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As a means of exactly derandomizing certain quantum information processing tasks, unitary designs have become an important concept in quantum information theory. A unitary design is a collection of unitary matrices that approximates the entire unitary group, much like a spherical design approximates the entire unit sphere. We use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. The tightness of these bounds is then considered, where specific unitary 2-designs are introduced that are analogous to SIC-POVMs and complete sets of MUBs in the complex projective case. Additionally, we catalogue the known constructions of unitary t-designs and give an upper bound on the size of the smallest weighted unitary t-design in U(d). This is joint work with Aidan Roy (Calgary): \'Unitary designs and codes,\' arXiv:0809.3813.

Abstract: Adapting the concept of Wigner functions to finite dimensional systems is no simple matter. Basic issues with existing proposals are that they are either over-complete in the sense that the degrees of freedom in the Wigner function does not match the degrees of freedom in the original density matrix, or that they work only for restricted dimensions, namely for prime powers. We propose a new way to define a Wigner function and associated quantum phase space for some non-prime power dimensions. This allows to perform quantum state tomography from the measurement statistics of a minimal number of yes/no measurements.

By definition, SIC-POVMs are symmetric in the sense that the magnitude of the inner product between any pair of vectors is constant. All known constructions are based on additional symmetries, mainly with respect to the Weyl-Heisenberg group. Analyzing solutions for small dimensions, Zauner has identified an additional symmetry of order three and conjectured that these symmetries can be used to construct SIC-POVMs for all dimensions. Appleby has confirmed that all numerical solutions of Renes et al. indeed have that additional symmetry. This leads to the main questions addressed in the talk: Do all SIC-POVMs necessarily possess these symmetries, or can we construct SIC-POVMs without or with other symmetries?

One of the most challenging problems in theoretical physics today is the so called cosmological constant problem. While current observational constraints are consistent with the predictions of GR with a tiny cosmological constant, often referred to as the dark energy, it remains possible that it\'s the deviation of the law of gravity at large distance from Einstein\'s theory that resolves the puzzle. In this talk, I will briefly review some of the theoretical attempts made along this line, including the simple massive gravity, large extra dimensions, Unimodular gravity, classically constrained gravity, as well as their difficulties. I will then focus on some most recent study on the theory of massive graviton in de Sitter space, which may be more closely related to the reality both today and during the inflationary epoch. In particular, I would describe a model, in which one is able to open up the forbidden mass range of the graviton on a de Sitter background discovered by Higuchi.

In an asymptotically anti-de Sitter space, three-dimensional topologically massive gravity has some remarkable properties, which suggest interesting applications to quantum gravity. Unfortunately, though, the theory appears to be unstable, even at the special \'chiral\' value of the coupling. I will discuss recent work, and recent controversies, in this field.