Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision

On stochasticity in nearly-elastic systems

Nearly-elastic model systems with one or two degrees of freedom are considered: the system is undergoing a small loss of energy in each collision with the "wall". We show that instabilities in this purely deterministic system lead to stochasticity of its long-time behavior. Various ways to give a rigorous meaning to the last statement are considered. All of them, if applicable, lead to the same stochasticity which is described explicitly. So that the stochasticity of the long-time behavior is an intrinsic property of the deterministic systems.Comment: 35 pages, 12 figures, already online at Stochastics and Dynamic

Cauchy's formulas for random walks in bounded domains

Cauchy's formula was originally established for random straight paths crossing a body $B \subset \mathbb{R}^{n}$ and basically relates the average chord length through $B$ to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length travelled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in $B$. Similar results are obtained also for the average number of collisions performed by the walker in $B$, and an extension to absorbing media is discussed.Comment: 12 pages, 6 figure

How often is a random quantum state k-entangled?

The set of trace preserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of k-positive maps, where k=2,...,d. Working with the measure induced by the Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of (k+1)-positive maps forms a small fraction of the outer set of k-positive maps. These results are related to analogous bounds for the relative volume of the sets of k-entangled states describing a bipartite d X d system.Comment: 19 pages in latex, 1 figure include

Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a "fat" one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm's theorem and its proof on illuminating convex bodies of constant width to the family of "fat" spindle convex bodies.Comment: 17 page

Convex hulls of random walks, hyperplane arrangements, and Weyl chambers

We give an explicit formula for the probability that the convex hull of an n-step random walk in Rd does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (Skand Aktuarietidskr 32:27–36, 1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments. This probabilistic problem is shown to be equivalent to either of the two geometric ones: (1) Find the number of Weyl chambers of type Bn intersected by a generic linear subspace of Rn of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type Bn. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (Discrete Comput Geom 46(3):417–426, 2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement. We obtain analogous distribution-free results for Weyl chambers of type An−1 (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type Dn, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form B1 ×···× B1 recovers the Wendel formula (Math Scand 11:109–111, 1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin. We also give an asymptotic analysis of the obtained absorption probabilities as n → ∞, in both cases of fixed and increasing dimension d

21 de setiembre: día de la ciencia y la técnica

Con motivo del 3O aniversario de la instauración del 21 de setiembre como día de la Ciencia y la Técnica en la Provincia de Buenos Aires, la CIC le ha solicitado a un grupo selecto de científicos, su opinión sobre aspectos esenciales del quehacer científico y tecnológico. El presente texto contiene los siguientes trabajos: - Consideraciones acerca de la investigacion técnico-científica sobre materiales, por Eduardo R. Abril. - Ciencia y tecnología de alimentos, por Pedro Cattáneo - Ciencia y tecnología en química orgánica por Venancio Deulofeu. - Ciencia y tecnología - recursos humanos por Ernesto E. Galloni. - Discurso de apertura de la reunion sobre ciencia y tecnologia realizada en Mar del Plata del 6 al 8 de julio de 1978 por Luis F. Leloir. - Incidencia de la investigación en el desarrollo de la metalurgia por Roberto F. Recoder. - Importancia de la investigación matemática en la sociedad contemporánea por Luis A. Santaló. - La investigación en fisicoquímica y su proyección en el desarrollo científico y tecnológico por Hans J. Schumacher

Integral geometry of complex space forms

We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.Comment: 68 pages; minor change

Local Anisotropy of Fluids using Minkowski Tensors

Statistics of the free volume available to individual particles have previously been studied for simple and complex fluids, granular matter, amorphous solids, and structural glasses. Minkowski tensors provide a set of shape measures that are based on strong mathematical theorems and easily computed for polygonal and polyhedral bodies such as free volume cells (Voronoi cells). They characterize the local structure beyond the two-point correlation function and are suitable to define indices $0\leq \beta_\nu^{a,b}\leq 1$ of local anisotropy. Here, we analyze the statistics of Minkowski tensors for configurations of simple liquid models, including the ideal gas (Poisson point process), the hard disks and hard spheres ensemble, and the Lennard-Jones fluid. We show that Minkowski tensors provide a robust characterization of local anisotropy, which ranges from $\beta_\nu^{a,b}\approx 0.3$ for vapor phases to $\beta_\nu^{a,b}\to 1$ for ordered solids. We find that for fluids, local anisotropy decreases monotonously with increasing free volume and randomness of particle positions. Furthermore, the local anisotropy indices $\beta_\nu^{a,b}$ are sensitive to structural transitions in these simple fluids, as has been previously shown in granular systems for the transition from loose to jammed bead packs