357 research outputs found
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
Factorizing Kernel Operators
Consider an operator T : X--->Y between Banach function
spaces having adequate order continuity and Fatou properties.
Assume that T can be factorized through a Banach space as T =SR,
where R and the adjoint of S are p-th power and q-th power factorable,
respectively. Then a canonical factorization scheme can be given for T.
We show that it provides a tool for analyzing T that becomes specially
useful for the case of kernel operators. In particular, we show that this
square factorization scheme for T is equivalent to some inequalities for
the bilinear form defined by T. Kernel operators are studied from this
point of view.Support of the Ministerio de Economia y Competitividad under project # MTM2012-36740-C02-02 (Spain) is gratefully acknowledged.Galdames, O.; Sánchez Pérez, EA. (2013). Factorizing Kernel Operators. Integral Equations and Operator Theory. 75(1):13-29. https://doi.org/10.1007/s00020-012-2019-zS1329751Bennett C., Sharpley R.: Interpolation of Operators. Academic Press, Boston (1988)Calabuig J.M., Delgado O., Sánchez-Pérez E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Defant A.: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant A., Sánchez Pérez E.A.: Maurey-Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1977)Fernández A., Mayoral F., Naranjo F., Sáez C., Sánchez-Pérez E.A.: Spaces of integrable functions with respect to a vector measure and factorizations through L p and Hilbert spaces. J. Math. Anal. Appl. 330, 1249–1263 (2007)Galdames O., Sánchez Pérez E.A.: Optimal range theorems for operators with p-th power factorable adjoints. Banach J. Math. Anal. 6(1), 63–71 (2012)Lindenstrauss J., Tzafriri L.: Classical Banach Spaces II. Springer, Berlin (1979)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)Zaanen, A.C.: Integration, 2nd revised edn. North Holland, Amsterdam; Interscience, New York (1967
On the Bohr inequality
The Bohr inequality, first introduced by Harald Bohr in 1914, deals with
finding the largest radius , , such that holds whenever in the unit disk
of the complex plane. The exact value of this largest radius,
known as the \emph{Bohr radius}, has been established to be This paper
surveys recent advances and generalizations on the Bohr inequality. It
discusses the Bohr radius for certain power series in as well as
for analytic functions from 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 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 -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
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
Unbounded violation of tripartite Bell inequalities
We prove that there are tripartite quantum states (constructed from random
unitaries) that can lead to arbitrarily large violations of Bell inequalities
for dichotomic observables. As a consequence these states can withstand an
arbitrary amount of white noise before they admit a description within a local
hidden variable model. This is in sharp contrast with the bipartite case, where
all violations are bounded by Grothendieck's constant. We will discuss the
possibility of determining the Hilbert space dimension from the obtained
violation and comment on implications for communication complexity theory.
Moreover, we show that the violation obtained from generalized GHZ states is
always bounded so that, in contrast to many other contexts, GHZ states do in
this case not lead to extremal quantum correlations. The results are based on
tools from the theories of operator spaces and tensor norms which we exploit to
prove the existence of bounded but not completely bounded trilinear forms from
commutative C*-algebras.Comment: Substantial changes in the presentation to make the paper more
accessible for a non-specialized reade
Large violation of Bell inequalities with low entanglement
In this paper we obtain violations of general bipartite Bell inequalities of
order with inputs, outputs and
-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
Extendibility of bilinear forms on banach sequence spaces
[EN] We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize c(0) in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.The second author was supported by MICINN Project MTM2011-22417.DANIEL CARANDO; Sevilla Peris, P. (2014). Extendibility of bilinear forms on banach sequence spaces. Israel Journal of Mathematics. 199(2):941-954. https://doi.org/10.1007/s11856-014-0003-9S9419541992F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Graduate Texts in Mathematics, Vol. 233, Springer, New York, 2006.R. Arens, The adjoint of a bilinear operation, Proceedings of the American Mathematical Society 2 (1951), 839–848.R. Arens, Operations induced in function classes, Monatshefte für Mathematik 55 (1951), 1–19.R. M. Aron and P. D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bulletin de la Société Mathématique de France 106 (1978), 3–24.S. Banach, Sur les fonctionelles linéaires, Studia Mathematica 1 (1929), 211–216.S. Banach, Théorie des opérations linéaires, (Monogr. Mat. 1) Warszawa: Subwncji Funduszu Narodowej. VII, 254 S., Warsaw, 1932.D. Carando, Extendible polynomials on Banach spaces, Journal of Mathematical Analysis and Applications 233 (1999), 359–372.D. Carando, Extendibility of polynomials and analytic functions on l p, Studia Mathematica 145 (2001), 63–73.D. Carando, V. Dimant and P. Sevilla-Peris, Limit orders and multilinear forms on lp spaces, Publications of the Research Institute for Mathematical Sciences 42 (2006), 507–522.J. M. F. Castillo, R. García, A. Defant, D. Pérez-García and J. Suárez, Local complementation and the extension of bilinear mappings, Mathematical Proceedings of the Cambridge Philosophical Society 152 (2012), 153–166.J. M. F. Castillo, R. García and J. A. Jaramillo, Extension of bilinear forms on Banach spaces, Proceedings of the American Mathematical Society 129 (2001), 3647–3656.P. Cembranos and J. Mendoza, The Banach spaces ℓ ∞(c 0) and c 0(ℓ ∞) are not isomorphic, Journal of Mathematical Analysis and Applications 367 (2010), 461–463.A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Mathematics Studies, Vol. 176, North-Holland Publishing Co., Amsterdam, 1993.A. Defant and C. Michels, Norms of tensor product identities, Note di Matematica 25 (2005/06), 129–166.J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press, Cambridge, 1995.D. J. H. Garling, On symmetric sequence spaces, Proceedings of the London Mathematical Society (3) 16 (1966), 85–106.A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79.H. Hahn, Über lineare Gleichungssysteme in linearen Räumen, Journal für die Reine und Angewandte Mathematik 157 (1927), 214–229.R. C. James, Bases and reflexivity of Banach spaces, Annals of Mathematics (2) 52 (1950), 518–527.H. Jarchow, C. Palazuelos, D. Pérez-García and I. Villanueva, Hahn-Banach extension of multilinear forms and summability, Journal of Mathematical Analysis and Applications 336 (2007), 1161–1177.W. B. Johnson and L. Tzafriri, On the local structure of subspaces of Banach lattices, Israel Journal of Mathematics 20 (1975), 292–299.P. Kirwan and R. A. Ryan, Extendibility of homogeneous polynomials on Banach spaces, Proceedings of the American Mathematical Society 126 (1998), 1023–1029.J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in Lp-spaces and their applications, Studia Mathematica 29 (1968), 275–326.J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Vol. 97, Springer-Verlag, Berlin, 1979. Function spaces.G. Pisier, 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.M. Fernndez-Unzueta and A. Prieto, Extension of polynomials defined on subspaces, Mathematical Proceedings of the Cambridge Philosophical Society 148 (2010), 505–518.W. L. C. Sargent, Some sequence spaces related to the lp spaces, Journal of the London Mathematical Society 35 (1960), 161–171.N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 38, Longman Scientific & Technical, Harlow, 1989
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