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.

翻译：自由张量是指经过基变换后具有自由支撑的张量：其支撑集中任意两个不同元素至少在两个坐标上存在差异。这类张量在双线性复杂度理论中具有特殊地位，特别是在Strassen关于渐近秩的对偶理论中。在量子信息理论中，张量被解释为多粒子量子态，自由性对应于一类多粒子施密特分解。具体而言，若某量子态在给定基下是自由的，则其约化密度矩阵呈对角形式。尽管通过参数计数可知，当$n \geq 4$时$\mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^n$中的一般张量均为非自由张量，但此前尚未发现显式的非自由张量实例。我们通过为每个$n \geq 3$构造显式的非自由张量，解决了这一"草堆寻针"问题。特别地，这证实了在$\mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^n$中存在非一般性的非自由张量。为证明非自由性，我们运用几何不变量理论与矩多面体理论的相关结果。具体而言，我们证明：若张量$T$是自由的，则在其GL轨道闭包中存在张量$S$，该张量具有自由支撑且其矩映射像恰为$T$的矩多面体中的最小范数点。这一结论将检验非自由性的问题，从对$T$的任意基变换转化为对$S$的酉基变换。结合矩映射的酉等变性，以及自由支撑张量的矩映射像必为对角矩阵这一事实，可进一步限制相关基变换的搜索范围。