General monogamy relation of multi-qubit systems in terms of squared R\'{e}nyi-$\alpha$ entanglement

TL;DR

This paper establishes a general monogamy inequality for squared Rényi-$\alpha$ entanglement in multi-qubit systems, revealing hierarchical structures and enabling robust multipartite entanglement indicators beyond traditional measures.

Contribution

It introduces a new monogamy relation for squared Rényi-$\alpha$ entanglement applicable to arbitrary multi-qubit states and explores its hierarchical structure and practical entanglement indicators.

Findings

Proves monogamy inequality for SR$\alpha$E in $N$-qubit states.

Derives analytical relation between Rényi-$\alpha$ and squared concurrence.

Constructs effective multipartite entanglement indicators.

Abstract

We prove that the squared R\'{e}nyi-$\alpha$ entanglement (SR$\alpha$E), which is the generalization of entanglement of formation (EOF), obeys a general monogamy inequality in an arbitrary $N$-qubit mixed state. Furthermore, for a class of R\'{e}nyi-$\alpha$ entanglement, we prove that the monogamy relations of the SR$\alpha$E have a hierarchical structure when the $N$-qubit system is divided into $k$ parties. As a byproduct, the analytical relation between the R\'{e}nyi-$\alpha$ entanglement and the squared concurrence is derived for bipartite $2\otimes d$ systems. Based on the monogamy properties of SR$\alpha$E, we can construct the corresponding multipartite entanglement indicators which still work well even when the indicators based on the squared concurrence and EOF lose their efficacy. In addition, the monogamy property of the $\mu$-th power of R\'{e}nyi-$\alpha$ entanglement is analyzed.