相关论文: Multiprojective witness sets and a trace test
A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical…
The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of…
In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These…
One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some…
Given a positive integer $n$ and a partition $(n_1,\ldots,n_r)$ of $n$, one can consider the associated $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$. These multiprojective spaces are…
In this paper we present a new method for entanglement witnesses construction. We show that to construct such an object we can deal with maps which are not positive on the whole domain, but only on a certain sub-domain. In our approach…
We derive a general framework that connects every positive map with a corresponding witness for partial separability in multipartite quantum systems. We show that many previous approaches were intimately connected to the witnesses derived…
We provide a novel tool which may be used to construct new examples of positive maps in matrix algebras (or, equivalently, entanglement witnesses). It turns out that this can be used to prove positivity of several well known maps (such as…
Numerical algebraic geometry has a close relationship to intersection theory from algebraic geometry. We deepen this relationship, explaining how rational or algebraic equivalence gives a homotopy. We present a general notion of witness set…
A new family of positive, trace-preserving maps is introduced. It is defined using the mutually unbiased measurements, which generalize the notion of mutual unbiasedness of orthonormal bases. This family allows one to define entanglement…
We derive a set of algebraic equations, the so-called multipartite separability eigenvalue equations. Based on their solutions, we introduce a universal method for the construction of multipartite entanglement witnesses. We witness…
We propose a family of positive maps constructed from a recently introduced class of symmetric measurements. These maps are used to define entanglement witnesses, which include other popular approaches with mutually unbiased bases and…
We report on a new class of dimension witnesses, based on quantum random access codes, which are a function of the recorded statistics and that have different bounds for all possible decompositions of a high-dimensional physical system.…
Entanglement in multipartite systems is a key resource for quantum information and communication protocols, making its verification in complex systems a necessity. Because an exact calculation of arbitrary entanglement probes is impossible,…
We provide a systematic method for nonlinear entanglement detection based on trace polynomial inequalities. In particular, this allows to employ multi-partite witnesses for the detection of bipartite states, and vice versa. We identify…
We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic…
Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in…
Generative reconstruction methods compute the 3D configuration (such as pose and/or geometry) of a shape by optimizing the overlap of the projected 3D shape model with images. Proper handling of occlusions is a big challenge, since the…
Our focus is upon {\it irreducible} nonnegative $n$-by-$n$ matrix realizations of nonnegatively realizable spectra or, equivalently, characteristic polynomials. After giving some general background, we make some useful new observations and…
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a…