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SUMMARY:Apoorva Khare (Indian Institute of Science\, India)
DTSTART;VALUE=DATE-TIME:20201118T120000Z
DTEND;VALUE=DATE-TIME:20201118T130000Z
DTSTAMP;VALUE=DATE-TIME:20230208T064708Z
UID:FAOT/1
DESCRIPTION:Title: Ent
rywise positivity preservers in fixed dimension: I\nby Apoorva Khare (
Indian Institute of Science\, India) as part of Functional Analysis and Op
erator Theory Webinar\n\n\nAbstract\nWhich functions preserve positive sem
idefiniteness (psd) when applied entrywise to\nthe entries of psd matrices
? This question has a long history beginning with Schur\,\nSchoenberg\, an
d Rudin\, who classified the positivity preservers of matrices of all dime
nsions. The study of positivity preservers in fixed dimension is harder\,
and a complete\ncharacterization remains elusive to date. In fact until re
cent work\, it was not known if\nthere exists any analytic preserver with
negative coefficients.\n\nIn my first talk\, I will explain the classical
history and modern motivations of this\nproblem\, followed by a “restric
ted” solution in every dimension. Central to the proof\nare novel determ
inantal identities involving Schur polynomials. I will conclude with a\nfe
w outstanding questions.\n\n(Based on two papers: with Alexander Belton\,
Dominique Guillot\, and Mihai Putinar\, Adv. Math. 2016\; and with Terence
Tao\, Amer. J. Math.\, in press.)\n
LOCATION:https://researchseminars.org/talk/FAOT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Apoorva Khare (Indian Institute of Science\, India)
DTSTART;VALUE=DATE-TIME:20201125T120000Z
DTEND;VALUE=DATE-TIME:20201125T130000Z
DTSTAMP;VALUE=DATE-TIME:20230208T064708Z
UID:FAOT/2
DESCRIPTION:Title: Ent
rywise positivity preservers in fixed dimension: II\nby Apoorva Khare
(Indian Institute of Science\, India) as part of Functional Analysis and O
perator Theory Webinar\n\n\nAbstract\nThe second talk in this series will
(after a quick introduction) focus on how to\nresolve the outstanding ques
tions from the first talk\, using additional tools from symmetric function
theory and type A representation theory. These tools help extend prior\nr
esults from entrywise polynomial preservers to finite and infinite sums of
real powers\,\nacting on positive matrices with positive entries. We conc
lude with a novel characterization of weak majorization of real tuples\, v
ia Schur polynomials and Vandermonde\ndeterminants\, and use it to strengt
hen and extend the Cuttler–Greene–Skandera/Sra\ncharacterization of ma
jorization to all real tuples.\n\n(Based on two papers: with Alexander Bel
ton\, Dominique Guillot\, and Mihai Putinar\, Adv. Math. 2016\; and with T
erence Tao\, Amer. J. Math.\, in press.)\n
LOCATION:https://researchseminars.org/talk/FAOT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karol Życzkowski (Jagiellonian University and Polish Academy of S
ciences\, Poland)
DTSTART;VALUE=DATE-TIME:20201216T120000Z
DTEND;VALUE=DATE-TIME:20201216T130000Z
DTSTAMP;VALUE=DATE-TIME:20230208T064708Z
UID:FAOT/3
DESCRIPTION:Title: The
set of quantum states analyzed by numerical range and numerical shadow of
an operator\nby Karol Życzkowski (Jagiellonian University and Polish
Academy of Sciences\, Poland) as part of Functional Analysis and Operator
Theory Webinar\n\n\nAbstract\nThe set $\\Omega_N$ of density matrices - p
ositive hermitian matrices of order N with trace equal to unity - plays a
key role in the theory of quantum information processing. It is a convex s
et embedded in $\\mathbb{R}^{N^2-1}$ with an involved structure\, which fo
r $N=2$ reduces to the 3-ball.\n\nNumerical range $W(X)$ (also called fiel
d of values) of an operator \n$X$ of size $N$ can be considered as a proje
ction 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 operator -
a probability measure \non the complex plane\, $P_X(z)$\, supported by th
e 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 $z$\, where u s
tands for a normalized $N$-dimensional random complex vector. In the case
of $N = 2$ the numerical shadow of a non-normal operator can be interpret
ed as a shadow\nof a hollow sphere projected on a plane.\n\nStudying joint
numerical range of three hermitian operators\, $W(H_1\,H_2\,H_3)$\, one c
an analyze projections of $\\Omega_N$ into a 3-space. A classification\nof
possible shapes of 3D numerical ranges of three hermitian operators of or
der three is presented.\n
LOCATION:https://researchseminars.org/talk/FAOT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark M. Wilde (Louisiana State University\, USA)
DTSTART;VALUE=DATE-TIME:20210120T130000Z
DTEND;VALUE=DATE-TIME:20210120T140000Z
DTSTAMP;VALUE=DATE-TIME:20230208T064708Z
UID:FAOT/4
DESCRIPTION:Title: α-
Logarithmic negativity\nby Mark M. Wilde (Louisiana State University\,
USA) as part of Functional Analysis and Operator Theory Webinar\n\n\nAbst
ract\nThe logarithmic negativity of a bipartite quantum state is a widely
employed entanglement measure in quantum information theory\, due to the f
act that it is easy to compute and serves as an upper bound on distillable
entanglement. More recently\, the $\\kappa$-entanglement of a bipartite s
tate was shown to be the first entanglement measure that is both easily co
mputable and has a precise information-theoretic meaning\, being equal to
the exact entanglement cost of a bipartite quantum state when the free ope
rations are those that completely preserve the positivity of the partial t
ranspose [Wang and Wilde\, Phys. Rev. Lett. 125(4):040502\, July 2020]. \n
\nIn this talk\, we discuss a non-trivial link between these two entanglem
ent 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 family\, t
he original logarithmic negativity is recovered as the smallest with $\\al
pha=1$\, and the $\\kappa$-entanglement is recovered as the largest with $
\\alpha=\\infty$. We prove that the $\\alpha$-logarithmic negativity satis
fies the following properties: entanglement monotone\, normalization\, fai
thfulness\, and subadditivity. We also prove that it is neither convex nor
monogamous. Finally\, we define the $\\alpha$-logarithmic negativity of a
quantum channel as a generalization of the notion for quantum states\, an
d we show how to generalize many of the concepts to arbitrary resource the
ories.\n
LOCATION:https://researchseminars.org/talk/FAOT/4/
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BEGIN:VEVENT
SUMMARY:Andreas Winter ((Universitat Autònoma de Barcelona\, Spain))
DTSTART;VALUE=DATE-TIME:20210217T120000Z
DTEND;VALUE=DATE-TIME:20210217T130000Z
DTSTAMP;VALUE=DATE-TIME:20230208T064708Z
UID:FAOT/5
DESCRIPTION:Title: Ent
ropy inequalities – beyond strong subadditivity(?)\nby Andreas Winte
r ((Universitat Autònoma de Barcelona\, Spain)) as part of Functional Ana
lysis and Operator Theory Webinar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FAOT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marcus Huber (Insititute for Quantum Optics and Quantum Informatio
n Vienna\, Austria)
DTSTART;VALUE=DATE-TIME:20210303T120000Z
DTEND;VALUE=DATE-TIME:20210303T130000Z
DTSTAMP;VALUE=DATE-TIME:20230208T064708Z
UID:FAOT/6
DESCRIPTION:Title: The
Bloch representation for qudits: overview and applications\nby Marcus
Huber (Insititute for Quantum Optics and Quantum Information Vienna\, Aus
tria) as part of Functional Analysis and Operator Theory Webinar\n\n\nAbst
ract\nThe Bloch representation for qubits is taught in every basic quantum
mechanics course and for a good reason. It manages to visualise and elega
ntly describe important features of two-dimensional Hilbert spaces. Going
to higher-dimensional or multipartite systems\, the visualisation is of co
urse more challenging\, but a lot of convenient properties remain and can
also be used to derive various results in quantum information. I will give
a brief overview of the Bloch representation for qudits\, showcase its mo
st important properties and present two simple\, yet powerful\, applicatio
ns in entanglement theory and entropy inequalities.\n
LOCATION:https://researchseminars.org/talk/FAOT/6/
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