Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

We consider questions posed in a recent paper of Mandayam et al. (2014) on the nature of unextendible mutually unbiased bases. We describe a conceptual framework to study these questions, using a connection proved by the author in Thas (2009) between the set of nonidentity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d a prime, and the geometry of non-degenerate alternating bilinear forms of rank N over finite fields F d We then supply alternative and short proofs of results obtained in Mandayam et al. (2014), as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of Mandayam et al. (2014) and speculate on variations of this conjecture

Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings

In this paper, we show that the building at infinity of a two-dimensional affine R-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type G~_2, and it reproves the results obtained mainly by the second author for types A~_2 and C~_2. The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional R-building)

SU(2) nonstandard bases: the case of mutually unbiased bases

This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU(2) corresponding to an irreducible representation of SU(2). The representation theory of SU(2) is reconsidered via the use of two truncated deformed oscillators. This leads to replace the familiar scheme {j^2, j_z} by a scheme {j^2, v(ra)}, where the two-parameter operator v(ra) is defined in the enveloping algebra of the Lie algebra su(2). The eigenvectors of the commuting set of operators {j^2, v(ra)} are adapted to a tower of chains SO(3) > C(2j+1), 2j integer, where C(2j+1) is the cyclic group of order 2j+1. In the case where 2j+1 is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases. Some useful relations on generalized quadratic Gauss sums are exposed in three appendices.Comment: 33 pages; version2: rescaling of generalized Hadamard matrices, acknowledgment and references added, misprints corrected; version 3: published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ (22 pages

Characterisations and classifications in the theory of parapolar spaces

This thesis in incidence geometry is divided into two parts, which can both be linked to the geometries of the Freudenthal-Tits magic square. The first and main part consists of an axiomatic characterisation of certain plane geometries, defined via the Veronese mapping using degenerate quadratic alternative algebras (over any field) with a radical that is (as a ring) generated by a single element. This extends and complements earlier results of Schillewaert and Van Maldeghem, who considered such geometries over non-degenerate quadratic alternative algebras. The second and smaller part deals with a classification of parapolar spaces exhibiting the feature that the dimensions of intersections of pairs of symplecta cannot take all possible sensible values, with the only further requirement that, if the parapolar spaces have symplecta of rank 2, then they are strong. This part is based on a joint work with Schillewaert, Van Maldeghem and Victoor

Finite Mathematics, Finite Quantum Theory and Applications to Gravity and Particle Theory

We argue that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete and even finite, is described by classical mathematics involving the notions of infinitely small, continuity etc. Moreover, since classical mathematics has its own foundational problems which cannot be resolved (as follows, in particular , from Gödel's incompleteness theorems), the ultimate physical theory cannot be based on that mathematics. In the first part of the work we discuss inconsistencies in standard quantum theory and reformulate the theory such that it can be naturally generalized to a formulation based on finite mathematics. It is shown that: a) as a consequence of inconsistent definition of standard position operator, predictions of the theory contradict the data on observations of stars; b) the cosmological acceleration and gravity can be treated simply as kinematical manifestations of quantum de Sitter symmetry, i.e. the cosmological constant problem does not exist, and for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed. In the second part we first prove that classical mathematics is a special degenerate case of finite mathematics in the formal limit when the characteristic p of the field or ring in the latter goes to infinity. This result fundamentally changes the standard paradigm on what mathematics and what physics are the most fundamental. Then we consider a quantum theory based on finite mathematics with a large p. In this approach the de Sitter gravitational constant depends on p and disappears in the formal limit p → ∞, i.e. gravity is a consequence of finiteness of nature. The application to particle theory gives that the notion of a particle and its antiparticle is only approximate and, as a consequence: a) the electric charge and the baryon and lepton quantum numbers can be only approximately conserved; b) particles which in standard theory are treated as neutral (i.e. coinciding with their antiparticles) cannot be elementary. We argue that only Dirac singletons can be true elementary particles and the main reasons are: a) massless and massive particles can be constructed from singletons; b) while irreducible representations describing massless and massive particles are necessarily over a field, irreducible representations describing singletons can be constructed over a ring. Finally we discuss a conjecture that classical time t manifests itself as a consequence of the fact that p changes, i.e. p and not t is the true evolution parameter