Elementary test for non-classicality based on measurements of position and momentum

We generalise a non-classicality test described by Kot et al. [Phys. Rev. Lett. 108, 233601 (2010)], which can be used to rule out any classical description of a physical system. The test is based on measurements of quadrature operators and works by proving a contradiction with the classical description in terms of a probability distribution in phase space. As opposed to the previous work, we generalise the test to include states without rotational symmetry in phase space. Furthermore, we compare the performance of the non-classicality test with classical tomography methods based on the inverse Radon transform, which can also be used to establish the quantum nature of a physical system. In particular, we consider a non-classicality test based on the so-called filtered back-projection formula. We show that the general non-classicality test is conceptually simpler, requires less assumptions on the system and is statistically more reliable than the tests based on the filtered back-projection formula. As a specific example, we derive the optimal test for a quadrature squeezed single photon state and show that the efficiency of the test does not change with the degree of squeezing

Partitioning planar graphs: a fast combinatorial approach for max-cut

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. and that of Berman et al. The running time of the former can be bounded by O(|V|^(3/2)log|V|). The latter algorithm is more generally for determining T-joins in graphs. Although it has a slightly larger bound on the running time of O(V{\ensuremath{|}}{\^{ }}(3/2)(log{\ensuremath{|}}V{\ensuremath{|}}){\^{ }}(3/2))alpha({\ensuremath{|}}V{\ensuremath{|}}), where alpha({\ensuremath{|}}V{\ensuremath{|}}) is the inverse Ackermann function, it can solve large instances in practice. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time is bounded by O({\ensuremath{|}}V{\ensuremath{|}}{\^{ }}(3/2)log{\ensuremath{|}}V{\ensuremath{|}}), similar to the bound achieved by Shih et al. It can easily determine maximum cuts in huge random as well as real-world graphs with up to 10{\^{ }}6 nodes. We present experimental results for our method using two different matching implementations. We furthermore compare our approach with those of Shih et al. and Berman et al. It turns out that our algorithm is considerably faster in practice than the one by Shih et al. Moreover, it yields a much smaller associated graph. Its expanded graph size is comparable to that of Berman et al. However, whereas the procedure of generating the expanded graph in Berman et al. is very involved (thus needs a sophisticated implementation), implementing our approach is an easy and straightforward task

On the Inversion of High Energy Proton

Inversion of the K-fold stochastic autoconvolution integral equation is an elementary nonlinear problem, yet there are no de facto methods to solve it with finite statistics. To fix this problem, we introduce a novel inverse algorithm based on a combination of minimization of relative entropy, the Fast Fourier Transform and a recursive version of Efron's bootstrap. This gives us power to obtain new perspectives on non-perturbative high energy QCD, such as probing the ab initio principles underlying the approximately negative binomial distributions of observed charged particle final state multiplicities, related to multiparton interactions, the fluctuating structure and profile of proton and diffraction. As a proof-of-concept, we apply the algorithm to ALICE proton-proton charged particle multiplicity measurements done at different center-of-mass energies and fiducial pseudorapidity intervals at the LHC, available on HEPData. A strong double peak structure emerges from the inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios, 2D

On Kahan's Rules for Determining Branch Cuts

In computer algebra there are different ways of approaching the mathematical concept of functions, one of which is by defining them as solutions of differential equations. We compare different such approaches and discuss the occurring problems. The main focus is on the question of determining possible branch cuts. We explore the extent to which the treatment of branch cuts can be rendered (more) algorithmic, by adapting Kahan's rules to the differential equation setting.Comment: SYNASC 2011. 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. (2011

Fuzzy Weighted Average: Analytical Solution

An algorithm is presented for the computation of analytical expressions for the extremal values of the α-cuts of the fuzzy weighted average, for triangular or trapeizoidal weights and attributes. Also, an algorithm for the computation of the inverses of these expressions is given, providing exact membership functions of the fuzzy weighted average. Up to now, only algorithms exist for the computation of the extremal values of the α-cuts for a fixed value of α. To illustrate the power of our algorithms, they are applied to several examples from the literature, providing exact membership functions in each case