Radial Convex Geometry of Quantum States and Its Relation to Best Separable Approximation

Abstract

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 $G(\rho)=[1-L(\rho)]/L(\rho)$ from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree $Q(\rho)=[(1-p)/p]/G(\rho)$ 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 $(1-p)L_B/[p(1-L_B)]\leq p_0\leq Q(\rho)$ 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 $(1-p)/(2p)\leq p_0\leq Q(\rho)$. These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.