Entanglement measures for multipartite quantum systems:limitations and new approaches

Abstract

In this research, we investigate the distribution of entanglement within two entangled $n$-qubit systems using the one-tangle and $\pi$-tangle. Our analysis reveals that for certain quantum states such as the generalized $W$ state, where the probability coefficients depend on the number of qubits, increasing the system size causes these traditional measures to gradually diminish, with the monogamy relation tending toward equality. This behavior highlights a limitation of conventional tangle-based measures in capturing multipartite correlations in large systems. To overcome this, we introduce two alternative measures: the sum of squared one-tangles and the generalized residual entanglement. Unlike the standard one-tangle and $\pi$-tangle, these new quantities remain robust as the number of qubits increases. Moreover, we propose a strong monogamy relation that preserves nontrivial entanglement distribution even in the large-$n$ limit. Our results provide a refined framework for quantifying entanglement in high-dimensional multipartite systems.