitly, those channels that possess an operator-valued 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 CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (‘generalized transition probability’) of Uhlmann, is topologically equivalent to the trace-norm distance. 2000 Mathematics Subject Classification. 46L07, 46L55, 46L60, 47L07.

### 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 trace-densities for quantum states, we restore one-to-one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian-positive operatorvalued ..."

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In celebration of the 100th anniversary of the birth of John von Neumann Abstract. Introducing contravariant trace-densities for quantum states, 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 CB-norm distance between two quantum operations is explicitly expressed in terms of these densities as the supremum over the input states. A larger C-distance is given as the natural norm-distance 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 trace-densities. 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.