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

Grassmann embeddings of polar Grassmannians

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians associated to a possibly degenerate Hermitian, alternating or quadratic form with possibly non-maximal Witt index. Moreover, in the characteristic $2$ case, when the form is quadratic and non-degenerate with bilinearization of minimal Witt index, we define a generalization of the so-called Weyl embedding (see [I. Cardinali and A. Pasini, Grassmann and Weyl embeddings of orthogonal Grassmannians. J. Algebr. Combin. 38 (2013), 863-888]) and prove that the Grassmann embedding is a quotient of this generalized "Weyl-like" embedding. We also estimate the dimension of the latter.Comment: 25 pages/revised version after revie

Quadrangles embedded in metasymplectic spaces

During the final steps in the classification of the Moufang quadrangles by Jacques Tits and Richard Weiss a new class of Moufang quadrangles unexpectedly turned up. Subsequently Bernhard Muhlherr and Hendrik Van Maldeghem showed that this class arises as the fixed points and hyperlines of certain involutions of a metasymplectic space (or equivalently a building of type F_4). In the same paper they also showed that other types of Moufang quadrangles can be embedded in a metasymplectic space as points and hyperlines. In this paper, we reverse the question: given a (thick) quadrangle embedded in a metasymplectic space as points and hyperlines, when is such a quadrangle a Moufang quadrangle

A geometric approach to alternating kk-linear forms

Given an $n$-dimensional vector space $V$ over a field $\mathbb K$, let $2\leq k < n$. There is a natural correspondence between the alternating $k$-linear forms $\varphi$ of $V$ and the linear functionals $f$ of $\bigwedge^kV$. Let $\varepsilon_k:{\mathcal G}_k(V)\rightarrow {\mathrm{PG}}(\bigwedge^kV)$ be the Plucker embedding of the $k$-Grassmannian ${\mathcal G}_k(V)$ of $V$. Then $\varepsilon_k^{-1}(\ker(f)\cap\varepsilon_k(\mathcal{G}_k(V)))$ is a hyperplane of the point-line geometry ${\mathcal G}_k(V)$. All hyperplanes of ${\mathcal G}_k(V)$ can be obtained in this way. For a hyperplane $H$ of ${\mathcal G}_k(V)$, let $R^\uparrow(H)$ be the subspace of ${\mathcal G}_{k-1}(V)$ formed by the $(k-1)$-subspaces $A\subset V$ such that $H$ contains all $k$-subspaces that contain $A$. In other words, if $\varphi$ is the (unique modulo a scalar) alternating $k$-linear form defining $H$, then the elements of $R^\uparrow(H)$ are the $(k-1)$-subspaces $A = \langle a_1,\ldots, a_{k-1}\rangle$ of $V$ such that $\varphi(a_1,\ldots, a_{k-1},x) = 0$ for all $x\in V$. When $n-k$ is even it might be that $R^\uparrow(H) = \emptyset$. When $n-k$ is odd, then $R^\uparrow(H) \neq \emptyset$, since every $(k-2)$-subspace of $V$ is contained in at least one member of $R^\uparrow(H)$. If every $(k-2)$-subspace of $V$ is contained in precisely one member of $R^\uparrow(H)$ we say that $R^\uparrow(H)$ is spread-like. In this paper we obtain some results on $R^\uparrow(H)$ which answer some open questions from the literature and suggest the conjecture that, if $n-k$ is even and at least $4$, then $R^\uparrow(H) \neq \emptyset$ but for one exception with ${\mathbb K}\leq{\mathbb R}$ and $(n,k) = (7,3)$, while if $n-k$ is odd and at least $5$ then $R^\uparrow(H)$ is never spread-like.Comment: 29 Page

Symbol correspondences for spin systems

The present monograph explores the correspondence between quantum and classical mechanics in the particular context of spin systems, that is, SU(2)-symmetric mechanical systems. Here, a detailed presentation of quantum spin-j systems, with emphasis on the SO(3)-invariant decomposition of their operator algebras, is followed by an introduction to the Poisson algebra of the classical spin system and a similarly detailed presentation of its SO(3)-invariant decomposition. Subsequently, this monograph proceeds with a detailed and systematic study of general quantum-classical symbol correspondences for spin-j systems and their induced twisted products of functions on the 2-sphere. This original systematic presentation culminates with the study of twisted products in the asymptotic limit of high spin numbers. In the context of spin systems, it shows how classical mechanics may or may not emerge as an asymptotic limit of quantum mechanics.Comment: Research Monograph, 171 pages (book format, preliminary version