The geometric measure of entanglement for a symmetric pure state with positive amplitudes

In this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement: 'for a symmetric pure state, the closest product state in terms of the fidelity can be chosen as a symmetric product state'. We show that this conjecture is true for symmetric pure states whose amplitudes are all non-negative in a computational basis. The more general conjecture is still open.Comment: Similar results have been obtained independently and with different methods by T-C. Wei and S. Severini, see arXiv:0905.0012v

Extension and splitting theorems for Fréchet spaces of type 2

Sin resume

On the Bohr inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius

The power of symmetric extensions for entanglement detection

In this paper, we present new progress on the study of the symmetric extension criterion for separability. First, we show that a perturbation of order O(1/N) is sufficient and, in general, necessary to destroy the entanglement of any state admitting an N Bose symmetric extension. On the other hand, the minimum amount of local noise necessary to induce separability on states arising from N Bose symmetric extensions with Positive Partial Transpose (PPT) decreases at least as fast as O(1/N^2). From these results, we derive upper bounds on the time and space complexity of the weak membership problem of separability when attacked via algorithms that search for PPT symmetric extensions. Finally, we show how to estimate the error we incur when we approximate the set of separable states by the set of (PPT) N -extendable quantum states in order to compute the maximum average fidelity in pure state estimation problems, the maximal output purity of quantum channels, and the geometric measure of entanglement.Comment: see Video Abstract at http://www.quantiki.org/video_abstracts/0906273

Smolyak's algorithm: A powerful black box for the acceleration of scientific computations

We provide a general discussion of Smolyak's algorithm for the acceleration of scientific computations. The algorithm first appeared in Smolyak's work on multidimensional integration and interpolation. Since then, it has been generalized in multiple directions and has been associated with the keywords: sparse grids, hyperbolic cross approximation, combination technique, and multilevel methods. Variants of Smolyak's algorithm have been employed in the computation of high-dimensional integrals in finance, chemistry, and physics, in the numerical solution of partial and stochastic differential equations, and in uncertainty quantification. Motivated by this broad and ever-increasing range of applications, we describe a general framework that summarizes fundamental results and assumptions in a concise application-independent manner

Survey on nonlocal games and operator space theory

This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states

Positron scattering from chiral enantiomers

We report on total cross section measurements for positron scattering from the chiral enantiomers (+)-methyl (R)-2-chloropropionate and (–)-methyl (S)-2-chloropropionate. The energy range of the present study was 0.1–50 eV, while the energy resolution of our incident positron beam was ∼0.25 eV (FWHM). As positrons emanating from β decay in radioactive nuclei have a high degree of spin polarization, which persists after moderation, we were particularly interested in probing whether the positron helicity differentiates between the measured total cross sections of the two enantiomers. No major differences were, however, observed. Finally, quantum chemical calculations, using the density functional theory based B3LYP-DGTZVP model within the gaussian 09 package, were performed as a part of this work in order to assist us in interpreting some aspects of our data

Amenability of algebras of approximable operators

We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space.Comment: 20 pages, to appear in Israel Journal of Mathematic

Factorization of Operators Through Orlicz Spaces

[EN] We study factorization of operators between quasi-Banach spaces. We prove the equivalence between certain vector norm inequalities and the factorization of operators through Orlicz spaces. As a consequence, we obtain the Maurey-Rosenthal factorization of operators into L-p-spaces. We give several applications. In particular, we prove a variant of Maurey's Extension Theorem.The research of the first author was supported by the National Science Centre (NCN), Poland, Grant No. 2011/01/B/ST1/06243. The research of the second author was supported by Ministerio de Economia y Competitividad, Spain, under project #MTM2012-36740-C02-02Mastylo, M.; Sánchez Pérez, EA. (2017). Factorization of Operators Through Orlicz Spaces. Bulletin of the Malaysian Mathematical Sciences Society. 40(4):1653-1675. https://doi.org/10.1007/s40840-015-0158-5S16531675404Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)Davis, W.J., Garling, D.J.H., Tomczak-Jaegermann, N.: The complex convexity of quasi-normed linear spaces. J. Funct. Anal. 55, 110–150 (1984)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Mastyło, M., Michels, C.: Orlicz norm estimates for eigenvalues of matrices. Isr. J. Math. 132, 45–59 (2002)Defant, A., Sánchez Pérez, E.A.: Maurey–Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Defant, A., Sánchez Pérez, E.A.: Domination of operators on function spaces. Math. Proc. Camb. Phil. Soc. 146, 57–66 (2009)Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin (1984)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Dilworth, S.J.: Special Banach lattices and their applications. In: Handbook of the Geometry of Banach Spaces, vol. 1. Elsevier, Amsterdam (2001)Figiel, T., Pisier, G.: Séries alétoires dans les espaces uniformément convexes ou uniformément lisses. Comptes Rendus de l’Académie des Sciences, Paris, Série A 279, 611–614 (1974)Kalton, N.J., Montgomery-Smith, S.J.: Set-functions and factorization. Arch. Math. (Basel) 61(2), 183–200 (1993)Kamińska, A., Mastyło, M.: Abstract duality Sawyer formula and its applications. Monatsh. Math. 151(3), 223–245 (2007)Kantorovich, L.V., Akilov, G.P.: Functional Analysis, 2nd edn. Pergamon Press, Oxford (1982)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices IV, Sibirsk. Mat. Z. 14, 140–155 (1973) (in Russian); English transl.: Siberian. Math. J. 14, 97–108 (1973)Lozanovskii, G.Ya.:Transformations of ideal Banach spaces by means of concave functions. In: Qualitative and Approximate Methods for the Investigation of Operator Equations, Yaroslavl, vol. 3, pp. 122–147 (1978) (Russian)Mastyło, M., Szwedek, R.: Interpolative constructions and factorization of operators. J. Math. Anal. Appl. 401, 198–208 (2013)Nikišin, E.M.: Resonance theorems and superlinear operators. Usp. Mat. Nauk 25, 129–191 (1970) (Russian)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, vol. 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986)Reisner, S.: On two theorems of Lozanovskii concerning intermediate Banach lattices, geometric aspects of functional analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 67–83. Springer, Berlin (1988)Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1991

Estimates for vector valued Dirichlet polynomials

[EN] We estimate the -norm of finite Dirichlet polynomials with coefficients in a Banach space. Our estimates quantify several recent results on Bohr's strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.A. Defant and P. Sevilla-Peris were supported by MICINN Project MTM2011-22417.Defant, A.; Schwarting, U.; Sevilla Peris, P. (2014). Estimates for vector valued Dirichlet polynomials. Monatshefte f�r Mathematik. 175(1):89-116. https://doi.org/10.1007/s00605-013-0600-4S891161751Balasubramanian, R., Calado, B., Queffélec, H.: The Bohr inequality for ordinary Dirichlet series. Studia Math. 175(3), 285–304 (2006)Bayart, F.: Hardy spaces of Dirichlet series and their composition operators. Monatsh. Math. 136(3), 203–236 (2002)Bennett, G.: Inclusion mappings between $$l^{p}$$ l p spaces. J. Funct. Anal. 13, 20–27 (1973)Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. (2) 32(3), 600–622 (1931)Bohr, H.: Über die Bedeutung der Potenzreihen unendlich vieler Variablen in der Theorie der Dirichlet–schen Reihen $$\sum \frac{a_n}{n^s}$$ ∑ a n n s . Nachr. Ges. Wiss. Göttingen Math. Phys. Kl., Heft 4, 441–488 (1913)Bohr, H.: Über die gleichmäßige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913)Carl, B.: Absolut- $$(p,\,1)$$ ( p , 1 ) -summierende identische Operatoren von $$l_{u}$$ l u in $$l_{v}$$ l v . Math. Nachr. 63, 353–360 (1974)Carlson, F.: Contributions à la théorie des séries de Dirichlet. Note i. Ark. fö”r Mat., Astron. och Fys. 16(18), 1–19 (1922)de la Bretèche, R.: Sur l’ordre de grandeur des polynômes de Dirichlet. Acta Arith. 134(2), 141–148 (2008)Defant, A., Frerick, L., Ortega-Cerdà, J., Ounaïes, M., Seip, K.: The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive. Ann. Math. (2) 174(1), 485–497 (2011)Defant, A., García, D., Maestre, M., Pérez-García, D.: Bohr’s strip for vector valued Dirichlet series. Math. Ann. 342(3), 533–555 (2008)Defant, A., García, D., Maestre, M., Sevilla-Peris, P.: Bohr’s strips for Dirichlet series in Banach spaces. Funct. Approx. Comment. Math. 44(part 2), 165–189 (2011)Defant, A., Maestre, M., Schwarting, U.: Bohr radii of vector valued holomorphic functions. Adv. Math. 231(5), 2837–2857 (2012)Defant, A., Popa, D., Schwarting, U.: Coordinatewise multiple summing operators in Banach spaces. J. Funct. Anal. 259(1), 220–242 (2010)Defant, A., Sevilla-Peris, P.: Convergence of Dirichlet polynomials in Banach spaces. Trans. Am. Math. Soc. 363(2), 681–697 (2011)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995)Harris, L.A.: Bounds on the derivatives of holomorphic functions of vectors. In: Analyse fonctionnelle et applications (Comptes Rendus Colloq. Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972), pp. 145–163. Actualités Aci. Indust., No. 1367. Hermann, Paris (1975)Hedenmalm, H., Lindqvist, P., Seip, K.: A Hilbert space of Dirichlet series and systems of dilated functions in $$L^2(0,1)$$ L 2 ( 0 , 1 ) . Duke Math. J. 86(1), 1–37 (1997)Kahane, J.-P.: Some Random Series of Functions. Cambridge Studies in Advanced Mathematics, vol. 5, 2nd edn. Cambridge University Press, Cambridge (1985)Konyagin, S.V., Queffélec, H.: The translation $$\frac{1}{2}$$ 1 2 in the theory of Dirichlet series. Real Anal. Exch. 27(1):155–175 (2001/2002)Kwapień, S.: Some remarks on $$(p,\, q)$$ ( p , q ) -absolutely summing operators in $$l_{p}$$ l p -spaces. Studia Math. 29, 327–337 (1968)Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes, reprint of the 1991 edn. Classics in Mathematics. Springer, Berlin (2011)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92. Springer, Berlin (1977)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. II, Function Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97. Springer, Berlin (1979)Maurizi, B., Queffélec, H.: Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl. 16, 676–692 (2010)Prachar, K.: Primzahlverteilung. Springer, Berlin (1957)Queffélec, H.: H. Bohr’s vision of ordinary Dirichlet series; old and new results. J. Anal. 3, 43–60 (1995)Tomczak-Jaegermann, N.: Banach–Mazur Distances and Finite-Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38. Longman Scientific & Technical, Harlow (1989