Pairs Of Orthonormal Bases Research Articles

On a class of weak R-duals and the duality relations

The concept of R-duals of a sequence was first introduced with the motivation to obtain a general version of duality principle in Gabor analysis. Since then, various R-duals (types II, III, IV) and some relaxations of the R-dual setup have been introduced and studied by some mathematicians. All these “R-duals” provide a powerful tool in the analysis of duality relations in general frame theory. It is of independent interest in mathematics and far beyond the duality principle in Gabor analysis. Observe that the underlying sequences of a R-dual are a pair of orthonormal bases. In this paper we introduce the concept of weak R-duals based on a pair of Parseval frames. It is a new relaxation of the R-dual setup. We obtain a characterization of frames based on their weak R-duals, and prove that the weak R-dual of a frame (Riesz basis) is a frame sequence (frame). We also characterize (unitarily) equivalent frames in terms of weak R-duals. Finally, we present an explicit expression of the canonical duals of weak R-duals.

Graph-state formalism for mutually unbiased bases

A pair of orthonormal bases is called mutually unbiased if all mutual overlaps between any element of one basis and an arbitrary element of the other basis coincide. In case the dimension, $d$, of the considered Hilbert space is a power of a prime number, complete sets of $d+1$ mutually unbiased bases (MUBs) exist. Here we present a method based on the graph-state formalism to construct such sets of MUBs. We show that for $n$ $p$-level systems, with $p$ being prime, one particular graph suffices to easily construct a set of ${p}^{n}+1$ MUBs. In fact, we show that a single $n$-dimensional vector, which is associated with this graph, can be used to generate a complete set of MUBs and demonstrate that this vector can be easily determined. Finally, we discuss some advantages of our formalism regarding the analysis of entanglement structures in MUBs, as well as experimental realizations.

A generalized uncertainty principle and sparse representation in pairs of bases

An elementary proof of a basic uncertainty principle concerning pairs of representations of R/sup N/ vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pairs of orthonormal bases as overcomplete dictionaries. The main contribution in this paper is the improvement of an important result due to Donoho and Huo (2001) concerning the replacement of the l/sub 0/ optimization problem by a linear programming (LP) minimization when searching for the unique sparse representation.