Blocking sets of the Hermitian unital

It is known that the classical unital arising from the Hermitian curve in PG(2,9) does not have a 2-coloring without monochromatic lines. Here we show that for q≥4 the Hermitian curve in PG(2,q2) does possess 2-colorings without monochromatic lines. We present general constructions and also prove a lower bound on the size of blocking sets in the classical unital

Unitals in PG(2,q2)PG(2,q^2) with a large 2-point stabiliser

Let \cU be a unital embedded in the Desarguesian projective plane \PG(2,q^2). Write $M$ for the subgroup of \PGL(3,q^2) which preserves \cU. We show that \cU is classical if and only if \cU has two distinct points $P,Q$ for which the stabiliser $G=M_{P,Q}$ has order $q^2-1$.Comment: Revised version - clarified the case mu\neq\lambda^{q+1} - 7 page

Ruling out higher-order interference from purity principles

As first noted by Rafael Sorkin, there is a limit to quantum interference. The interference pattern formed in a multi-slit experiment is a function of the interference patterns formed between pairs of slits, there are no genuinely new features resulting from considering three slits instead of two. Sorkin has introduced a hierarchy of mathematically conceivable higher-order interference behaviours, where classical theory lies at the first level of this hierarchy and quantum theory theory at the second. Informally, the order in this hierarchy corresponds to the number of slits on which the interference pattern has an irreducible dependence. Many authors have wondered why quantum interference is limited to the second level of this hierarchy. Does the existence of higher-order interference violate some natural physical principle that we believe should be fundamental? In the current work we show that such principles can be found which limit interference behaviour to second-order, or "quantum-like", interference, but that do not restrict us to the entire quantum formalism. We work within the operational framework of generalised probabilistic theories, and prove that any theory satisfying Causality, Purity Preservation, Pure Sharpness, and Purification---four principles that formalise the fundamental character of purity in nature---exhibits at most second-order interference. Hence these theories are, at least conceptually, very "close" to quantum theory. Along the way we show that systems in such theories correspond to Euclidean Jordan algebras. Hence, they are self-dual and, moreover, multi-slit experiments in such theories are described by pure projectors.Comment: 18+8 pages. Comments welcome. v2: Minor correction to Lemma 5.1, main results are unchange

A characterization of Hermitian varieties as codewords

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $PG(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of $PG(r,q^2)$ of the same size as a non-singular Hermitian variety of $PG(r,q^2)$, having the same intersection sizes with the hyperplanes of $PG(r,q^2)$. In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of $PG(2,q^2)$ is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in $PG(3,q^2)$, $q=p^{h}$, as well as in $PG(r,q^2)$, $q=p$ prime, or $q=p^2$, $p$ prime, and $r\geq 4$