47 research outputs found

    Corners of normal matrices

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    We study various conditions on matrices BB and CC under which they can be the off-diagonal blocks of a partitioned normal matrix.Comment: 7 page

    A 3x3 dilation counterexample

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    We define four 3x3 commuting contractions which do not dilate to commuting isometries. However they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three 3×33\times3 commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries.Comment: 11 page

    Higher-Rank Numerical Ranges and Compression Problems

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    We consider higher-rank 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 higher-rank numerical ranges, and give a complete description in the Hermitian case. We also consider associated projection compression problems.Comment: 14 pages, 3 figures, to appear in Linear Algebra and its Application

    Numerical ranges of the powers of an operator

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    The numerical range W(A) of a bounded linear operator A on a Hilbert space is the collection of complex numbers of the form (Av, v) with v ranging over the unit vectors in the Hilbert space. In terms of the location of W(A). inclusion regions are obtained for W(A(k)) for positive integers k, and also for negative integers k if A(-1) exists. Related inequalities on the numerical radius w(A) = sup{vertical bar u vertical bar: mu is an element of EW(A)} and the Crawford number c(A) = inf{vertical bar u vertical bar: mu is an element of W(A)} are deduced. (C) 2009 Elsevier Inc. All rights reserved

    On unital qubit channels

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    A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients p1,…,pmp_1, \dots, p_m as long as 2(p1,…,pm)2(p_1, \dots, p_m) is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.Comment: 15 page

    Transversal zeros and positive semidefinite forms

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    Symmetry in the Cuntz Algebra on two generators

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    We investigate the structure of the automorphism of O2\mathcal{O}_{2} which exchanges the two canonical isometries. Our main observation is that the fixed point C*-subalgebra for this action is isomorphic to O2\mathcal{O}_{2} and we detail the relationship between the crossed-product and fixed point subalgebra.Comment: 14 Pages. Minor changes and additions to the references sectio
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