显式非自由张量
Explicit non-free tensors
AI总结：
本文通过几何不变量理论和矩多面体方法，为每个n≥3显式构造了C^n⊗C^n⊗C^n中的非自由张量，解决了此前缺乏显式非自由张量的问题，并确立了非自由张量的存在性。
摘要：
自由张量是指在基变换后具有自由支撑的张量：其支撑中任意两个不同元素至少有两个坐标不同。自由张量在双线性复杂度理论中扮演着特殊角色，特别是在Strassen关于渐近秩的对偶理论中。在量子信息论的背景下，张量被解释为多粒子量子态，自由性对应于一种多粒子Schmidt分解。特别地，如果某个态在给定基下是自由的，则其约化密度矩阵是对角的。尽管通过参数计数可知，对于n≥4，C^n⊗C^n⊗C^n中的一般张量是非自由的，但此前并未已知任何显式的非自由张量。我们解决了这一“大海捞针”问题，为每个n≥3构造了显式的非自由张量。这特别确立了非自由张量在C^n⊗C^n⊗C^n中的存在性，而它们并非一般张量。为确立非自由性，我们使用了几何不变量理论和矩多面体理论的结果。特别地，我们证明若张量T是自由的，则存在T的GL轨道闭包中的张量S，其支撑是自由的，且其矩映射像是T的矩多面体的最小范数点。这给出了从T的任意基变换到S的酉基变换来检验非自由性的约化。矩映射的酉等变性可与自由支撑张量具有对角矩映射像这一事实结合，以进一步限制相关基变换的集合。
英文摘要：
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.
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