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23
Geometries of Quantum States
, 1995
"... The quantum analogue of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cram'erRao type inequalities or the generalization ..."
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Cited by 34 (6 self)
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The quantum analogue of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cram'erRao type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and the FubiniStudy metric shows up there.
Compendium of the foundations of classical statistical physics
 Philosophy of Science Archive, 2006. http://philsciarchive.pitt.edu/archive/00002691/01/UffinkFinal.pdf
"... Roughly speaking, classical statistical physics is the branch of theoretical physics that aims to account for the thermal behaviour of macroscopic bodies in terms of a classical mechanical model of their microscopic constituents, with the help of probabilistic assumptions. In the last century and a ..."
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Cited by 9 (0 self)
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Roughly speaking, classical statistical physics is the branch of theoretical physics that aims to account for the thermal behaviour of macroscopic bodies in terms of a classical mechanical model of their microscopic constituents, with the help of probabilistic assumptions. In the last century and a half, a fair number of approaches have been developed to meet this aim. This study of their foundations assesses their coherence and analyzes the motivations for their basic assumptions, and the interpretations of their central concepts. The most outstanding foundational problems are the explanation of timeasymmetry in thermal behaviour, the relative autonomy of thermal phenomena from their microscopic underpinning, and the meaning of probability. A more or less historic survey is given of the work of Maxwell, Boltzmann and Gibbs in statistical physics, and the problems and objections to which their work gave rise. Next, we review some modern approaches to (i) equilibrium statistical mechanics, such as ergodic theory and the theory of the thermodynamic limit; and to (ii) nonequilibrium statistical mechanics as provided by Lanford’s work on the Boltzmann equation, the socalled BogolyubovBornGreenKirkwoodYvon approach, and stochastic approaches such as ‘coarsegraining ’ and the ‘open systems ’ approach. In all cases,
A general approach to sparse basis selection: Majorization, concavity, and affine scaling
 IN PROCEEDINGS OF THE TWELFTH ANNUAL CONFERENCE ON COMPUTATIONAL LEARNING THEORY
, 1997
"... Measures for sparse best–basis selection are analyzed and shown to fit into a general framework based on majorization, Schurconcavity, and concavity. This framework facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed concentration measures use ..."
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Cited by 6 (3 self)
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Measures for sparse best–basis selection are analyzed and shown to fit into a general framework based on majorization, Schurconcavity, and concavity. This framework facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed concentration measures useful for sparse basis selection. It also allows one to define new concentration measures, and several general classes of measures are proposed and analyzed in this paper. Admissible measures are given by the Schurconcave functions, which are the class of functions consistent with the socalled Lorentz ordering (a partial ordering on vectors also known as majorization). In particular, concave functions form an important subclass of the Schurconcave functions which attain their minima at sparse solutions to the best basis selection problem. A general affine scaling optimization algorithm obtained from a special factorization of the gradient function is developed and proved to converge to a sparse solution for measures chosen from within this subclass.
Quantum state transformations and the Schubert calculus.” Ann
, 2005
"... Recent developments in mathematics have provided powerful tools for comparing the eigenvalues of matrices related to each other via a moment map. In this paper we survey some of the more concrete aspects of the approach with a particular focus on applications to quantum information theory. After dis ..."
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Cited by 4 (0 self)
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Recent developments in mathematics have provided powerful tools for comparing the eigenvalues of matrices related to each other via a moment map. In this paper we survey some of the more concrete aspects of the approach with a particular focus on applications to quantum information theory. After discussing the connection between Horn’s Problem and Nielsen’s Theorem, we move on to characterizing the eigenvalues of the partial trace of a matrix. 1
Entropy as a fixed point
"... We study complexity and information and introduce the idea that while complexity is relative to a given class of processes, information is process independent: Information is complexity relative to the class of all conceivable processes. In essence, the idea is that information is an extension of ..."
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Cited by 4 (2 self)
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We study complexity and information and introduce the idea that while complexity is relative to a given class of processes, information is process independent: Information is complexity relative to the class of all conceivable processes. In essence, the idea is that information is an extension of the concept algorithmic complexity from a class of desirable and concrete processes, such as those represented by binary decision trees, to a class more general that can only in pragmatic terms be regarded as existing in the conception. It is then precisely the fact that information is defined relative to such a large class of processes that it becomes an eective tool for analyzing phenomena in a wide range of disciplines. We test
Minimization of convex functionals over frame operators
 Adv. Comput. Math
"... Abstract. We present results about minimization of convex functionals defined over a finite set of vectors in a finite dimensional Hilbert space, that extend several known results for the BenedettoFickus frame potential. Our approach depends on majorization techniques. We also consider some perturb ..."
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Cited by 4 (2 self)
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Abstract. We present results about minimization of convex functionals defined over a finite set of vectors in a finite dimensional Hilbert space, that extend several known results for the BenedettoFickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one. 1.
The local form of doubly stochastic maps and joint majorization in II1 factors
 Integral Equations Operator Theory
, 2008
"... Dedicated to our families Abstract. We find a description of the restriction of doubly stochastic maps to separable abelian C ∗subalgebras of a II1 factor M. We use this local form of doubly stochastic maps to develop a notion of joint majorization between ntuples of mutually commuting selfadjoin ..."
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Cited by 2 (1 self)
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Dedicated to our families Abstract. We find a description of the restriction of doubly stochastic maps to separable abelian C ∗subalgebras of a II1 factor M. We use this local form of doubly stochastic maps to develop a notion of joint majorization between ntuples of mutually commuting selfadjoint operators that extends those of Kamei (for single selfadjoint operators) and Hiai (for single normal operators) in the II1 factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian C ∗subalgebra of M can be embedded into a separable abelian C ∗subalgebra of M with diffuse spectral measure. 1.
A technique for verifying measurements
 24th Conference on Mathematical Foundations of Programming Semantics, Electronic Notes in Theoretical Computer Science
, 2008
"... We give a technique that can be used to prove that a given function is a measurement. We demonstrate its applicability by using it to resolve three notoriously difficult cases: capacity in information theory, entropy in quantum mechanics and global time in general relativity. We then show that this ..."
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Cited by 2 (1 self)
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We give a technique that can be used to prove that a given function is a measurement. We demonstrate its applicability by using it to resolve three notoriously difficult cases: capacity in information theory, entropy in quantum mechanics and global time in general relativity. We then show that this technique provides a new and surprising characterization of measurement. Thus, in principle, it can always be used. Keywords: domain theory, measurement, information theory, quantum mechanics, general relativity
PerronFrobenius Theory For Positive Maps On Trace Ideals
"... . This article provides sufficient conditions for positive maps on the Schatten classes Jp ; 1 p < 1 of bounded operators on a separable Hilbert space such that a corresponding PerronFrobenius theorem holds. With applications in quantum information theory in mind sufficient conditions are given f ..."
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. This article provides sufficient conditions for positive maps on the Schatten classes Jp ; 1 p < 1 of bounded operators on a separable Hilbert space such that a corresponding PerronFrobenius theorem holds. With applications in quantum information theory in mind sufficient conditions are given for a trace preserving, positive map on J1 , the space of trace class operators, to have a unique, strictly positive density matrix which is left invariant under the map. Conversely to any given strictly positive density matrix there are trace preserving, positive maps for which the density matrix is the unique PerronFrobenius vector. Dedicated to S. Doplicher and J.E. Roberts on the occasion of their 60th birthday 1. INTRODUCTION In the theory of quantum information the transmission through noisy channels plays an important role. Usually it is described by what physicists either call a quantum operation (see e.g. [25]) or a stochastic map ([1], see also [19]) or a superoperator (see e.g. ...