preprint unpublished Sevag Gharibian author 715410000-0002-9992-3379 Carsten Hecht author Dorian Rudolph author 57863 We prove quantitative lower bounds on the semidefinite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. 2026 eng 2609.09033 S. Gharibian, C. Hecht, D. Rudolph, ArXiv:2609.09033 (n.d.). 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. 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>. S. Gharibian, C. Hecht, and D. Rudolph, “Semidefinite extension complexity of the separable set, with applications to approximate disentanglers,” <i>arXiv:2609.09033</i>. . Gharibian S, Hecht C, Rudolph D. Semidefinite extension complexity of the separable set, with applications to approximate disentanglers. <i>arXiv:260909033</i>. @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} } Gharibian, Sevag, et al. “Semidefinite Extension Complexity of the Separable Set, with Applications to Approximate Disentanglers.” <i>ArXiv:2609.09033</i>. 670882026-09-09T12:06:36Z2026-09-09T12:07:31Z