Quantifying Multipartite Entanglement Based on unified entropy

Abstract

In this paper, we investigate the characterization of multipartite entanglement based on the unified $(q,s)$-entropy framework. For bipartite quantum states, we propose an entanglement measure $E_{q,s}^{A|B}(\rho)$ based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures $A^k_{q,s}(\rho)$ and $G^k_{q,s}(\rho)$ for quantifying $k$-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters $q$ and $s$ are in certain ranges, $A^k_{q,s}(\rho)$ and $G^k_{q,s}(\rho)$ also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures $A^k_{q,s}(\rho)$ and $G^k_{q,s}(\rho)$ with respect to the parameters $q$ and $s$ for fixed quantum states. In addition, for fixed $q$ and $s$, we also give their order relations for any two pure states.