Hidden Variables or Positive Probabilities?

Despite claims that Bell's inequalities are based on the Einstein locality condition, or equivalent, all derivations make an identical mathematical assumption: that local hidden-variable theories produce a set of positive-definite probabilities for detecting a particle with a given spin orientation. The standard argument is that because quantum mechanics assumes that particles are emitted in a superposition of states the theory cannot produce such a set of probabilities. We examine a paper by Eberhard, and several similar papers, which claim to show that a generalized Bell inequality, the CHSH inequality, can be derived solely on the basis of the locality condition, without recourse to hidden variables. We point out that these authors nonetheless assumes a set of positive-definite probabilities, which supports the claim that hidden variables or "locality" is not at issue here, positive-definite probabilities are. We demonstrate that quantum mechanics does predict a set of probabilities that violate the CHSH inequality; however these probabilities are not positive-definite. Nevertheless, they are physically meaningful in that they give the usual quantum-mechanical predictions in physical situations. We discuss in what sense our results are related to the Wigner distribution.Comment: 19 pages, 2 ps files This is a second replacement. In this version we include an analysis of yet another version of Bell's theorem which has been brought to our attention. We also discuss in what sense our results are related to the Wigner distributio

Decomposing 1-Sperner hypergraphs

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of $1$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of $1$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that $1$-Sperner hypergraphs are both threshold and equilizable. The study of $1$-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper

On the convex hull of convex quadratic optimization problems with indicators

We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are ``finitely generated." In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets

Positive maps and trace polynomials from the symmetric group

With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials $X_{\alpha_1} \cdots X_{\alpha_r}$ and their traces $\operatorname{tr}(X_{\alpha_1} \cdots X_{\alpha_r})$. Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.Comment: 28 pages, 3 figures, 2 tables. Extensively rewritten: asymmetric maps, proof for Motzkin matrix polynomial, and connections to QIT added. Comments welcome

What drives private saving around the world?

The authors investigate the policy and non-policy factors behind saving disparities, using a large panel data set and an encompassing approach including several relevant determinants of private saving. They extend the literature in several dimensions, by: 1) Using the largest data set on aggregate saving assembled to date. 2) Using panel instrumental variable techniques to correct for endogeneity and heterogeneity. 3) Performing robustness checks on changes in estimation procedures, data samples, and model specification. Their main empirical findings: a) Private saving rates show considerable inertia (are highly serially correlated even after controlling for other relevant factors). b) Private sector rates rise with the level and growth rate of real per capita income. So policies that spur development are in indirect but effective way to raise private saving rates. c) Predictions of the life-cycle hypothesis are supported in that dependency ratios generally have a negative effect on private saving rates. d) The precautionary motive for saving is supported by the finding that inflation - conventionally taken as a summary measure of macroeconomic volatility - has a positive impact on private saving, holding other facts constant. e) Fiscal policy is a moderately effective tool for raising national saving. F) the direct effect of financial liberalization are largely detrimental to private saving rates. Greater availability of credit reduces the private saving rate; financial depth and higher real interest rates do not increase saving.Environmental Economics&Policies,Banks&Banking Reform,Economic Theory&Research,Payment Systems&Infrastructure,Contractual Savings,Economic Theory&Research,Achieving Shared Growth,Environmental Economics&Policies,Inequality,Banks&Banking Reform

How does income inequality influence international migration?

The increasing importance of highly-skilled migration in times of so-called ?skills shortages? is leading to a growing interest in the determinants and characteristics of highly-skilled migration. However, migration theory with regard to the highly-skilled is not well developed. An important strand of literature that clearly serves for the derivation of empirically testable hypotheses about the determinants of particular types of migrants is self-selection theory. This theory dates back to Roy (1951) and has been adopted by Borjas (1987) for the analysis of the relation between the income distribution and the skills of migrants. He concludes that a relatively more equal income distribution in the host country vis-à-vis the source country leads to a negative self-selection of migrants (i.e. the lowly-skilled will be particularly attracted) and vice versa. Borjas has confirmed this hypothesis with data on immigration to the US. Chiswick (1999) and others, however, have questioned these results. Sample-selection biases may arise in single-country analyses and in all studies based on host-country data, due to the impact of host-country specifics such as migration policy, network migration, and the like. Due to a lack of internationally comparable data, however, international empirical studies with data from the origin countries have not been undertaken to resolve the dispute between Borjas and Chiswick. Furthermore, data on the intentions to emigrate (as opposed to actual migration data) has the distinct advantage of being free from the above-mentioned selection-bias problem. This paper sheds some new light on the self-selection controversy by analysing the relationship between country-specific emigration propensities and each country?s score on the Gini-Index on inequality. The 1995 International Social Survey Programme (ISSP) conducted a survey on national identity, which gathers the necessary data in a rich international microdata set. We run probit-regressions with two samples, one including all surveyed persons and one including only the sub-sample of the highly-skilled. By relying on the Gini index as a proxy for wage inequality, the paper follows Borjas? (1987) approach. Borjas, however, proxies skills differentials by income differentials. Chiswick (1999) argues that these two differentials may only be poorly related. The analysis presented here partly avoids this criticism, as we compare the sub-sample of highly-skilled persons with all surveyed individuals and with the medium- and lowly-skilled. A strong positive correlation between skills and income, as predicted by standard economic theory, therefore suffices for the validity of our approach. The main result of this paper is that, ceteris paribus, a more egalitarian income distribution is associated with lower emigration propensities, while income inequality does not have any impact on the emigration propensities of particularly highly-skilled persons. These results seem to contradict Borjas? prediction that the highly-skilled should be particularly attracted by countries which have relatively high returns to skills. Thus, our analysis is more in line with the arguments put forward by Chiswick (1999) and others.