R\'enyi relative entropy based monogamy of entanglement in tripartite systems

A comprehensive investigation of the entanglement characteristics is carried out on tripartite spin-1/2 systems, examining prototypical tripartite states, the thermal Heisenberg model, and the transverse field Ising model. The entanglement is computed using the R\'enyi relative entropy. In the traditional R\'enyi relative entropy, the generalization parameter $\alpha$ can take values only in the range $0 \leq \alpha \leq 2$ due to the requirements of joint convexity of the measure. To use the R\'enyi relative entropy over a wider range of $\alpha$, we use the sandwiched form which is jointly convex in the regime $0.5 \leq \alpha \leq \infty$. In prototypical tripartite states, we find that GHZ states are monogamous, but surprisingly so are W states. On the other hand, star states exhibit polygamy, due to the higher level of purity of the bipartite subsystems. For spin models, we study the dependence of entanglement on various parameters such as temperature, spin-spin interaction, and anisotropy, and identify regions where entanglement is the largest. The R\'enyi parameter $\alpha$ scales the amount of entanglement in the system. The entanglement measure based on the traditional and the sandwiched R\'enyi relative entropies obey the Araki-Lieb-Thirring inequality. In the Heisenberg models, namely the XYZ, XXZ, and XY models, the system is always monogamous. However, in the transverse field Ising model, the state is initially polygamous and becomes monogamous with temperature and coupling.
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