Hyperbolic reflections as fundamental building blocks for multilayer optics

We reelaborate on the basic properties of lossless multilayers by using bilinear transformations. We study some interesting properties of the multilayer transfer function in the unit disk, showing that hyperbolic geometry turns out to be an essential tool for understanding multilayer action. We use a simple trace criterion to classify multilayers into three classes that represent rotations, translations, or parallel displacements. Moreover, we show that these three actions can be decomposed as a product of two reflections in hyperbolic lines. Therefore, we conclude that hyperbolic reflections can be considered as the basic pieces for a deeper understanding of multilayer optics.Comment: 7 pages, 7 figures, accepted for publication in J. Opt. Soc. Am.

General unit-disk representation for periodic multilayers

We suggest a geometrical framework to discuss periodic layered structures in the unit disk. The band gaps appear when the point representing the system approaches the unit circle. We show that the trace of the matrix describing the basic period allows for a classification in three families of orbits with quite different properties. The laws of convergence of the iterates to the unit circle can be then considered as universal features of the reflection.Comment: 3 pages, 2 eps-figures. To be published in Optics Letter

Simple trace criterion for classification of multilayers

The action of any lossless multilayer is described by a transfer matrix that can be factorized in terms of three basic matrices. We introduce a simple trace criterion that classifies multilayers in three classes with properties closely related with one (and only one) of these three basic matrices.Comment: To be published in Optics Letter

On the structure of the sets of mutually unbiased bases for N qubits

For a system of N qubits, spanning a Hilbert space of dimension d=2^N, it is known that there exists d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive the four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of sixteen sets of bases,and show some of them, and their interrelations, as examples. The extension of the method to the general case of N qubits is outlined.Comment: 16 pages, 10 tables, 1 figur

Assessing the Polarization of a Quantum Field from Stokes Fluctuation

We propose an operational degree of polarization in terms of the variance of the projected Stokes vector minimized over all the directions of the Poincar\'e sphere. We examine the properties of this degree and show that some problems associated with the standard definition are avoided. The new degree of polarization is experimentally determined using two examples: a bright squeezed state and a quadrature squeezed vacuum.Comment: 4 pages, 2 figures. Comments welcome

Graph states in phase space

The phase space for a system of $n$ qubits is a discrete grid of $2^{n} \times 2^{n}$ points, whose axes are labeled in terms of the elements of the finite field \Gal{2^n} to endow it with proper geometrical properties. We analyze the representation of graph states in that phase space, showing that these states can be identified with a class of non-singular curves. We provide an algebraic representation of the most relevant quantum operations acting on these states and discuss the advantages of this approach.Comment: 14 pages. 2 figures. Published in Journal of Physics