{"abstracts":[{"sha1":"e939fa972d492114c5d1cbf04ba660eddcbf7d85","content":"The asymptotic restriction problem for tensors is to decide, given tensors\ns and t, whether the nth tensor power of s can be obtained from the\n(n+o(n))th tensor power of t by applying linear maps to the tensor legs (this\nwe call restriction), when n goes to infinity. In this context, Volker\nStrassen, striving to understand the complexity of matrix multiplication,\nintroduced in 1986 the asymptotic spectrum of tensors. Essentially, the\nasymptotic restriction problem for a family of tensors X, closed under direct\nsum and tensor product, reduces to finding all maps from X to the reals that\nare monotone under restriction, normalised on diagonal tensors, additive under\ndirect sum and multiplicative under tensor product, which Strassen named\nspectral points. Strassen created the support functionals, which are spectral\npoints for oblique tensors, a strict subfamily of all tensors.\n Universal spectral points are spectral points for the family of all tensors.\nThe construction of nontrivial universal spectral points has been an open\nproblem for more than thirty years. We construct for the first time a family of\nnontrivial universal spectral points over the complex numbers, using quantum\nentropy and covariants: the quantum functionals. In the process we connect the\nasymptotic spectrum to the quantum marginal problem and to the entanglement\npolytope.\n To demonstrate the asymptotic spectrum, we reprove (in hindsight) recent\nresults on the cap set problem by reducing this problem to computing asymptotic\nspectrum of the reduced polynomial multiplication tensor, a prime example of\nStrassen. A better understanding of our universal spectral points construction\nmay lead to further progress on related questions. We additionally show that\nthe quantum functionals characterise asymptotic slice rank for complex tensors.","mimetype":"text/plain","lang":"en"},{"sha1":"2e2fa4045e0a7efd1019a63f09d42c57cfa8e06e","content":"The asymptotic restriction problem for tensors is to decide, given tensors\n$s$ and $t$, whether the nth tensor power of $s$ can be obtained from the\n$(n+o(n))$th tensor power of t by applying linear maps to the tensor legs (this\nwe call restriction), when $n$ goes to infinity. In this context, Volker\nStrassen, striving to understand the complexity of matrix multiplication,\nintroduced in 1986 the asymptotic spectrum of tensors. Essentially, the\nasymptotic restriction problem for a family of tensors $X$, closed under direct\nsum and tensor product, reduces to finding all maps from $X$ to the reals that\nare monotone under restriction, normalised on diagonal tensors, additive under\ndirect sum and multiplicative under tensor product, which Strassen named\nspectral points. Strassen created the support functionals, which are spectral\npoints for oblique tensors, a strict subfamily of all tensors.\n Universal spectral points are spectral points for the family of all tensors.\nThe construction of nontrivial universal spectral points has been an open\nproblem for more than thirty years. We construct for the first time a family of\nnontrivial universal spectral points over the complex numbers, using quantum\nentropy and covariants: the quantum functionals. In the process we connect the\nasymptotic spectrum to the quantum marginal problem and to the entanglement\npolytope.\n To demonstrate the asymptotic spectrum, we reprove (in hindsight) recent\nresults on the cap set problem by reducing this problem to computing asymptotic\nspectrum of the reduced polynomial multiplication tensor, a prime example of\nStrassen. A better understanding of our universal spectral points construction\nmay lead to further progress on related questions. We additionally show that\nthe quantum functionals characterise asymptotic slice rank for complex tensors.","mimetype":"application/x-latex","lang":"en"}],"refs":[],"contribs":[{"index":0,"raw_name":"Matthias Christandl","role":"author"},{"index":1,"raw_name":"Péter Vrana","role":"author"},{"index":2,"raw_name":"Jeroen Zuiddam","role":"author"}],"license_slug":"ARXIV-1.0","language":"en","version":"v1","ext_ids":{"arxiv":"1709.07851v1"},"release_year":2017,"release_date":"2017-09-22","release_stage":"submitted","release_type":"article","webcaptures":[],"filesets":[],"files":[{"release_ids":["hpga2s5e45cczewcekiwnqdbb4"],"mimetype":"application/pdf","urls":[{"url":"https://arxiv.org/pdf/1709.07851v1.pdf","rel":"repository"},{"url":"https://web.archive.org/web/20191021011650/https://arxiv.org/pdf/1709.07851v1.pdf","rel":"webarchive"}],"sha256":"70f87cd7e9830d4142552450f365651e40192339d944a2ca04b68c2c12e77b10","sha1":"43811cdc11f9654b1df39d94c0b80c6bdc6c4192","md5":"226e3cd71dfe32771709ba6c83d19b57","size":880819,"revision":"21d7cfbd-cae1-4635-9ced-0d214da24767","ident":"55qyxk5frvcxtcgmfipdx2dvkm","state":"active"}],"work_id":"wgabr3caqvfbvh3n56o2cuh66i","title":"Universal points in the asymptotic spectrum of tensors","state":"active","ident":"hpga2s5e45cczewcekiwnqdbb4","revision":"d7cf9c87-6bb2-48c5-969f-97df8d213d1b","extra":{"arxiv":{"base_id":"1709.07851","categories":["math.CO","cs.CC","quant-ph"]},"superceded":true}}