Semidefinite extension complexity of the separable set, with applications to approximate disentanglers
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 $0<\theta<2/7$, there are constants $c_\theta,a_\theta>0$ such that, for sufficiently large $d$, any such SDP with uniform additive error $0<a\le a_\theta$ has size at least $d^{c_\theta\min\{a^{-1/3},d^\theta\}}$. The bound applies at sufficiently small constant error, is superpolynomial in $d$ whenever $a=o(1)$, and becomes $d^{\Omega(d^\theta)}$ when $a\le d^{-3\theta}$, 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.