Information and entanglement measures applied to the analysis of complexity in doubly excited states of helium

Abstract

Shannon entropy and Fisher information calculated from one-particle density distributions and von Neumann and linear entropies (the latter two as measures of entanglement) computed from the reduced one-particle density matrix are analyzed for the $^{1,3}S^{e},\phantom{\rule{4pt}{0ex}}^{1,3}P^{o}$, and $^{1,3}D^{e}$ Rydberg series of He doubly excited states below the second ionization threshold. In contrast with the Shannon entropy, we find that both the Fisher information and entanglement measures are able to discriminate low-energy resonances pertaining to different $_{2}(K,T)_{{n}_{2}}^{A}$ series according to the Herrick-Sinano\ifmmode \breve{g}\else \u{g}\fi{}lu-Lin classification. Contrary to bound states, which show a clear and unique asymptotic value for both Fisher information and entanglement measures in their Rydberg series $1sn\ensuremath{\ell}$ for $n\ensuremath{\rightarrow}\ensuremath{\infty}$ (which implies a loss of spatial entanglement), the variety of behaviors and asymptotic values of entanglement above the noninteracting limit value in the Rydberg series of doubly excited states $_{2}(K,T)_{{n}_{2}}^{A}$ indicates a signature of the intrinsic complexity and remnant entanglement in these high-lying resonances even with infinite excitation ${n}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}$, for which all known attempts of resonance classifications fail in helium.