标题： 多罗特斯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准则指出，一个状态$\rho$在$\mathbb{C}^d \otimes \mathbb{C}^d$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$(\Phi \otimes \mathsf{I})(\rho)$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$\rho$在存在大量随机化噪声的情况下仍然保持纠缠）超过$\exp(c d^3 / \log d)$。 该证明基于对由少量顶点或面的多面体近似状态集合（分别是对可分离状态的近似）的研究，并最终依赖于关于凸体几乎球形截面维度的Dvoretzky--Milman定理。 该结果可以被解释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 $\mathbb{C}^d \otimes \mathbb{C}^d$ is entangled if and only if there exists a positive map $\Phi : \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(\Phi \otimes \mathsf{I})(\rho)$ is not positive semi-definite. We show that 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 a study of the approximability of the set of states (resp. of separable states) by polytopes with few vertices or with few faces, and ultimately relies on the Dvoretzky--Milman theorem about the dimension of almost spherical sections of convex bodies. The result can be interpreted as a geometrical manifestation of the complexity of entanglement detection.