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arXiv · 2509.09463

Minimality of Tree Tensor Network Ranks

Jana Jovcheva·Tim Seynnaeve·Nick Vannieuwenhoven
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Abstract

For a given tree tensor network $G$, we call a tuple of bond dimensions minimal if there exists a tensor $T$ that can be represented by this network but not on the same tree topology with strictly smaller bond dimensions. We establish necessary and sufficient conditions on the bond dimensions of a tree tensor network to be minimal, generalizing a characterization of Carlini and Kleppe about existence of tensors with a given multilinear rank. We also show that in a minimal tree tensor network, the non-minimal tensors form a Zariski closed subset, so minimality is a generic property in this sense.

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Jana Jovcheva, Tim Seynnaeve, Nick Vannieuwenhoven. 2025-09-11. Minimality of Tree Tensor Network Ranks. https://arxiv.org/abs/2509.09463

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