Unextendible product bases and the construction of inseparable states

Let H[N] denote the tensor product of n finite dimensional Hilbert spaces H(r). A state |phi> of H[N] is separable if |phi> is the tensor product of states in the respective product spaces. An orthogonal unextendible product basis is a finite set B of separable orthonormal states |phi(k)> such that the non-empty space B9perp), the set of vectors orthogonal to B, contains no separable projection. Examples of orthogonal UPB sets were first constructed by Bennett et al [1] and other examples appear, for example, in [2] and [3]. If F denotes the set of convex combinations of the projections |phi(k)><phi(k)|, then F is a face in the set S of separable densities. In this note we show how to use F to construct families of positive partial transform states (PPT) which are not separable. We also show how to make an analogous construction when the condition of orthogonality is dropped. The analysis is motivated by the geometry of the faces of the separable states and leads to a natural construction of entanglement witnesses separating the inseparable PPT states from S.Comment: to appear in Lin. Alg. App

Comparison theorems for deformation functors via invariant theory

We compare deformations of algebras to deformations of schemes in the setting of invariant theory. Our results generalize comparison theorems of Schlessinger and the second author for projective schemes. We consider deformations (abstract and embedded) of a scheme $X$ which is a good quotient of a quasi-affine scheme $X^\prime$ by a linearly reductive group $G$ and compare them to invariant deformations of an affine $G$-scheme containing $X^\prime$ as an open invariant subset. The main theorems give conditions for when the comparison morphisms are smooth or isomorphisms.Comment: Minor improvements and corrections. Improved presentation. Changed some definitions and term

Cavefish Population Status and Environmental Quality in Cave Springs Cave, Arkansas - Final Report submitted to Arkansas Natural Heritage Commission

This report summarizes the continuing effort to monitor environmental quality in the Cave Springs Cave Natural Area and to implement the Ozark Cavefish Recovery Plan. Last year’s report (Brown et al., 1998) identified certain environmental stressors, including a trend over 15 years of increasing nutrient pollution, a low cavefish population count of only 106, and the presence of heavy metals in the cave water and one semi-volatile organic compound (the phthalate DEHP at 500 ppb) in resident crayfish tissue. This year’s monitoring effort demonstrates that fecal coliforms continue to exceed Arkansas State Water Quality Standards (Regulation 2), sometimes by a factor of 1000. The presence of heavy metals is confirmed, in not only the cave water and sediments, but in crayfish tissue, which implies that it may be bioaccumulating in the cave foodweb. Furthermore, beryllium, copper, lead, selenium, and zinc are present in concentrations in the cave water that exceeded the Regulation 2 standards for chronic, and sometimes acute, toxicity to aquatic life. Significant amounts of nitrate are also present (with a yearly average of over 5 mg NO3-N/ L), and phosphate concentrations occasionally exceed Regulation 2 standards. Concentrations of nitrate, ortho-phosphate, total phosphate, total coliforms, and several dissolved metals (Al, Ba, Cu, Fe, and Pb) were all highly correlated with discharge, and concentrations were highest during stormflow events. No pesticides were detected in cave water, crayfish tissue or bat guano. Phthalates were again detected in crayfish tissue (DEP and DEHP at 1 ppb each), as well as the cave water (DEHP at 0.7 ppb). While the effects of these phthalates upon aquatic organisms are unknown, the U. S. Environmental Protection Agency considers phthalates to be human carcinogens and hormone disrupters. Ironically, while the cavefish habitat appears to be quite polluted, this year’s population survey counted 166 Ozark cavefish, the most ever counted for this (or any other) habitat. In order to integrate these pollution concerns and other data about this cave complex, a geographic information system was created for the Cave Springs Cave recharge zone. Preliminary analyses have detected several sensitive areas and pollution sources. The cave complex was determined to extend outside of the Natural Area boundary, and several sinkholes were identified. Photo-lineaments and fracture traces were identified, and other studies in Benton County demonstrate that these features, commonly expressed as intermittent streams on the surface, allow surface pollutants to rapidly enter the groundwater. Protection of these adjacent lands, sinkholes, and streams is recommended. The reduction or cessation of the land application of sewage sludge and agricultural waste in the recharge zone is also recommended

Complete Separability and Fourier representations of n-qubit states

Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and sufficient conditions for full separability for a particular set of $n$-qubit states whose densities all satisfy the Peres condition

The Signature of Proper Motion in the Microwave Sky

The cosmic microwave background radiation defines a preferred cosmic rest frame, and inflationary cosmological theories predict that the microwave background temperature fluctuations should be statistically isotropic in this rest frame. For observers moving with respect to the rest frame, the temperature fluctuations will no longer be isotropic, due to the preferred direction of motion. The most prominent effect is a dipole temperature variation, which has long been observed with an amplitude of a part in a thousand of the mean temperature. An observer's velocity with respect to the rest frame will also induce changes in the angular correlation function and creation of non-zero off-diagonal correlations between multipole moments. We calculate both of these effects, which are part-in-a-thousand corrections to the rest frame power spectrum and correlation function. Both should be detectable in future full-sky microwave maps from the Planck satellite. These signals will constrain cosmological models in which the cosmic dipole arises partly from large-scale isocurvature perturbations, as suggested by recent observations.Comment: 5 pages, no figures. Submitted to Physical Review Letter

Mutually Unbiased Bases, Generalized Spin Matrices and Separability

A collection of orthonormal bases for a complex dXd Hilbert space is called mutually unbiased (MUB) if for any two vectors v and w from different bases the square of the inner product equals 1/d: || ^{2}=1/d. The MUB problem is to prove or disprove the the existence of a maximal set of d+1 bases. It has been shown in [W. K. Wootters, B. D. Fields, Annals of Physics, 191, no. 2, 363-381, (1989)] that such a collection exists if d is a power of a prime number p. We revisit this problem and use dX d generalizations of the Pauli spin matrices to give a constructive proof of this result. Specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the MUB problem. Additionally we give formulas from which the orthogonal bases can be readily computed. We show how the techniques developed here provide a natural way to analyze the separability of the bases. The techniques used require properties of algebraic field extensions, and the relevant part of that theory is included in an Appendix

Model Predictive Regulation

We show how optimal nonlinear regulation can be achieved in a model predictive control fashion