Coarse-grained entanglement classification through orthogonal arrays

Classification of entanglement in multipartite quantum systems is an open problem solved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of $N$ subsystems with an arbitrary number of internal levels each, based on properties of orthogonal arrays with $N$ columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four and five qubits, as well as heterogeneous tripartite systems consisting of two qubits and one qutrit or one qubit and two qutrits.Comment: 38 pages, 1 figur

Aberration in qualitative multilevel designs

Generalized Word Length Pattern (GWLP) is an important and widely-used tool for comparing fractional factorial designs. We consider qualitative factors, and we code their levels using the roots of the unity. We write the GWLP of a fraction ${\mathcal F}$ using the polynomial indicator function, whose coefficients encode many properties of the fraction. We show that the coefficient of a simple or interaction term can be written using the counts of its levels. This apparently simple remark leads to major consequence, including a convolution formula for the counts. We also show that the mean aberration of a term over the permutation of its levels provides a connection with the variance of the level counts. Moreover, using mean aberrations for symmetric $s^m$ designs with $s$ prime, we derive a new formula for computing the GWLP of ${\mathcal F}$. It is computationally easy, does not use complex numbers and also provides a clear way to interpret the GWLP. As case studies, we consider non-isomorphic orthogonal arrays that have the same GWLP. The different distributions of the mean aberrations suggest that they could be used as a further tool to discriminate between fractions.Comment: 16 pages, 1 figur

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Fresnel Interferometric Imager: ground-based prototype

The Fresnel Interferometric Imager is a space-based astronomical telescope project yielding milli-arc second angular resolution and high contrast images with loose manufacturing constraints. This optical concept involves diffractive focusing and formation flying: a first "primary optics" space module holds a large binary Fresnel Array, and a second "focal module" holds optical elements and focal instruments that allow for chromatic dispersion correction. We have designed a reduced-size Fresnel Interferometric Imager prototype and made optical tests in our lab, in order to validate the concept for future space missions. The Primary module of this prototype consists of a square, 8 cm side, 23 m focal length Fresnel array. The focal module is composed of a diaphragmed small telescope used as "field lens", a small cophased diverging Fresnel Zone Lens (FZL) that cancels the dispersion and a detector. An additional module collimates the artificial targets of various shapes, sizes and dynamic ranges to be imaged. In this paper, we describe the experimental setup, different designs of the primary Fresnel array, and the cophased Fresnel Zone Lens that achieves rigorous chromatic correction. We give quantitative measurements of the diffraction limited performances and dynamic range on double sources. The tests have been performed in the visible domain, lambda = 400 - 700 nm. In addition, we present computer simulations of the prototype optics based on Fresnel propagation, that corroborate the optical tests. This numerical tool has been used to simulate the large aperture Fresnel arrays that could be sent to space with diameters of 3 to 30 m, foreseen to operate from Lyman-alpha (121 nm) to mid I.R. (25 microns).Comment: 10 pages, 13 figures; accepted for publication in Applied Optic

Orthogonal methods based ant colony search for solving continuous optimization problems

Research into ant colony algorithms for solving continuous optimization problems forms one of the most significant and promising areas in swarm computation. Although traditional ant algorithms are designed for combinatorial optimization, they have shown great potential in solving a wide range of optimization problems, including continuous optimization. Aimed at solving continuous problems effectively, this paper develops a novel ant algorithm termed "continuous orthogonal ant colony" (COAC), whose pheromone deposit mechanisms would enable ants to search for solutions collaboratively and effectively. By using the orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and e±ciently. By implementing an "adaptive regional radius" method, the proposed algorithm can reduce the probability of being trapped in local optima and therefore enhance the global search capability and accuracy. An elitist strategy is also employed to reserve the most valuable points. The performance of the COAC is compared with two other ant algorithms for continuous optimization of API and CACO by testing seventeen functions in the continuous domain. The results demonstrate that the proposed COAC algorithm outperforms the others