A 3x3 dilation counterexample

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 $3\times3$ 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

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

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

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 $p_1, \dots, p_m$ as long as $2(p_1, \dots, p_m)$ 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

Symmetry in the Cuntz Algebra on two generators

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

A method to find quantum noiseless subsystems

We develop a structure theory for decoherence-free subspaces and noiseless subsystems that applies to arbitrary (not necessarily unital) quantum operations. The theory can be alternatively phrased in terms of the superoperator perspective, or the algebraic noise commutant formalism. As an application, we propose a method for finding all such subspaces and subsystems for arbitrary quantum operations. We suggest that this work brings the fundamental passive technique for error correction in quantum computing an important step closer to practical realization.Comment: 5 pages, to appear in Physical Review Letter

Every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues

We show that for every invertible n x n complex matrix A there is an n x n diagonal invertible D such that AD has distinct eigenvalues. Using this result, we affirm a conjecture of Feng, Li, and Huang that an is x is matrix is not diagonally equivalent to a matrix with distinct eigenvalues if and only if it is singular and all its principal minors of size n - 1 are zero. (C) 2011 Elsevier Inc. All rights reserved