Two-body cluster entangled structure in the $\mathcal{H}_1\otimes \mathcal{H}_2$ Hilbert space
Fei-Long Xu, Xi-Guang Cao, Yu-Gang Ma
TL;DR
This work addresses how to quantify nonlocal quantum correlations in a simple nuclear two-cluster system by recasting the problem in the $\mathcal{H}_{1}\otimes \mathcal{H}_{2}$ framework. It develops a local transformation from the usual center-of-mass/relative coordinate basis to the particle basis, enabling a Schmidt-decomposition and von Neumann entropy as entanglement measures, exemplified by the $0^+$ state of $^8$Be with $\psi=\phi_{cm}(R_{cm})u_{l_r}(r)Y_{00}=\sum_\alpha \Phi_\alpha(r_1,r_2)[Y_{\alpha,-m}\otimes Y_{\alpha,m}]_{\{\alpha,00\}}$. Applying the method, the authors find substantial entanglement arising from mixing of angular-momentum channels, quantified by $S\approx 2.77$ with dominant contributions from a handful of channels; they also introduce spatially resolved entropy $S(r'_1,r'_2)$ that reveals fm-scale entanglement structure and a rise in entanglement with larger inter-cluster separation, vanishing as $r'_1,r'_2\to 0$ to yield an $S$-wave separable state. This framework bridges nuclear cluster theory with quantum information, providing a tool to analyze nonlocal correlations in nuclei and to study how spatial configuration governs entanglement in cluster dynamics.
Abstract
This study introduces a quantum information perspective to analyze the internal structure of atomic nuclei, focusing on the quantum entanglement between $α$ clusters in the 0$^+$ state of $^8$Be. A wave function based on angular momentum coupling is developed to transform the two-cluster wave function from the conventional center of mass and relative coordinate basis ($\mathcal{H}_{R_{c.m.}} \otimes \mathcal{H}_r$) into the individual particle basis ($\mathcal{H}_1 \otimes \mathcal{H}_2$), which is essential for a precise quantification of entanglement. Within this method, the von Neumann entropy is employed to quantify the entanglement arising from the mixing of angular momentum channels. Additionally, we introduce the concept of spatially resolved entropy, which measures entanglement as a function of the radial separation between clusters. Our analysis reveals that the entanglement is strongly correlated with the spatial configuration at the femtometer scale. As the inter cluster separation vanishes, the system approaches a separable $S$-wave state, indicating that entanglement is dynamically generated during the spatial separation of the clusters. This research provides a new tool for investigating nonlocal quantum correlations in nuclear structure, complementing existing descriptions.