1,368 research outputs found
Complementary vertices and adjacency testing in polytopes
Our main theoretical result is that, if a simple polytope has a pair of
complementary vertices (i.e., two vertices with no facets in common), then it
has at least two such pairs, which can be chosen to be disjoint. Using this
result, we improve adjacency testing for vertices in both simple and non-simple
polytopes: given a polytope in the standard form {x \in R^n | Ax = b and x \geq
0} and a list of its V vertices, we describe an O(n) test to identify whether
any two given vertices are adjacent. For simple polytopes this test is perfect;
for non-simple polytopes it may be indeterminate, and instead acts as a filter
to identify non-adjacent pairs. Our test requires an O(n^2 V + n V^2)
precomputation, which is acceptable in settings such as all-pairs adjacency
testing. These results improve upon the more general O(nV) combinatorial and
O(n^3) algebraic adjacency tests from the literature.Comment: 14 pages, 5 figures. v1: published in COCOON 2012. v2: full journal
version, which strengthens and extends the results in Section 2 (see p1 of
the paper for details
Bell inequalities from variable elimination methods
Tight Bell inequalities are facets of Pitowsky's correlation polytope and are
usually obtained from its extreme points by solving the hull problem. Here we
present an alternative method based on a combination of algebraic results on
extensions of measures and variable elimination methods, e.g., the
Fourier-Motzkin method. Our method is shown to overcome some of the
computational difficulties associated with the hull problem in some non-trivial
cases. Moreover, it provides an explanation for the arising of only a finite
number of families of Bell inequalities in measurement scenarios where one
experimenter can choose between an arbitrary number of different measurements
A bound on the k-gonality of facets of the hypermetric cone and related complexity problems
AbstractWe give a bound on gh(n), the largest integer such that there is a gh(n)-gonal facet of the hypermetric cone Hypn, gh(n) ⩽ 2n−2 (n−1)! This proves simultaneously the polyhedrality of the hypermetric cone. We give complete description of Delaunay polytopes related to facets of Hypn. We prove that the problem determining hypermetricity lies in co-NP and give some related NP-hard problem
Limb-darkening functions as derived from along-track operation of the ERBE scanning radiometers for August 1985
During August 1985, the scanning radiometers of the Earth Radiation Budget Experiment aboard the Earth Radiation Budget Satellite (ERBS) and the NOAA-9 satellite were operated in along-track scanning modes. These data were analyzed to produce limb darkening functions for Earth-emitted radiation, which relates the radiance in any given direction to the radiant exitence. Limb darkening functions are presented and shown as figures for day and night for each spacecraft. The scene types were computed using measurements within 10 deg of zenith. The models have values near zenith of 1.02 to 1.09, with values near 1.06 being typical. The typical value of the model is 1.06 for both day and night for ERBS, and for NOAA-9, the typical value at zenith is 1.06 for day and 1.05 for night. Mean models are formed for the ERBS and for the NOAA-9 results and are found to differ less than 1 percent, the ERBS results being the higher. The models vary about 1 percent with latitude near zenith
On the Relationship between Convex Bodies Related to Correlation Experiments with Dichotomic Observables
In this paper we explore further the connections between convex bodies
related to quantum correlation experiments with dichotomic variables and
related bodies studied in combinatorial optimization, especially cut polyhedra.
Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J.
Phys. A: Math. Gen. 38 10971-87) with respect to Bell inequalities. We show
that several well known bodies related to cut polyhedra are equivalent to
bodies such as those defined by Tsirelson (1993 Hadronic J. S. 8 329-45) to
represent hidden deterministic behaviors, quantum behaviors, and no-signalling
behaviors. Among other things, our results allow a unique representation of
these bodies, give a necessary condition for vertices of the no-signalling
polytope, and give a method for bounding the quantum violation of Bell
inequalities by means of a body that contains the set of quantum behaviors.
Optimization over this latter body may be performed efficiently by semidefinite
programming. In the second part of the paper we apply these results to the
study of classical correlation functions. We provide a complete list of tight
inequalities for the two party case with (m,n) dichotomic observables when
m=4,n=4 and when min{m,n}<=3, and give a new general family of correlation
inequalities.Comment: 17 pages, 2 figure
Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph
In this paper, we address the problem of enumerating all induced subtrees in
an input k-degenerate graph, where an induced subtree is an acyclic and
connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for
any its induced subgraph has a vertex whose degree is less than or equal to k,
and many real-world graphs have small degeneracies, or very close to small
degeneracies. Although, the studies are on subgraphs enumeration, such as
trees, paths, and matchings, but the problem addresses the subgraph
enumeration, such as enumeration of subgraphs that are trees. Their induced
subgraph versions have not been studied well. One of few example is for
chordless paths and cycles. Our motivation is to reduce the time complexity
close to O(1) for each solution. This type of optimal algorithms are proposed
many subgraph classes such as trees, and spanning trees. Induced subtrees are
fundamental object thus it should be studied deeply and there possibly exist
some efficient algorithms. Our algorithm utilizes nice properties of
k-degeneracy to state an effective amortized analysis. As a result, the time
complexity is reduced to O(k) time per induced subtree. The problem is solved
in constant time for each in planar graphs, as a corollary
The CONEstrip algorithm
Uncertainty models such as sets of desirable gambles and (conditional) lower previsions can be represented as convex cones. Checking the consistency of and drawing inferences from such models requires solving feasibility and optimization problems. We consider finitely generated such models. For closed cones, we can use linear programming; for conditional lower prevision-based cones, there is an efficient algorithm using an iteration of linear programs. We present an efficient algorithm for general cones that also uses an iteration of linear programs
Looking for symmetric Bell inequalities
Finding all Bell inequalities for a given number of parties, measurement
settings, and measurement outcomes is in general a computationally hard task.
We show that all Bell inequalities which are symmetric under the exchange of
parties can be found by examining a symmetrized polytope which is simpler than
the full Bell polytope. As an illustration of our method, we generate 238885
new Bell inequalities and 1085 new Svetlichny inequalities. We find, in
particular, facet inequalities for Bell experiments involving two parties and
two measurement settings that are not of the
Collins-Gisin-Linden-Massar-Popescu type.Comment: Joined the associated website as an ancillary file, 17 pages, 1
figure, 1 tabl
Nonlocality as a Benchmark for Universal Quantum Computation in Ising Anyon Topological Quantum Computers
An obstacle affecting any proposal for a topological quantum computer based
on Ising anyons is that quasiparticle braiding can only implement a finite
(non-universal) set of quantum operations. The computational power of this
restricted set of operations (often called stabilizer operations) has been
studied in quantum information theory, and it is known that no
quantum-computational advantage can be obtained without the help of an
additional non-stabilizer operation. Similarly, a bipartite two-qubit system
based on Ising anyons cannot exhibit non-locality (in the sense of violating a
Bell inequality) when only topologically protected stabilizer operations are
performed. To produce correlations that cannot be described by a local hidden
variable model again requires the use of a non-stabilizer operation. Using
geometric techniques, we relate the sets of operations that enable universal
quantum computing (UQC) with those that enable violation of a Bell inequality.
Motivated by the fact that non-stabilizer operations are expected to be highly
imperfect, our aim is to provide a benchmark for identifying UQC-enabling
operations that is both experimentally practical and conceptually simple. We
show that any (noisy) single-qubit non-stabilizer operation that, together with
perfect stabilizer operations, enables violation of the simplest two-qubit Bell
inequality can also be used to enable UQC. This benchmarking requires finding
the expectation values of two distinct Pauli measurements on each qubit of a
bipartite system.Comment: 12 pages, 2 figure
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