BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Apoorva Khare (Indian Institute of Science\, India) DTSTART:20201118T120000Z DTEND:20201118T130000Z DTSTAMP:20260404T111133Z UID:FAOT/1 DESCRIPTION:Title: Entrywise positivity preservers in fixed dimension: I\nby Apoorva Khare (Indian Institute of Science\, India) as part of Functional Analysis and Operator Theory Webinar\n\n\nAbstract\nWhich functions preserve posit ive semidefiniteness (psd) when applied entrywise to\nthe entries of psd m atrices? This question has a long history beginning with Schur\,\nSchoenbe rg\, and Rudin\, who classified the positivity preservers of matrices of a ll dimensions. The study of positivity preservers in fixed dimension is ha rder\, and a complete\ncharacterization remains elusive to date. In fact u ntil recent work\, it was not known if\nthere exists any analytic preserve r with negative coefficients.\n\nIn my first talk\, I will explain the cla ssical history and modern motivations of this\nproblem\, followed by a “ restricted” solution in every dimension. Central to the proof\nare novel determinantal identities involving Schur polynomials. I will conclude wit h a\nfew outstanding questions.\n\n(Based on two papers: with Alexander Be lton\, Dominique Guillot\, and Mihai Putinar\, Adv. Math. 2016\; and with Terence Tao\, Amer. J. Math.\, in press.)\n LOCATION:https://stable.researchseminars.org/talk/FAOT/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Apoorva Khare (Indian Institute of Science\, India) DTSTART:20201125T120000Z DTEND:20201125T130000Z DTSTAMP:20260404T111133Z UID:FAOT/2 DESCRIPTION:Title: Entrywise positivity preservers in fixed dimension: II\nby Apoorva Khare (Indian Institute of Science\, India) as part of Functional Analysi s and Operator Theory Webinar\n\n\nAbstract\nThe second talk in this serie s will (after a quick introduction) focus on how to\nresolve the outstandi ng questions from the first talk\, using additional tools from symmetric f unction theory and type A representation theory. These tools help extend p rior\nresults from entrywise polynomial preservers to finite and infinite sums of real powers\,\nacting on positive matrices with positive entries. We conclude with a novel characterization of weak majorization of real tup les\, via Schur polynomials and Vandermonde\ndeterminants\, and use it to strengthen and extend the Cuttler–Greene–Skandera/Sra\ncharacterizatio n of majorization to all real tuples.\n\n(Based on two papers: with Alexan der Belton\, Dominique Guillot\, and Mihai Putinar\, Adv. Math. 2016\; and with Terence Tao\, Amer. J. Math.\, in press.)\n LOCATION:https://stable.researchseminars.org/talk/FAOT/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Karol Życzkowski (Jagiellonian University and Polish Academy of S ciences\, Poland) DTSTART:20201216T120000Z DTEND:20201216T130000Z DTSTAMP:20260404T111133Z UID:FAOT/3 DESCRIPTION:Title: The set of quantum states analyzed by numerical range and numerical sh adow of an operator\nby Karol Życzkowski (Jagiellonian University and Polish Academy of Sciences\, Poland) as part of Functional Analysis and O perator Theory Webinar\n\n\nAbstract\nThe set $\\Omega_N$ of density matri ces - positive hermitian matrices of order N with trace equal to unity - p lays a key role in the theory of quantum information processing. It is a c onvex set embedded in $\\mathbb{R}^{N^2-1}$ with an involved structure\, w hich for $N=2$ reduces to the 3-ball.\n\nNumerical range $W(X)$ (also call ed field of values) of an operator \n$X$ of size $N$ can be considered as a projection of $\\Omega_N$ into a 2-plane. Further structure of the set $ \\Omega_N$ of quantum states is revealed by the numerical shadow of an ope rator - a probability measure \non the complex plane\, $P_X(z)$\, supporte d by the numerical range $W(X)$. The shadow of $X$ at point $z$ is defined as the probability that the inner product $(Xu\, u)$ is equal to $z$\, wh ere u stands for a normalized $N$-dimensional random complex vector. In t he case of $N = 2$ the numerical shadow of a non-normal operator can be in terpreted as a shadow\nof a hollow sphere projected on a plane.\n\nStudyin g joint numerical range of three hermitian operators\, $W(H_1\,H_2\,H_3)$\ , one can analyze projections of $\\Omega_N$ into a 3-space. A classificat ion\nof possible shapes of 3D numerical ranges of three hermitian operator s of order three is presented.\n LOCATION:https://stable.researchseminars.org/talk/FAOT/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Mark M. Wilde (Louisiana State University\, USA) DTSTART:20210120T130000Z DTEND:20210120T140000Z DTSTAMP:20260404T111133Z UID:FAOT/4 DESCRIPTION:Title: α-Logarithmic negativity\nby Mark M. Wilde (Louisiana State Unive rsity\, USA) as part of Functional Analysis and Operator Theory Webinar\n\ n\nAbstract\nThe logarithmic negativity of a bipartite quantum state is a widely employed entanglement measure in quantum information theory\, due t o the fact that it is easy to compute and serves as an upper bound on dist illable entanglement. More recently\, the $\\kappa$-entanglement of a bipa rtite state was shown to be the first entanglement measure that is both ea sily computable and has a precise information-theoretic meaning\, being eq ual to the exact entanglement cost of a bipartite quantum state when the f ree operations are those that completely preserve the positivity of the pa rtial transpose [Wang and Wilde\, Phys. Rev. Lett. 125(4):040502\, July 20 20]. \n\nIn this talk\, we discuss a non-trivial link between these two en tanglement measures\, by showing that they are the extremes of an ordered family of $\\alpha$-logarithmic negativity entanglement measures\, each of which is identified by a parameter $\\alpha\\in[1\,\\infty]$. In this fam ily\, the original logarithmic negativity is recovered as the smallest wit h $\\alpha=1$\, and the $\\kappa$-entanglement is recovered as the largest with $\\alpha=\\infty$. We prove that the $\\alpha$-logarithmic negativit y satisfies the following properties: entanglement monotone\, normalizatio n\, faithfulness\, and subadditivity. We also prove that it is neither con vex nor monogamous. Finally\, we define the $\\alpha$-logarithmic negativi ty of a quantum channel as a generalization of the notion for quantum stat es\, and we show how to generalize many of the concepts to arbitrary resou rce theories.\n LOCATION:https://stable.researchseminars.org/talk/FAOT/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Andreas Winter ((Universitat Autònoma de Barcelona\, Spain)) DTSTART:20210217T120000Z DTEND:20210217T130000Z DTSTAMP:20260404T111133Z UID:FAOT/5 DESCRIPTION:Title: Entropy inequalities – beyond strong subadditivity(?)\nby Andrea s Winter ((Universitat Autònoma de Barcelona\, Spain)) as part of Functio nal Analysis and Operator Theory Webinar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/FAOT/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Marcus Huber (Insititute for Quantum Optics and Quantum Informatio n Vienna\, Austria) DTSTART:20210303T120000Z DTEND:20210303T130000Z DTSTAMP:20260404T111133Z UID:FAOT/6 DESCRIPTION:Title: The Bloch representation for qudits: overview and applications\nby Marcus Huber (Insititute for Quantum Optics and Quantum Information Vienn a\, Austria) as part of Functional Analysis and Operator Theory Webinar\n\ n\nAbstract\nThe Bloch representation for qubits is taught in every basic quantum mechanics course and for a good reason. It manages to visualise an d elegantly describe important features of two-dimensional Hilbert spaces. Going to higher-dimensional or multipartite systems\, the visualisation i s of course more challenging\, but a lot of convenient properties remain a nd can also be used to derive various results in quantum information. I wi ll give a brief overview of the Bloch representation for qudits\, showcase its most important properties and present two simple\, yet powerful\, app lications in entanglement theory and entropy inequalities.\n LOCATION:https://stable.researchseminars.org/talk/FAOT/6/ END:VEVENT END:VCALENDAR