Related papers: Concavity of the quantum body for any given dimens…

Measurements in quantum theory exhibit incompatibility, i.e., they can fail to be jointly measurable. An intuitive way to represent the (in)compatibility relations among a set of measurements is via a hypergraph representing their joint…

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric…

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are…

For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull…

We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we present explicitly a given a set of non jointly…

We prove sharp inequalities for the average number of affine diameters through the points of a convex body $K$ in ${\mathbb R}^n$. These inequalities hold if $K$ is either a polytope or of dimension two. An example shows that the proof…

The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central…

Consider a scenario where $N$ separated quantum systems are measured, each with one among two possible dichotomic observables. Assume that the $N$ events corresponding to the choice and performance of the measurement in each site are…

Let $\mu$ be a probability distribution on $\mathbb{R}^d$ which assigns measure zero to every hyperplane and $S$ a set of points sampled independently from $\mu$. What can be said about the expected combinatorial structure of the convex…

According to Popescu's recent analysis [Phys. Rev. Lett. {\bf72}, 797 (1994)], {\it nonideal} measurements, rather than ideal ones, may be more sensitive to reveal nonlocal correlations between distant parts of composite quantum systems.…

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $\mathrm{diam}(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error…

Let us consider the set of joint quantum correlations arising from two-outcome local measurements on a bipartite quantum system. We prove that no finite dimension is sufficient to generate all these sets. We approach the problem in two…

We introduce a measure of ''quantumness'' for any quantum state in a finite dimensional Hilbert space, based on the distance between the state and the convex set of classical states. The latter are defined as states that can be written as a…

Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum state space may thus be thought…

We introduce the concept of quantum weight as a ground state property of quantum many-body systems that is encoded in the static structure factor and characterizes density fluctuation at long wavelengths. The quantum weight carries a wealth…

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a…

We put forward and demonstrate experimentally a {\it quantum-inspired} protocol that allows to quantify the degree of similarity between two spatial shapes embedded in two optical beams without the need to measure the amplitude and phase…

A three-dimensional convex body is said to have Rupert's property if its copy can be passed through a straight hole inside that body. In this work we construct a polyhedron which is provably not Rupert, thus we disprove a conjecture from…

We analyze the convex structure of the set of positive operator valued measures (POVMs) representing quantum measurements on a given finite dimensional quantum system, with outcomes in a given locally compact Hausdorff space. The extreme…

Let $K$ be a convex body in $\mathbb{R}^n$ with Santal\'o point at 0\. We show that if $K$ has a point on the boundary with positive generalized Gau{\ss} curvature, then the volume product $|K| |K^\circ|$ is not minimal. This means that a…