[{"type":"preprint","external_id":{"arxiv":["2609.09033"]},"date_created":"2026-09-09T12:06:36Z","abstract":[{"text":"We prove quantitative lower bounds on the semidefinite extension complexity of the set of separable quantum states on $\\mathbb{C}^d\\otimes\\mathbb{C}^d$. We consider semidefinite programs (SDPs) that approximate the maximum acceptance probability of a measurement over separable states, the optimization problem underlying QMA(2). In the extended-formulation model of Harrow, Natarajan, and Wu (HNW), all measurements share a common feasible region and an objective-independent embedding of product states that exactly reproduces their acceptance probabilities. For every $0<θ<2/7$, there are constants $c_θ,a_θ>0$ such that, for sufficiently large $d$, any such SDP with uniform additive error $0<a\\le a_θ$ has size at least $d^{c_θ\\min\\{a^{-1/3},d^θ\\}}$. The bound applies at sufficiently small constant error, is superpolynomial in $d$ whenever $a=o(1)$, and becomes $d^{Ω(d^θ)}$ when $a\\le d^{-3θ}$, improving HNW's quasipolynomial bound at inverse- square error. The same bound holds for any SDP-representable convex set of states that contains all separable states and lies within trace distance $a$ of them, giving a quantitative counterpart to Fawzi's theorem that the separable set has no exact semidefinite representation. Our proof combines the quantitative pseudo-density theorem of Lee, Raghavendra, and Steurer with explicit block-positive operators and Chebyshev amplification. Our main results are supported by Lean proofs.","lang":"eng"}],"publication":"arXiv:2609.09033","citation":{"ama":"Gharibian S, Hecht C, Rudolph D. Semidefinite extension complexity of the separable set, with applications to approximate disentanglers. <i>arXiv:260909033</i>.","bibtex":"@article{Gharibian_Hecht_Rudolph, title={Semidefinite extension complexity of the separable set, with applications to approximate disentanglers}, journal={arXiv:2609.09033}, author={Gharibian, Sevag and Hecht, Carsten and Rudolph, Dorian} }","mla":"Gharibian, Sevag, et al. “Semidefinite Extension Complexity of the Separable Set, with Applications to Approximate Disentanglers.” <i>ArXiv:2609.09033</i>.","short":"S. Gharibian, C. Hecht, D. Rudolph, ArXiv:2609.09033 (n.d.).","chicago":"Gharibian, Sevag, Carsten Hecht, and Dorian Rudolph. “Semidefinite Extension Complexity of the Separable Set, with Applications to Approximate Disentanglers.” <i>ArXiv:2609.09033</i>, n.d.","apa":"Gharibian, S., Hecht, C., &#38; Rudolph, D. (n.d.). Semidefinite extension complexity of the separable set, with applications to approximate disentanglers. In <i>arXiv:2609.09033</i>.","ieee":"S. Gharibian, C. Hecht, and D. Rudolph, “Semidefinite extension complexity of the separable set, with applications to approximate disentanglers,” <i>arXiv:2609.09033</i>. ."},"user_id":"71541","_id":"67088","language":[{"iso":"eng"}],"date_updated":"2026-09-09T12:07:31Z","publication_status":"unpublished","status":"public","year":"2026","title":"Semidefinite extension complexity of the separable set, with applications to approximate disentanglers","author":[{"orcid":"0000-0002-9992-3379","last_name":"Gharibian","first_name":"Sevag","full_name":"Gharibian, Sevag","id":"71541"},{"full_name":"Hecht, Carsten","first_name":"Carsten","last_name":"Hecht"},{"id":"57863","last_name":"Rudolph","first_name":"Dorian","full_name":"Rudolph, Dorian"}]}]