No abstract

Structure of Classical Groups 69 2.1 Semisimple Elements 69 2.1.1 Toral Groups 70 2.1.2 Maximal Torus in a Classical Group 72 2.1.3 Exercises 76 2.2 Unipotent Elements 77 2.2.1 Low-Rank Examples 77 2.2.2 Unipotent Generation of Classical Groups 2.2.3 Connected Groups 81 2.2.4 Exercises 2.3 Regular Representations of SL(2, C) 2.3.1 Irreducible Representations of .5((2, C) 2.3.2 Irreducible Regular Representations of SL(2, C) 2.3.3 Complete Reducibility of SL(2, (C) 2.3.4 Exercises 2.4 The Adjoint Representation 2.4.1 Roots with Respect to a Maximal Torus 2.4.2 Commutation Relations of Root Spaces 95 2.4.3 Structure of Classical Root Systems 99 2.4.4 Irreducibility of the Adjoint Representation 2.4.5 Exercises 2.5 Semisimple Lie Algebras 2.5.1 Solvable Lie Algebras 2.5.2 Root Space Decomposition 2.5.3 Geometry of Root Systems 2.5.4 Conjugacy of Cartan Subalgebras 2.5.5 Exercises x Contents

We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin-1 2 particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate the extent to which it quantifies global entanglement. We also apply it to track the evolution of entanglement during a quantum computation.

No abstract

Let ∆ n−1 denote the (n − 1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of ∆ n−1 obtained by starting with the full (k − 1)-dimensional skeleton of ∆ n−1 and then adding each k-simplex independently with probability p. Let H k−1 (Y ; R) denote the (k − 1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ 1 and for any function ω(n) that tends to infinity