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arXiv:2503.22650

Explicit non-free tensors

Maxim van den Berg, Matthias Christandl, Vladimir Lysikov, Harold A. Nieuwboer, Michael Walter, Jeroen Zuiddam

Published Mar 28, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2503.22650

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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 Cn⊗Cn⊗Cn\mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^nCn⊗Cn⊗Cn are non-free for n≥4n \geq 4n≥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≥3n \geq 3n≥3. In particular, this establishes that non-free tensors exist in Cn⊗Cn⊗Cn\mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^nCn⊗Cn⊗Cn, 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 TTT is free, then there is a tensor SSS in the GL-orbit closure of TTT, whose support is free and whose moment map image is the minimum-norm point of the moment polytope of TTT. This implies a reduction for checking non-freeness from arbitrary basis changes of TTT to unitary basis changes of SSS. 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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Upper and lower quantum functionals, introduced by Christandl, Vrana and Zuiddam (STOC 2018, J. Amer. Math. Soc. 2023), are families of monotone functions of tensors indexed by a weighting on the set of subsets of the tensor legs. Inspired by quantum information theory, they were crafted as obstructions to asymptotic tensor transformations, relevant in algebraic complexity theory. For tensors of order three, and more generally for weightings on singletons for higher-order tensors, the upper and lower quantum functionals coincide and are spectral points in Strassen's asymptotic spectrum. Moreover, the singleton quantum functionals characterize the asymptotic slice rank, whereas general weightings provide upper bounds on asymptotic partition rank. It has been an open question whether the upper and lower quantum functionals also coincide for other cases, or more generally, how to construct further spectral points, especially for higher-order tensors. In this work, we show that upper and lower quantum functionals generally do not coincide, but that they anchor new spectral points. With this we mean that there exist new spectral points, which equal the quantum functionals on the set of tensors on which upper and lower coincide. The set is shown to include embedded three-tensors and W-like states and concerns all laminar weightings, significantly extending the singleton case. Moreover, it is shown that these spectral points provide obstructions to asymptotic restriction beyond the previously known spectral points.
The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor QQQ with an error allowance ΔΔΔ is to find vectors ϕiϕ^iϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ\|Q-\sum_{i=1}^R ϕ^i\otimes ϕ^i\cdots \otimes ϕ^i\|^2 \leq Δ∥Q−∑i=1R​ϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all possible such ϕiϕ^iϕi is an interesting quantity which measures the amount of possible decompositions for a tensor QQQ within an allowance. While it would be difficult to evaluate this quantity for each QQQ, we find an explicit formula for a similar quantity by integrating over all QQQ of unit norm. The expression as a function of ΔΔΔ is given by the product of a hypergeometric function and a power function. We also extend the formula to generic decompositions of non-symmetric tensors. The derivation depends on the existence (convergence) of the partition function of a matrix model which appeared in the context of the CTM.
Motivated by the problem of constructing fast matrix multiplication algorithms, Strassen (FOCS 1986, Crelle 1987–1991) introduced and developed the theory of asymptotic spectra of tensors. For any sub-semiring X \mathcal {X} of tensors (under direct sum and tensor product), the duality theorem that is at the core of this theory characterizes basic asymptotic properties of the elements of X \mathcal {X} in terms of the asymptotic spectrum of X \mathcal {X} , which is defined as the collection of semiring homomorphisms from X \mathcal {X} to the non-negative reals with a natural monotonicity property. The asymptotic properties characterized by this duality encompass fundamental problems in complexity theory, combinatorics and quantum information. Universal spectral points are elements in the asymptotic spectrum of the semiring of all tensors. Finding all universal spectral points suffices to find the asymptotic spectrum of any sub-semiring. The construction of non-trivial universal spectral points has been an open problem for more than thirty years. We construct, for the first time, a family of non-trivial universal spectral points over the complex numbers, called quantum functionals . We moreover prove that the quantum functionals precisely characterise the asymptotic slice rank of complex tensors. Our construction, which relies on techniques from quantum information theory and representation theory, connects the asymptotic spectrum of tensors to the quantum marginal problem and entanglement polytopes.
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