Large violation of Bell inequalities with low entanglement

In this paper we obtain violations of general bipartite Bell inequalities of order $\frac{\sqrt{n}}{\log n}$ with $n$ inputs, $n$ outputs and $n$-dimensional Hilbert spaces. Moreover, we construct explicitly, up to a random choice of signs, all the elements involved in such violations: the coefficients of the Bell inequalities, POVMs measurements and quantum states. Analyzing this construction we find that, even though entanglement is necessary to obtain violation of Bell inequalities, the Entropy of entanglement of the underlying state is essentially irrelevant in obtaining large violation. We also indicate why the maximally entangled state is a rather poor candidate in producing large violations with arbitrary coefficients. However, we also show that for Bell inequalities with positive coefficients (in particular, games) the maximally entangled state achieves the largest violation up to a logarithmic factor.Comment: Reference [16] added. Some typos correcte

The metasomatic record in the shallow peridotite mantle beneath Grenada (Lesser Antilles arc)

The composition and geochemical signatures of the mantle wedge beneath the Lesser Antilles arc are documented by the ultramafic xenoliths included in alkali basalts (M-series) on Grenada. Xenoliths consist of harzburgites, lherzolites, dunites and subordinate wehrlites and pyroxenites. Primary minerals phases are olivine, low-Al and high-Al orthopyroxene, clinopyroxene and Cr-Spinel. In addition to the primary assemblage, Grenada xenoliths contain metasomatic phases such as Al-rich clinopyroxene, plagioclase, Al-rich spinel, pargasitic amphibole and Si- and Al-rich glasses. The trace-element signatures of pyroxenes and glasses have been determined on selected samples by LA-ICP-MS. Pyroxenes from both lherzolite and harzburgite xenoliths have U-shaped rare earth element (REE) profiles, unusually high Th, U and Sr concentrations and large negative Nb, Ta and Zr, and Hf anomalies. The geochemical signatures of metasomatic clinopyroxene are different from those reported for clinopyroxene from fluid-metasomatised mantle wedge, and are clearly distinct from those of clinopyroxene in equilibrium with host lavas. Si-rich glasses show a narrow compositional range, with trace-element characteristics closely similar to those of reacted pyroxenes. This, along with the general lack of chemical gradients of LILE and LREE over more compatible elements suggests dacitic glasses represent the products of in-situ melting caused by temperature increase before and during the uptake of xenoliths by host lavas. Dacitic melts are believed to represent local re-melts of regions metasomatically enriched by earlier arc magmas that had stalled, fractionated, and solidified in the upper mantle. These local re-melts thus reflect the metasomatic component formed by earlier arc-related metasomatic agents and liable to be re-mobilised. This also appears to be the easiest way to explain the compositional similarities between erupted arc lavas and the metasomatised peridotites. The results of this study suggest that the mantle wedge beneath the Lesser Antilles underwent complex peridotite-melt reaction processes operated by sub-arc melts and, later on, by magmas similar in compositions to the host alkali basalts. The majority of the compositional range of erupted Grenada magmas, but adakites found at surface, can be obtained by the interaction of basalts, possibly formed by hydrous melting of MORB-source mantle, with the overlying mantle wedge