Quantum entanglement in helium-like ions

Recently, there have been considerable interests to investigate quantum entanglement in two-electron atoms [1-3]. Here we investigate quantum entanglement for the ground and excited states of helium-like ions using correlated wave functions, concentrating on the particle-particle entanglement coming from the continuous spatial degrees of freedom. We use the two-electron wave functions constructed by employing $B$-spline basis to calculate the linear entropy of the reduced density matrix $L=1-Tr_A (\rho _A^2 )$ as a measure of the spatial entanglement. Here$\rho _A =Tr_B (\left| \varphi \right\rangle _{AB} { }_{AB}\left\langle \varphi \right|)$ is the one-electron reduced density matrix obtained after tracing the two-electron density matrix over the degrees of freedom of the other electron. We have investigated the spatial entanglement for the helium-like systems with $Z$=1 to $Z$=10. For the helium atoms (Z=2), we have calculated the linear entropy for the ground state and the 1$s$n$s ^{1}S^{e}$ (n=2-10) excited states. Results are compared with other calculations [1-3]. \\[4pt] [1] J. P. Coe and I. D'Amico, \textit{J. Phys.: Conf. Ser.} \textbf{254}, 012010 (2010) \\[0pt] [2] D. Manzano \textit{et. al.}, \textit{J. Phys. A: Math. Theor.} \textbf{43}, 275301 (2010) \\[0pt] [3] J. S. Dehesa \textit{et. al.}, \textit{J. Phys. B}\textbf{ 45}, 015504 (2012)