可分集的半定扩展复杂度及其在近似解纠缠器中的应用

中文摘要 AI 辅助

我们证明了在$\mathbb{C}^d\otimes\mathbb{C}^d$上可分量子态集合的半定扩展复杂度的定量下界。我们考虑半定规划（SDP），它近似测量在可分态上的最大接受概率，这是QMA(2)背后的优化问题。在Harrow、Natarajan和Wu（HNW）的扩展公式模型中，所有测量共享一个共同的可行区域和一个与目标无关的乘积态嵌入，该嵌入精确重现其接受概率。对于每个$0<\theta<2/7$，存在常数$c_\theta,a_\theta>0$，使得对于足够大的$d$，任何具有均匀加性误差$0<a\le a_\theta$的此类SDP的大小至少为$d^{c_\theta\min\{a^{-1/3},d^\theta\}}$。该下界在足够小的常数误差下适用，当$a=o(1)$时在$d$上是超多项式的，并且当$a\le d^{-3\theta}$时变为$d^{\Omega(d^\theta)}$，改进了HNW在平方反比误差下的拟多项式界。同样的下界适用于任何SDP可表示的凸状态集，该集合包含所有可分态并且与它们处于迹距离$a$内，从而为Fawzi定理（可分集没有精确的半定表示）提供了定量对应。我们的证明结合了Lee、Raghavendra和Steurer的定量伪密度定理与显式块正算子和切比雪夫放大。我们的主要结果由Lean证明支持。

英文摘要

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.

发表机构

• Paderborn University（帕德博恩大学）
• PhoQS（量子物理与量子系统研究中心）

机构由 AI 辅助整理，请以论文原文为准。