Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations

TL;DR

This work analyzes how border ranks can diverge from standard ranks in multipartite tensor decompositions under positive and invariant constraints, revealing gaps for many natural decompositions (including MPS/MPO, cyclic, and ti variants) and linking these gaps to nonclosed sets of quantum correlations. Using the ( Omega,G) framework, the authors establish precise correspondences between tensor decompositions and density/marginal distributions, and show that border-rank gaps imply nonclosed correlation sets, with notable implications for membership testing and resource quantification. While standard and symmetric decompositions exhibit gaps, the paper also proves nonexistence of border-rank gaps for nonnegative/separable standard decompositions and for tree-structured decompositions, illustrating a nuanced landscape across geometries. The results illuminate the instability of ranks under approximation, the special behavior of bipartite versus multipartite systems, and provide a foundation for deeper study of correlation complexity in quantum and classical settings.

Abstract

The matrix rank and its positive versions are robust for small approximations, i.e. they do not decrease under small perturbations. In contrast, the multipartite tensor rank can collapse for arbitrarily small errors, i.e. there may be a gap between rank and border rank, leading to instabilities in the optimization over sets with fixed tensor rank. Can multipartite positive ranks also collapse for small perturbations? In this work, we prove that multipartite positive and invariant tensor decompositions exhibit gaps between rank and border rank, including tensor rank purifications and cyclic separable decompositions. We also prove a correspondence between positive decompositions and membership in certain sets of multipartite probability distributions, and leverage the gaps between rank and border rank to prove that these correlation sets are not closed. It follows that testing membership of probability distributions arising from resources like translational invariant Matrix Product States is impossible in finite time. Overall, this work sheds light on the instability of ranks and the unique behavior of bipartite systems.

Figures (9)

• Figure 1: Border rank. Given a tensor $T$ in an $n$-fold tensor product space and a certain type of rank $\mathop{\mathrm{\operatorname{t-rank}}}\nolimits$, if there exists a family of tensors $(T_{\varepsilon})_{\varepsilon > 0}$ such that $T_{\varepsilon} \to T$ for $\varepsilon \to 0$ and $\mathop{\mathrm{\operatorname{t-rank}}}\nolimits(T_{\varepsilon}) < \mathop{\mathrm{\operatorname{t-rank}}}\nolimits(T)$ for all $\varepsilon > 0$, we say that $\mathop{\mathrm{\operatorname{t-rank}}}\nolimits$ exhibits a gap between rank and border rank.
• Figure 2: Is there a gap between rank and border rank in an $n$-fold tensor product space? This table summarizes known results and the contributions of this paper (marked boldface): We prove that gaps persist when imposing positivity constrains corresponding to quantum correlation scenarios (second row), and that certain gaps disappear for stronger positivity constrains corresponding to classical correlation scenarios (third row). The types of ranks and of decompositions are defined in \ref{['sec:posTensorDec']}.
• Figure 3: Implications of border ranks for correlations. (a) The translational invariant (t.i.) cyclic psd-rank characterizes the minimal bond dimension to generate this distribution via a t.i. MPS together with local measurements, where $P$ is a nonnegative tensor representing an $n$-partite probability distribution. The gap between rank and border rank implies that the set of probability distributions generated in such a way is not closed. The same applies when replacing the cyclic graph and translational invariance by other decomposition geometries and symmetries (see \ref{['thm:quantumCorrMainText']}). (b) Gaps between border rank and rank also imply that the set of $n$-partite density matrices arising via MPS of bond dimension $r$ and local quantum channels is not closed.
• Figure 4: Abbreviations of all the different notions of ranks. The columns represent different arrangements of indices in the decomposition. The rows distinguish between the positivity constraints imposed on the decomposition. The term $\mathop{\mathrm{\operatorname{osr}}}\nolimits$ stands for the operator Schmidt rank, the abbreviation ti stands for translational invariant, and symm stands for symmetric.
• Figure 5: Examples of decomposition types. Hypergraph structures $\Omega$ give rise to different tensor decompositions. Each vertex corresponds to a local space and each (hyper-)edge corresponds to an index in the decomposition. If $\Omega$ is the $n$-simplex, we obtain standard tensor decomposition (a), for the $n$-cycle, we obtain the cyclic decomposition (b), for a line of length $n$ we get the decomposition in (c) and for a tree, we obtain a tree tensor network (d).

Theorems & Definitions (14)

• Lemma 1
• Conjecture 2
• Theorem 3
• Theorem 4
• Lemma 5
• Theorem 6
• Theorem 7
• proof
• Definition 8
• Definition 9