A Note on Pseudo-reflections

In this note, we show that if V is a finite dimensional vector space equipped with a non-degenerate bilinear form, and one has a set of pseudo-reflections on V, preserving the form and having no non-zero common fixed vector, then the group G generated by this set is ‘sufficiently large’ in the sense that for every linear transformation T : V → V, there exists an element g ∈ G such that g − G is invertible

The BMM symmetrising trace conjecture for the exceptional 2-reflection groups of rank 2

We prove the symmetrising trace conjecture of Brou\'e, Malle and Michel for the generic Hecke algebra associated to the exceptional irreducible complex reflection group $G_{13}$. Our result completes the proof of the conjecture for the exceptional 2-reflection groups of rank 2.Comment: 17 page

A categorification of non-crossing partitions

We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix together with a particular choice of a Coxeter element. Examples arise from Grothendieck groups of hereditary artin algebras.Comment: 34 pages. Substantially revised and final version, accepted for publication in Journal of the European Mathematical Societ

Reflections on the Role of Entanglement in the Explanation of Quantum Computational Speedup

Of the many and varied applications of quantum information theory, perhaps the most fascinating is the sub-field of quantum computation. In this sub-field, computational algorithms are designed which utilise the resources available in quantum systems in order to compute solutions to computational problems with, in some cases, exponentially fewer resources than any known classical algorithm. While the fact of quantum computational speedup is almost beyond doubt, the source of quantum speedup is still a matter of debate. In this paper I argue that entanglement is a necessary component for any explanation of quantum speedup and I address some purported counter-examples that some claim show that the contrary is true. In particular, I address Biham et al.'s mixed-state version of the Deutsch-Jozsa algorithm, and Knill \& Laflamme's deterministic quantum computation with one qubit (DQC1) model of quantum computation. I argue that these examples do not demonstrate that entanglement is unnecessary for the explanation of quantum speedup, but that they rather illuminate and clarify the role that entanglement does play