标题： 多沃雷茨基定理与纠缠检测的复杂性 显示英文标题

标题： Dvoretzky's Theorem and the Complexity of Entanglement Detection

摘要： 众所周知的Horodecki准则断言，一个状态$\rho$在$\mathbf{C}^d \otimes \mathbf{C}^d$上是纠缠的当且仅当存在一个正映射$\Phi : \mathsf{M}_d \to \mathsf{M}_d$，使得算子$(\Phi \otimes \mathrm{Id})(\rho)$不是半正定的。 我们证明了需要这样的映射数量来检测所有鲁棒纠缠态（即状态$\rho$即使在存在大量随机噪声的情况下仍保持纠缠）超过$\exp(c d^3 / \log d)$。 该证明基于Figiel--Lindenstrauss--Milman于1977年提出的不等式，该不等式最终依赖于关于凸体几乎球面截面的Dvoretzky定理。 我们将该不等式解释为关于用顶点或面数量较少的多面体近似凸体的陈述，并将其应用于研究量子态集合和可分态集合的精细性质。 我们的结果可以被视为纠缠检测复杂性的几何表现。 显示英文摘要

摘要： The well-known Horodecki criterion asserts that a state $\rho$ on $\mathbf{C}^d \otimes \mathbf{C}^d$ is entangled if and only if there exists a positive map $\Phi : \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(\Phi \otimes \mathrm{Id})(\rho)$ is not positive semi-definite. We show that the number of such maps needed to detect all the robustly entangled states (i.e., states $\rho$ which remain entangled even in the presence of substantial randomizing noise) exceeds $\exp(c d^3 / \log d)$. The proof is based on the 1977 inequality of Figiel--Lindenstrauss--Milman, which ultimately relies on Dvoretzky's theorem about almost spherical sections of convex bodies. We interpret that inequality as a statement about approximability of convex bodies by polytopes with few vertices or with few faces and apply it to the study of fine properties of the set of quantum states and that of separable states. Our results can be thought of as geometrical manifestations of the complexity of entanglement detection.