  Strassen (Strassen, J. Reine Angew. Math., 375/376, 1987) introduced the
subrank of a tensor as a natural extension of matrix rank to tensors. Subrank
measures the largest diagonal tensor that can be obtained by applying linear
operations to the different indices (legs) of the tensor (just like the matrix
rank measures the largest diagonal matrix that can be obtained using row and
column operations). Motivated by problems in combinatorics and complexity
theory we introduce the new notion of symmetric subrank of tensors by
restricting these linear operations to be the same for each index.
  We prove precise relations and separations between subrank and symmetric
subrank. We prove that for symmetric tensors the subrank and the symmetric
subrank are asymptotically equal. This proves the asymptotic subrank analogon
of a conjecture known as Comon's conjecture in the theory of tensors. This
result allows us to prove a strong connection between the general and symmetric
version of an asymptotic duality theorem of Strassen. We introduce a
representation-theoretic method to asymptotically bound the symmetric subrank
called the symmetric quantum functional in analogy with the quantum functionals
(Christandl, Vrana, Zuiddam, J. Amer. Math. Soc., 2021), and we study the
relations between these functionals.
