Separable Ball around any Full-Rank Multipartite Product State

We show that around any $m$-partite product state $\rho_{\rm prod}=\rho_1\otimes...\otimes\rho_m$ of full rank (that is ${\rm det}(\rho_{\rm prod})\neq 0)$, there exists a finite-sized closed ball of separable states centered around $\rho_{\rm prod}$ whose radius is $\beta:=2^{1-m/2}\lambda_{\rm min}(\rho_{\rm prod})$. Here, $\lambda_{\rm min}(\rho_{\rm prod})$ is the smallest eigenvalue of $\rho_{\rm prod}$. We are assuming that the total Hilbert space is finite dimensional and we use the notion of distance induced by the Frobenius norm. Applying a scaling relation, we also give a new and simple sufficient criterion for multipartite separability based on trace: ${\rm Tr}[\rho\rho_{\rm prod}]^2/{\rm Tr}[\rho^2]\geq {\rm Tr}[\rho_{\rm prod}^2]-\beta^2$. Using the separable balls around the full-rank product states, we discuss the existence and possible sizes of separable balls around any multipartite separable states, which are important features for the set of all separable states. We discuss the implication of these separable balls on entanglement dynamics.