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SUMMARY:Karol Życzkowski (Jagiellonian University and Polish Academy of S
 ciences\, Poland)
DTSTART:20201216T120000Z
DTEND:20201216T130000Z
DTSTAMP:20260423T021219Z
UID:FAOT/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/FAOT/3/">The
  set of quantum states analyzed by numerical range and numerical shadow of
  an operator</a>\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/
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