Polynomial-Time Algorithms for Nuclear Tensor Norms and Multipartite Separability

Abstract:We study a multilinear optimization problem for tensors with bounded Frobenius norm, where each of the $k$ local factors of dimension $d$ is chosen from a convex set. We give a deterministic algorithm running in time $d^{O(k)}$ for constant additive approximations. As applications, we obtain the first polynomial-time algorithms at constant accuracy for weak membership in the nuclear-norm unit ball of high-order tensors and for multipartite quantum separability in Frobenius norm. Our approach views tensor optimization as a cooperative multiprover game and combines this perspective with a recursive spectral compression procedure. We then show that related ideas admit a quantum implementation when the input state is given through copies, rather than an explicit classical description. In these settings we obtain an algorithm to solve Frobenius-norm separability testing using $\text{poly}(k)$ copies and $\text{poly}(k\log d)$ time.

arXiv:2610.03637 [quant-ph]

(or arXiv:2610.03637v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.03637 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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