Additivity and non-additivity of multipartite entanglement measures

We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. First, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interests have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state, and the generalization of D\"{u}r's multipartite bound entangled states. We also prove the additivity of other two measures for some of these examples. Second, we show the non-additivity of GM of all antisymmetric states of three or more parties, and provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In particular, we derive analytical formulae of the three measures of one copy and two copies of the antisymmetric projector states respectively. Third, we show, with a statistical approach, that almost all multipartite pure states with sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement. However, their GM is not strong additive; what's more surprising, for generic pure states with real entries in the computational basis, GM of one copy and two copies, respectively, are almost equal. Hence, more states may be suitable for universal quantum computation, if measurements can be performed on two copies of the resource states. We also show that almost all multipartite pure states cannot be produced reversibly with the combination multipartite GHZ states under asymptotic LOCC, unless relative entropy of entanglement is non-additive for generic multipartite pure states.Comment: 45 pages, 4 figures. Proposition 23 and Theorem 24 are revised by correcting a minor error from Eq. (A.2), (A.3) and (A.4) in the published version. The abstract, introduction, and summary are also revised. All other conclusions are unchange

Indicative conditionals, restricted quantification, and naive truth

This paper extends Kripke’s theory of truth to a language with a variably strict conditional operator, of the kind that Stalnaker and others have used to represent ordinary indicative conditionals of English. It then shows how to combine this with a different and independently motivated conditional operator, to get a substantial logic of restricted quantification within naive truth theory

Interview with Kenneth Arrow

Arrow argues that the biggest failures of economic theory are: our failure to explain the business cycle; the missing explanations for the size of fluctuations of prices; our failure to explain the causes of growth and of the spread of innovation. He then discusses several of the existing alternatives to the rational expectations paradigm. He tells the story of his dissertation, and how Koopmans wanted to decline his Nobel Prize.Finally, he discusses health care reform, and malaria in Africa.Health Care; Business Cycles; Fluctuations

Space Math: Black Hole Math

This booklet includes 11 problems of increasing difficulty that explore the properties of black holes. Students learn about the relationship between black hole size and mass, time dilation, energy extraction, accretion disks, and what’s inside a black hole by working with a series of math problems that feature simple algebra, scientific notation, exponential functions, and the Pythagorean Theorem. (8.5 x11, 28 pages, 11 color images, PDF file) Educational levels: Intermediate elementary, Middle school, High school

Noam Chomsky on Where Artificial Intelligence Went Wrong

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Why bad ideas are a good idea

What would happen if we wrote an Abstract that was the exact opposite of what the paper described? This is a bad idea, but it makes us think more carefully than usual about properties of Abstracts. This paper describes BadIdeas, a collection of techniques that uses ???bad??? or ???silly??? ideas to inspire creativity, explore design domains and teach critical thinking in interaction design. We describe the approach, some evidence, how it is performed in practice and experience in its use.published or submitted for publicationis peer reviewe