AI 中文总结
该研究从量子态凸几何出发，定义了几何纠缠空间量与相对纠缠度，将其应用于最佳可分近似（BSA），推导得到相关界并明确两量子比特态BSA纠缠分量的性质，为BSA等提供了几何描述。
AI 中文摘要
我们从量子态的凸几何角度研究双体纠缠。以最大混态为参考点，我们根据可分边界与量子态边界的相对位置定义几何纠缠空间量$G(\rho)=[1-L(\rho)]/L(\rho)$，并通过与最大混态的鲁棒性对比引入相对纠缠度$Q(\rho)=[(1-p)/p]/G(\rho)$。我们将该几何构造应用于最佳可分近似（BSA），对最优分解推导得到一般界$(1-p)L_B/[p(1-L_B)]\leq p_0 \leq Q(\rho)$，并证明BSA的纠缠分量的纠缠空间尺寸不小于原混态，即$[1-L_R]/L_R \leq [1-L_B]/L_B$。对于两量子比特态，BSA的纠缠分量是纯纠缠态，利用PPT准则可明确计算其几何参数，得到界$(1-p)/(2p)\leq p_0 \leq Q(\rho)$。这些结果为BSA、鲁棒性与量子态空间纠缠区域之间的关系提供了简洁的几何描述。
英文摘要
We study bipartite entanglement from the convex geometry of quantum states. Taking the maximally mixed state as a reference point, we define a geometric entangled-space quantity $G(ρ)=[1-L(ρ)]/L(ρ)$ from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree $Q(ρ)=[(1-p)/p]/G(ρ)$ by comparing it with the robustness relative to the maximally mixed state. We apply this geometric construction to the Best Separable Approximation (BSA). For the optimal decomposition, we derive the general bound $(1-p)L_B/[p(1-L_B)]\leq p_0\leq Q(ρ)$ and show that the entangled component of the BSA has an entangled-space size no smaller than that of the original mixed state, namely $[1-L_R]/L_R\leq[1-L_B]/L_B$. For two-qubit states, the BSA entangled component is a pure entangled state. Using the PPT criterion, its geometric parameter can be evaluated explicitly, giving the bound $(1-p)/(2p)\leq p_0\leq Q(ρ)$. These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.
