03688nam a22004695i 4500001001800000003000900018005001700027007001500044008004100059020001800100024003500118050001600153072001700169072002300186072001600209100007600225245013500301250001800436264006900454300004300523336002600566337002600592338003600618347002400654490005300678505051300731520154901244650002502793650001902818650002602837650003102863650002502894650002402919650002602943650001802969710003402987773002603021776003603047776003603083830005303119856004603172978-3-031-57201-2DE-He21320251030113038.0cr nn 008mamaa240913s2024    sz |    s    |||| 0|eng d  a97830315720127 a10.1007/978-3-031-57201-22doi 4aQA319-329.9 7aPBKF2bicssc 7aMAT0370002bisacsh 7aPBKF2thema1 aOertel, Frank.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut10aUpper Bounds for Grothendieck Constants, Quantum Correlation Matrices and CCP Functionsh[electronic resource] /cby Frank Oertel.  a1st ed. 2024. 1aCham :bSpringer Nature Switzerland :bImprint: Springer,c2024.  aXVI, 230 p. 1 illus.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aLecture Notes in Mathematics,x1617-9692 ;v23490 a- Introduction and motivation -- Complex Gaussian random vectors and their probability law -- A quantum correlation matrix version of the Grothendieck inequality -- Powers of inner products of random vectors, uniformly distributed on the sphere -- Completely correlation preserving functions -- The real case: towards extending Krivine's approach -- The complex case: towards extending Haagerup's approach -- A summary scheme of the main result -- Concluding remarks and open problems -- References -- Index.  aThis book concentrates on the famous Grothendieck inequality and the continued search for the still unknown best possible value of the real and complex Grothendieck constant (an open problem since 1953). It describes in detail the state of the art in research on this fundamental inequality, including Krivine's recent contributions, and sheds light on related questions in mathematics, physics and computer science, particularly with respect to the foundations of quantum theory and quantum information theory. Unifying the real and complex cases as much as possible, the monograph introduces the reader to a rich collection of results in functional analysis and probability. In particular, it includes a detailed, self-contained analysis of the multivariate distribution of complex Gaussian random vectors. The notion of Completely Correlation Preserving (CCP) functions plays a particularly important role in the exposition. The prerequisites are a basic knowledge of standard functional analysis, complex analysis, probability, optimisation and some number theory and combinatorics. However, readers missing some background will be able to consult the generous bibliography, which contains numerous references to useful textbooks. The book will be of interest to PhD students and researchers in functional analysis, complex analysis, probability, optimisation, number theory and combinatorics, in physics (particularly in relation to the foundations of quantum mechanics) and in computer science (quantum information and complexity theory). 0aFunctional analysis. 0aProbabilities. 0aMathematical physics. 0aMathematical optimization.14aFunctional Analysis.24aProbability Theory.24aMathematical Physics.24aOptimization.2 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303157200508iPrinted edition:z9783031572029 0aLecture Notes in Mathematics,x1617-9692 ;v234940uhttps://doi.org/10.1007/978-3-031-57201-2