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15
Higherrank numerical ranges and compression problems
 Linear Algebra Appl
"... Abstract. We consider higherrank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory for the higherrank numerical ranges, and give a complete description in the Hermitian case. ..."
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Abstract. We consider higherrank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory for the higherrank numerical ranges, and give a complete description in the Hermitian case. We also consider associated projection compression problems. 1.
Harmonic analysis of iterated function systems with overlap
 J. Math. Phys
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Entropy encoding, hilbert space, and karhunenloève transforms
 J. Math. Phys
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Cones with a Mapping Cone Symmetry in the FiniteDimensional Case
"... In the finitedimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio lkowskiChoi i ..."
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In the finitedimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio lkowskiChoi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.
APPROXIMATING ORTHOGONAL MATRICES BY PERMUTATION MATRICES
, 2005
"... Abstract. Motivated in part by a problem of combinatorial optimization and in part by analogies with quantum computations, we consider approximations of orthogonal matrices U by “noncommutative convex combinations”A of permutation matrices of the type A = ∑ Aσσ, where σ are permutation matrices an ..."
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Abstract. Motivated in part by a problem of combinatorial optimization and in part by analogies with quantum computations, we consider approximations of orthogonal matrices U by “noncommutative convex combinations”A of permutation matrices of the type A = ∑ Aσσ, where σ are permutation matrices and Aσ are positive semidefinite n × n matrices summing up to the identity matrix. We prove that for every n × n orthogonal matrix U there is a noncommutative convex combination A of permutation matrices which approximates U entrywise within an error of cn −1 2 ln n and in the Frobenius norm within an error of c lnn. The proof uses a certain procedure of randomized rounding of an orthogonal matrix to a permutation matrix. 1. Introduction and
The Measure of a Measurement
, 2007
"... Abstract. While finite noncommutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is a certain splitting of the total Hilbert space an ..."
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Abstract. While finite noncommutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is a certain splitting of the total Hilbert space and its recursive iterations to further iterated subdivisions. This paper explores some implications for associated probability measures (in the classical sense of measure theory), specifically their fractal components. We identify a fractal scale s in a family of Borel probability measures µ on the unit interval which arises independently in quantum information theory and in wavelet analysis. The scales s we find satisfy s ∈ R+ and s = 1, some s < 1 and some s> 1. We identify these scales s by considering the asymptotic properties of µ(J) / J  s where J are dyadic subintervals, and J  → 0.
QUANTUM CAUSAL HISTORIES AND THE DIRECTED GRAPH OPERATOR FRAMEWORK
, 2005
"... Abstract. A mathematical formalism called Quantum Causal Histories was recently invented as an attempt to describe causality within a quantum theory of gravity. Fundamental examples include quantum computers. We show there is a connection between this formalism and the directed graph operator framew ..."
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Abstract. A mathematical formalism called Quantum Causal Histories was recently invented as an attempt to describe causality within a quantum theory of gravity. Fundamental examples include quantum computers. We show there is a connection between this formalism and the directed graph operator framework from the theory of operator algebras. 1.
A.: Localized bases in L 2 (0,1) and their use in the analysis of Brownian motion
 J. Approx. Theory
, 2008
"... Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L 2 (0,1) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the Hilbert space of squareintegrable functions on the unitinterval ..."
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Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L 2 (0,1) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the Hilbert space of squareintegrable functions on the unitinterval, both with respect to Lebesgue measure, and also with respect to a wider class of selfsimilar measures µ. That is, we consider recursive and orthogonal decompositions for the Hilbert space L 2 (µ) where µ is some selfsimilar measure on [0,1]. Up to two specific reflection symmetries, our scheme produces infinite families of orthonormal bases in L 2 (0,1). Our approach is as versatile as the more traditional spline constructions. But while singly generated spline bases typically do not produce orthonormal bases, each of our present algorithms does.
A Framework of Quantuminspired MultiObjective Evolutionary Algorithms and its Convergence Properties
, 2007
"... In this paper, a general framework of quantuminspired multiobjective evolutionary algorithms is proposed based on the basic principles of quantum computing and general schemes of multiobjective evolutionary algorithms. One of the sufficient convergence conditions to Pareto optimal set is presente ..."
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In this paper, a general framework of quantuminspired multiobjective evolutionary algorithms is proposed based on the basic principles of quantum computing and general schemes of multiobjective evolutionary algorithms. One of the sufficient convergence conditions to Pareto optimal set is presented and it is proved under partially order set theory. Moreover, two algorithms are given as examples meeting this convergence condition, in which two improved Qgates are used. Their convergence properties are discussed. Additionally, one counterexample is also given.
A unified framework for graph algebras and quantum causal histories, preprint
, 2005
"... Abstract. We present a mathematical framework that unifies the quantum causal history formalism from theoretical high energy physics and the directed graph operator framework from the theory of operator algebras. The approach involves completely positive maps and directed graphs and leads naturally ..."
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Abstract. We present a mathematical framework that unifies the quantum causal history formalism from theoretical high energy physics and the directed graph operator framework from the theory of operator algebras. The approach involves completely positive maps and directed graphs and leads naturally to a new class of operator algebras. 1.