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Anita Buckley

Publications and source records attributed to Anita Buckley.

9 recordsLinked to original sources
arXiv2022

New examples of entangled states on $\mathbb{C}^3 \otimes \mathbb{C}^3$

We build apon our previous work, the Buckley-\vSivic method for simultaneous construction of families of positive maps on $3 \times 3$ self-adjoint matrices by prescribing a set of complex zeros to the associated forms. Positive maps that are not completely positive can be used to prove (witness) that certain mixed states are entangled. We obtain entanglement witnesses that are indecomposable and belong to extreme rays of the cone of positive maps. Consequently our semidefinite program returns new examples of entangled states whose entanglement cannot be certified by the transposition map nor by other well-known positive maps. The constructed states as well as the method of their construction offer some valuable insights for quantum information theory, in particular into the geometry of positive cones.

quant-ph↗
arXiv2020

New examples of extremal positive linear maps

New families of nonnegative biquadratic forms that have 8, 9 or 10 real zeros in $\mathbb{P}^2\times \mathbb{P}^2$ are constructed. These are the first examples with 8, 9 or 10 real zeros. It is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros; our examples show that the upper bound is obtained. Such biquadratic forms define positive linear maps on real symmetric $3\times 3$ matrices that are not completely positive. Our constructions are explicit, and moreover we are able to determine which of the examples are extremal. We extend the examples to positive maps on complex matrices and find families of extreme rays in the cone of positive maps.

math.RA↗
arXiv2016

Simple determinantal representations of up to quintic bivariate polynomials

For bivariate polynomials of degree $n\le 5$ we give fast numerical constructions of determinantal representations with $n\times n$ matrices. Unlike some other available constructions, our approach returns matrices of the smallest possible size $n\times n$ for all polynomials of degree $n$ and does not require any symbolic computation. We can apply these linearizations to numerically compute the roots of a system of two bivariate polynomials by using numerical methods for two-parameter eigenvalue problems.

math.NA↗
arXiv2015

Indecomposable Matrices Defining Plane Cubics

In this article we find all (decomposable and indecomposable) $6\times 6$ linear determinantal representations of smooth Weierstrass cubics. As a corollary we verify the Kippenhahn conjecture for $M_6$.

math.AG↗
arXiv2012

Ice cream and orbifold Riemann-Roch

We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold X,D, under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called "ice cream functions". This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [A. Buckley and B. Szendroi, Orbifold Riemann-Roch for 3-folds with an application to Calabi-Yau geometry, J. Algebraic Geometry 14 (2005) 601--622] and [Shengtian Zhou, Orbifold Riemann-Roch and Hilbert series, University of Warwick PhD thesis, March 2011, 91+vii pp.], although the correct statements are considerably trickier. We expect to return to this in future publications.

math.AG↗
arXiv2009

Elementary Transformations of Pfaffian Representations of Plane Curves

Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$. In this paper we consider elementary transformations of linear pfaffian representations of $C$. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on $C$ with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of $C$ can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.

math.AG↗
arXiv2008

Plane curves as Pfaffians

Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$. In this paper we parametrise all linear pfaffian representations of $F$ by an open subset in the moduli space $M_C(2,K_C)$. We construct an explicit correspondence between pfaffian representations of $C$ and rank 2 vector bundles on $C$ with canonical determinant and no sections.

math.AG↗
arXiv2007

Determinantal representations of smooth cubic surfaces

For every smooth (irreducible) cubic surface $S$ we give an explicit construction of a representative for each of the 72 equivalence classes of determinantal representations. Equivalence classes (under $\GL_3\times \GL_3$ action by left and right multiplication) of determinantal representations are in one to one correspondence with the sets of six mutually skew lines on $S$ and with the 72 (two-dimensional) linear systems of twisted cubic curves on $S$. Moreover, if a determinantal representation $M$ corresponds to lines $(a_1,...,a_6)$ then its transpose $M^t$ corresponds to lines $(b_1,...,b_6)$ which together form a Schläfli's double-six $a_1... a_6 \choose b_1... b_6$. We also discuss the existence of self-adjoint and definite determinantal representation for smooth real cubic surfaces. The number of these representations depends on the Segre type $F_i$. We show that a surface of type $F_i$, $i=1,2,3,4$ has exactly $2(i-1)$ nonequivalent self-adjoint determinantal representations none of which is definite, while a surface of type $F_5$ has 24 nonequivalent self-adjoint determinantal representations, 16 of which are definite.

math.AG↗