Explicit non-free tensors
Maxim van den Berg, Matthias Christandl, Vladimir Lysikov, Harold Nieuwboer, Michael Walter, Jeroen Zuiddam
TL;DR
The paper addresses the existence of explicit non-free tensors in $\mathbb{C}^n\otimes\mathbb{C}^n\otimes\mathbb{C}^n$ for all $n\ge 3$, key for understanding freeness in the context of moment polytopes and quantum multipartite states. It develops a geometric invariant theory framework built around the moment map and moment polytopes, establishing a criterion that reduces non-freeness questions to unitary actions on minimal-norm orbit representatives. The authors construct explicit witnesses: in 3×3×3, two concise non-free tensors $\mathsf{T}_2$ and $\mathsf{T}_5$; in general, a family $T^W$ supported on $\Gamma_n$ yielding non-free tensors for all $n\ge 3$, and a corresponding non-free $0/1$ tensor. These results illuminate the structure of non-free tensors, connect to Nurmiev’s classification, and provide concrete tools for exploring moment polytopes and potential separations in related asymptotic spectral theories.
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.