Separable Ball around any Full-Rank Multipartite Product State

TL;DR

This paper establishes the existence of a finite-sized separable ball around any full-rank multipartite product state in finite-dimensional Hilbert spaces, providing new criteria for separability and insights into entanglement dynamics.

Contribution

It introduces a novel finite-sized separable ball around full-rank product states and derives a simple sufficient separability criterion based on trace relations.

Findings

Existence of a finite separable ball around any full-rank product state.

A new simple criterion for multipartite separability based on trace ratios.

Implications for the size of separable regions and entanglement dynamics.

Abstract

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.