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SUMMARY:Universal points in the asymptotic spectrum of tensors - Matthias 
 Christandl\, University of Copenhagen
DTSTART:20171115T170000Z
DTEND:20171115T180000Z
UID:TALK86871@talks.cam.ac.uk
CONTACT:Steve Brierley
DESCRIPTION:The asymptotic restriction problem for tensors is to decide\, 
 given tensors s and t\, whether the nth tensor power of s can be obtained 
 from the (n+o(n))th tensor power of t by applying linear maps to the tenso
 r legs (this we call restriction)\, when n goes to infinity. In this conte
 xt\, Volker Strassen\, striving to understand the complexity of matrix mul
 tiplication\, introduced in 1986 the asymptotic spectrum of tensors. Essen
 tially\, the asymptotic restriction problem for a family of tensors X\, cl
 osed under direct sum and tensor product\, reduces to finding all maps fro
 m X to the reals that are monotone under restriction\, normalised on diago
 nal tensors\, additive under direct sum and multiplicative under tensor pr
 oduct\, which Strassen named spectral points. Strassen created the support
  functionals\, which are spectral points for oblique tensors\, a strict su
 bfamily of all tensors. \nUniversal spectral points are spectral points fo
 r the family of all tensors. The construction of nontrivial universal spec
 tral points has been an open problem for more than thirty years. We constr
 uct for the first time a family of nontrivial universal spectral points ov
 er the complex numbers\, using quantum entropy and covariants: the quantum
  functionals. In the process we connect the asymptotic spectrum to the qua
 ntum marginal problem and to the entanglement polytope. \nTo demonstrate t
 he asymptotic spectrum\, we reprove (in hindsight) recent results on the c
 ap set problem by reducing this problem to computing asymptotic spectrum o
 f the reduced polynomial multiplication tensor\, a prime example of Strass
 en. A better understanding of our universal spectral points construction m
 ay lead to further progress on related questions. We additionally show tha
 t the quantum functionals are an upper bound on the recently introduced (m
 ulti-)slice rank.
LOCATION:MR5\, Centre for Mathematical Sciences\, Wilberforce Road\, Cambr
 idge
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