Semidefinite extension complexity of the separable set, with applications to approximate disentanglers

RESUMO

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.