The Classical notions of Computation are rapidly entering Quantum Computation. People try to find Quantum counterparts for Classical concepts. In this essay, we will discuss about classical t-design, the quantum counterpart, Mutual Unbiased bases and a new technique for evaluating the expectation of a polynomial over Haar measure.

The latter, by definition, means that the Lie algebra of H is the span of eigenspaces of Ad(gt), t > 0, with eigenvalues bigger than 1 in absolute value. The space X def = G/Γ can be identified with the space of unimodular lattices in R, on which G acts by left translations. Denote by π the natural projection G → X, g 7→ gΓ, and for any z ∈ X let πz : G → X be defined by πz(g) = gz. Also denote...

Recall that a locally compact group G is called unimodular if the left Haar measure on G is equal to the right one. It is proved in this paper that G is unimodular iff it is approximable by finite quasigroups (Latin squares).

We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.

Let G be a connected Lie group with Lie algebra g and a 1 ; : : : ; a d 0 an algebraic basis of g. Further let A i denote the generators of left translations, acting on the L p-spaces L p (G ; dg) formed with left Haar measure dg, in the directions a i. We consider second-order operators

We investigate rigidity of measurable structure for higher rank abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a 0 entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.

We study the dimension of “random” Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from [LMS], to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much “weaker” randomness of “diagonal” subspaces (Corollary 1.4 and explanation after). We also add some relative information on “p...

In this paper we systematically define Haar-like wavelets over a tetrahedrical grid which is generated b y a regular subdivision method. These wavelets form an inconditional basis for LJ'(T,C,p), 1 < p < 00 , being p the Lebesgue measure and C the U-algebra of all tetrahedra generated from a tetrahedron T by the chosen subdivision method.

Face detection is a fundamental research area in computer vision field. Most of the face-related applications such as face recognition and face tracking assume that the face region is perfectly detected. To adopt a certain face detection algorithm in these applications, evaluation of its performance is needed. Unfortunately, it is difficult to evaluate the performance of face detection algorith...