John Holbrook

Geometry of higher-rank numerical ranges

We consider geometric aspects of higher-rank numerical ranges for arbitrary N  × N matrices. Of particular interest is the issue of convexity and a possible extension of the Toeplitz–Hausdorff Theorem. We derive a number of reductions and obtain partial results for the general problem. We also conduct graphical and computational experiments. Added in proof: Following acceptance of this paper, our subject has developed rapidly. First, Hugo Woerdeman established convexity of the higher-rank numerical ranges by combining Proposition 2.4 and Theorem 2.12 with the theory of algebraic Riccati equations. See Woerdeman, H., 2007, The higher rank numerical range is convex, Linear and Multilinear Algebra, to appear. Subsequently Chi-Kwong Li and Nung-Sing Sze followed a different approach that not only yields convexity but also provides important additional insights. See Li, C.-K. and Sze, N.-S., 2007, Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations, preprint. See also Li, C.-K., Poon, Y.-T., and Sze, N.-S., 2007, Condition for the higher rank numerical range to be non-empty, preprint.

Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction

AbstractThe effect of noise on a quantum system can be described by a set of operators obtained from the interaction Hamiltonian. Recently it has been shown that generalized quantum error correcting codes can be derived by studying the algebra of this set of operators. This led to the discovery of noiseless subsystems. They are described by a set of operators obtained from the commutant of the noise generators. In this paper we derive a general method to compute the structure of this commutant in the case of unital noise. PACS: 03.67.–a, 03.67.Pp

Numerical shadow and geometry of quantum states

The totality of normalised density matrices of order N forms a convex setQN in N 2 1 . Working with the at geometry induced by the Hilbert{Schmidt distance we consider images of orthogonal projections of QN onto a two{plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one denes its numerical shadow as a probability distribution supported on its numerical range W (A), induced by the unitarily invariant Fubini{ Study measure on the complex projective manifold P N 1 . We dene generalized, mixed{states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.

Spectral variation of normal matrices

Abstract It has been a durable conjecture that the distance (appropriately defined) between the spectra of two normal matrices is bounded by the operator norm of their difference. We report on some numerical studies that show that this conjecture is, in general, false. In fact, even in the 3 x 3 case, the spectral distance may exceed the norm distance by between 1 and 2 percent. We also describe the background of the conjecture and offer an analysis of extremal counterexamples.

Approximating commuting operators

The problem of approximating m-tuples of commuting n×n complex matrices by commuting m-tuples of generic matrices is studied. We narrow the gap for commuting triples by showing that they can be perturbed if n 29.

Restricted numerical shadow and the geometry of quantum entanglement

The restricted numerical range WR(A) of an operator A acting on a D-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A—a normalized probability distribution on the complex plane supported in WR(A). Its value at point z ∈ C is equal to the probability that the expectation value � ψ|A|ψ� is equal to z, where |ψ� represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini–Study measure. Studying restricted shadows of operators of a fixed size D = NANB we analyse the geometry of sets of separable and maximally entangled states of the NA × NB composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow, we study the dynamics of quantum entanglement. A similar analysis extended for operators onD = 2 3 -dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ andW quantum states of a three-qubit system.

Unitary invariance and spectral variation

Abstract We call a norm on operators or matrices weakly unitarily invariant if its value at operator A is not changed by replacing A by U∗AU , provided only that U is unitary. This class includes such norms as the numerical radius. We extend to all such norms an inequality that bounds the spectral variation when a normal operator A is replaced by another normal B in terms of the arclength of any normal path from A to B , computed using the norm in question. Related results treat the local metric geometry of the “manifold” of normal operators. We introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball, and identify this correspondence explicitly in certain cases.

Schur norms and the multivariate von Neumann inequality

Starting from some classical counterexamples to the von Neumann inequality for several variables, we are led to especially simple examples of this phenomenon. We display three commuting 4-dimensional contractions Ck and a polynomial p(zi, z2, z3) such that

Distortion coefficients for cryptocontractions

\parallel p({C_{1}},{C_{2}},{C_{3}})\parallel = \tfrac{6}{5}\max \{ |p({z_{1}},{z_{2}},{z_{3}})| : |{z_{k}}| \leqslant 1\} .