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CATEGORIES:CQIF Seminar
SUMMARY:Universal points in the asymptotic spectrum of ten
sors - Matthias Christandl\, University of Copenha
gen
DTSTART;TZID=Europe/London:20171115T170000
DTEND;TZID=Europe/London:20171115T180000
UID:TALK86871AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/86871
DESCRIPTION:The asymptotic restriction problem for tensors is
to decide\, given tensors s and t\, whether the nt
h tensor power of s can be obtained from the (n+o(
n))th tensor power of t by applying linear maps to
the tensor legs (this we call restriction)\, when
n goes to infinity. In this context\, Volker Stra
ssen\, striving to understand the complexity of ma
trix multiplication\, introduced in 1986 the asymp
totic spectrum of tensors. Essentially\, the asymp
totic restriction problem for a family of tensors
X\, closed under direct sum and tensor product\, r
educes to finding all maps from X to the reals tha
t are monotone under restriction\, normalised on d
iagonal tensors\, additive under direct sum and mu
ltiplicative under tensor product\, which Strassen
named spectral points. Strassen created the suppo
rt functionals\, which are spectral points for obl
ique tensors\, a strict subfamily of all tensors.
\nUniversal spectral points are spectral points fo
r the family of all tensors. The construction of n
ontrivial universal spectral points has been an op
en problem for more than thirty years. We construc
t for the first time a family of nontrivial univer
sal spectral points over the complex numbers\, usi
ng quantum entropy and covariants: the quantum fun
ctionals. In the process we connect the asymptotic
spectrum to the quantum marginal problem and to t
he entanglement polytope. \nTo demonstrate the asy
mptotic spectrum\, we reprove (in hindsight) recen
t results on the cap set problem by reducing this
problem to computing asymptotic spectrum of the re
duced polynomial multiplication tensor\, a prime e
xample of Strassen. A better understanding of our
universal spectral points construction may lead to
further progress on related questions. We additio
nally show that the quantum functionals are an upp
er bound on the recently introduced (multi-)slice
rank.
LOCATION:MR5\, Centre for Mathematical Sciences\, Wilberfor
ce Road\, Cambridge
CONTACT:Steve Brierley
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