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In a recent work [Ge {\it et al.}, arXiv: 2312. 17496 (2023)], we havederived the polygon relation of bipartite entanglement measures that is usefulto reveal the entanglement properties of discrete, continuous, and even hybridmultipartite quantum systems. In this work, with the information-theoreticalmeasures of R\'enyi and Tsallis entropies, we study the relationship betweenthe polygon relation and the subadditivity of entropy. In particular, theentropy-polygon relations are derived for pure multi-qubit states andgeneralized to multi-mode Gaussian states, by utilizing the known results fromthe quantum marginal problem. Moreover, the equivalence between the polygonrelation and subadditivity is established, in the sense that for all discreteor continuous multipartite states, the polygon relation holds if and only ifthe underlying entropy is subadditive. As byproduct, the subadditivity ofR\'enyi and Tsallis entropies is proven for all bipartite Gaussian states.Finally, the difference between polygon relations and monogamy relations isclarified, and generalizations of our results are discussed. Our work providesa better understanding of the rich structure of multipartite states, and henceis expected to be helpful for the study of multipartite entanglement.

The polygon relation and subadditivity of entropic measures for discrete and continuous multipartite entanglement