Metod Saniga

Three-qubit operators, the split Cayley hexagon of order two, and black holes

The set of 63 real generalized Pauli matrices of three-qubits can be factored into two subsets of 35 symmetric and 28 antisymmetric elements. This splitting is shown to be completely embodied in the properties of the Fano plane; the elements of the former set being in a bijective correspondence with the 7 points, 7 lines and 21 flags, whereas those of the latter set having their counterparts in 28 anti-flags of the plane. This representation naturally extends to the one in terms of the split Cayley hexagon of order two. 63 points of the hexagon split into 9 orbits of 7 points (operators) each under the action of an automorphism of order 7. 63 lines of the hexagon carry three points each and represent the triples of operators such that the product of any two gives, up to a sign, the third one. Since this hexagon admits a full embedding in a projective 5-space over GF(2), the 35 symmetric operators are also found to answer to the points of a Klein quadric in such space. The 28 antisymmetric matrices can be associated with the 28 vertices of the Coxeter graph, one of two distinguished subgraphs of the hexagon. The P SL2(7) subgroup of the automorphism group of the hexagon is discussed in detail and the Coxeter sub-geometry is found to be intricately related to the E7-symmetric black-hole entropy formula in string theory. It is also conjectured that the full geometry/symmetry of the hexagon should manifest itself in the corresponding black-hole solutions. Finally, an intriguing analogy with the case of Hopf sphere fibrations and a link with coding theory are briefly mentioned.

Mutually unbiased bases and finite projective planes

It is conjectured that the question of the existence of a set of d +1 mutu ally unbiased bases in a d-dimensional Hilbert space if d differs from a power of ap rimenumber is intimately linked with the problem of whether there exist projective planes whose order d is not a power of a prime number.

Comparing eclipse observations of the 2008 August 1 solar corona with an MHD model prediction

Context. The structure of the white-light and emission solar coronas and their MHD modelling are the context of our work. Aims. A comparison is made between the structure of the solar corona as observed during the 2008 August 1 total eclipse from Mongolia and that predicted by an MHD model. Methods. The model has an improved energy formulation, including the effect of coronal heating, conduction of heat parallel to the magnetic field, radiative losses, and acceleration by Alfven waves. Results. The white-light corona, which was visible up to 20 solar radii, was of an intermediate type with well-pronounced helmet streamers situated above a chain of prominences at position angles of 48, 130, 241, and 322 degrees. Two polar coronal holes, filled with a plethora of thin polar plumes, were observed. High-quality pictures of the green (530.3 nm, Fe XIV) corona were obtained with the help of two narrow-passband filters (centered at the line itself and the vicinity of 529.1 nm background), with a FWHM of 0.15 nm. Conclusions. The large-scale shape of both the white-light and green corona was found to agree well with that predicted by the model. In this paper we describe the morphological properties of the observed corona, and how it compares with that predicted by the model. A more detailed analysis of the quantitative properties of the corona will be addressed in a future publication.

The 2008 August 1 Eclipse Solar-Minimum Corona Unraveled

We discuss the results stemming from observations of the white-light and [Fe XIV] emission corona during the total eclipse of the Sun of 2008 August 1, in Mongolia (Altaj region) and in Russia (Akademgorodok, Novosibirsk, Siberia). Corresponding to the current extreme solar minimum, the white-light corona, visible up to 20 solar radii, was of a transient type with well pronounced helmet streamers situated above a chain of prominences at position angles 48°, 130°, 241°, and 322°. A variety of coronal holes, filled with a number of thin polar plumes, were seen around the poles. Furthering an original method of image processing, stars up to 12 mag, a Kreutz-group comet (C/2008 O1) and a coronal mass ejection (CME) were also detected, with the smallest resolvable structures being of, and at some places even less than, 1 arcsec. Differences, presumably motions, in the corona and prominences are seen even with the 19 minutes time difference between our sites. In addition to the high-resolution coronal images, which show the continuum corona (K-corona) that results from electron scattering of photospheric light, images of the overlapping green-emission-line (530.3 nm, [Fe XIV]) corona were obtained with the help of two narrow-passband filters (centered on the line itself and for the continuum in the vicinity of 529.1 nm, respectively), each with an FWHM of 0.15 nm. Through solar observations, on whose scheduling and details we consulted, with the Solar and Heliospheric Observatory, Hinodes XRT and SOT, Transition Region and Coronal Explorer, and STEREO, as well as Wilcox Solar Observatory and Solar and Heliospheric Observatory/Michelson Doppler Imager magnetograms, we set our eclipse observations in the context of the current unusually low and prolonged solar minimum.

PROJECTIVE RING LINE ENCOMPASSING TWO-QUBITS

We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF(2) × GF(2) (the “Mermin” part) and two lines over GF(4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications.

Quantum Entanglement and Projective Ring Geometry

The paper explores the basic geometrical properties of the observables charac- terizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum en- tanglement in such systems, we demonstrate that the 15 ◊ 15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of n copies of the Galois field GF(2), with n = 2,3 and 4.

Polar Plume Brightening During the 2006 March 29 Total Eclipse

We discuss a remarkable brightening in a polar plume, as inferred from unique coordinated observations of the white-light corona during the total eclipse of the Sun of 2006 March 29. The polar plume (also known as a polar ray, with distinctions that we discuss) was observed at the positional angle of 9°; the velocity at which the brightening propagated was about 65 km s−1, which is close to the values derived by modeling of mass/energy transfer in polar plumes/rays as well as to those acquired from images from the Extreme-ultraviolet Imaging Telescope on the European Space Agency/NASA Solar and Heliospheric Observatory (SOHO/EIT). Comparing our data with those from the SOHO/LASCO C2 coronagraph, we estimate the lifetime of the polar ray to be less than 24 hr.

Hjelmslev geometry of mutually unbiased bases

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p2 and rank r. The q vectors of a basis of correspond to the q points of a (so-called) neighbour class and the q + 1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.

STRUCTURE AND DYNAMICS OF THE 2010 JULY 11 ECLIPSE WHITE-LIGHT CORONA

The white-light corona (WLC) during the total solar eclipse on 2010 July 11 was observed by several teams in the Moons shadow stretching across the Pacific Ocean and a number of isolated islands. We present a comparison of the WLC as observed by eclipse teams located on the Tatakoto Atoll in French Polynesia and on Easter Island, 83?minutes later, combined with near-simultaneous space observations. The eclipse was observed at the beginning of the solar cycle, not long after solar minimum. Nevertheless, the solar corona shows a plethora of different features (coronal holes, helmet streamers, polar rays, very faint loops and radial-oriented thin streamers, a coronal mass ejection, and a puzzling curtain-like object above the north pole). Comparing the observations from the two sites enables us to detect some dynamic phenomena. The eclipse observations are further compared with a hairy-ball model of the magnetic field and near-simultaneous images from the Atmospheric Imaging Assembly on NASAs Solar Dynamics Observatory, the Extreme Ultraviolet Imager on NASAs Solar Terrestrial Relations Observatory, the Sun Watcher, using Active Pixel System Detector and Image Processing on ESAs PRoject for Onboard Autonomy, and the Naval Research Laboratorys Large Angle and Spectrometric Coronagraph on ESAs Solar and Heliospheric Observatory. The Ludendorff flattening coefficient is 0.156, matching the expected ellipticity of coronal isophotes at 2 R ?, for this rising phase of the solar-activity cycle.

Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators

Abstract The commutation relations of the generalized Pauli operators of a qubit–qutrit system are discussed in the newly established graph-theoretic and finite-geometrical settings. The dual of the Pauli graph of this system is found to be isomorphic to the projective line over the product ring