Explicit non-free tensors
Journal:
arXiv
Published Date:
Mar 28, 2025
Abstract
Free tensors are tensors which, after a change of bases, have free support:
any two distinct elements of its support differ in at least two coordinates.
They play a distinguished role in the theory of bilinear complexity, in
particular in Strassen's duality theory for asymptotic rank. Within the context
of quantum information theory, where tensors are interpreted as multiparticle
quantum states, freeness corresponds to a type of multiparticle Schmidt
decomposition. In particular, if a state is free in a given basis, the reduced
density matrices are diagonal. Although generic tensors in $\mathbb{C}^n
\otimes \mathbb{C}^n \otimes \mathbb{C}^n$ are non-free for $n \geq 4$ by
parameter counting, no explicit non-free tensors were known until now. We solve
this hay in a haystack problem by constructing explicit tensors that are
non-free for every $n \geq 3$. In particular, this establishes that non-free
tensors exist in $\mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^n$,
where they are not generic.
To establish non-freeness, we use results from geometric invariant theory and
the theory of moment polytopes. In particular, we show that if a tensor $T$ is
free, then there is a tensor $S$ in the GL-orbit closure of $T$, whose support
is free and whose moment map image is the minimum-norm point of the moment
polytope of $T$. This implies a reduction for checking non-freeness from
arbitrary basis changes of $T$ to unitary basis changes of $S$. The unitary
equivariance of the moment map can then be combined with the fact that tensors
with free support have diagonal moment map image, in order to further restrict
the set of relevant basis changes.
