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

Quantum entanglement in (d−1)-spherium

“…We observe that entanglement of (d − 1)-dimensional spherium tends to increase with energy. A similar behaviour has been observed in other models, such as the Crandall and the Hooke ones [5], as well as for the singlet states of the Helium atom employing highquality, state-of-the-art wavefunctions [12] (although for more general states of Helium the energy-entanglement connection seems to be much more complicated [28,29]). …”

“…The value (28) of the integral J 0 is straightforward since it is the product of the volumes of the hyperspheres for each electron. To obtain the values (29) and (30) of J 1 and J 2 , respectively, we use the Cohl expansion (24) for r p 12 in terms of the Gegenbauer polynomials, C α m (x), and then we apply the orthogonality property of these polynomials which reads [45] as which is equal to the wanted value (30). Finally, by performing the sum in (21) with J 0 , J 1 and J 2 , we arrive at final expression (31) for the normalization, N 1 .…”

Shannon Information Entropy in Position Space for the Ground and Singly Excited States of Helium with Finite Confinements

“…There have been considerable research activities being carried out on investigations of quantum entanglement in model, artificial, and natural atoms [2][3][4][5][6][7][8][9][10]. In our group, we have also carried out quantification of entanglement entropies in model [11,12] and natural atoms [13][14][15][16][17][18][19][20][21], such as the helium atom and the hydrogen and positronium negative ions.…”

Entanglement in helium atom confined in an impenetrable cavity

“…Generally, the von Neumann (vN) entropy [2] and the linear entropy [3] are used to quantify the amount of entanglement in composite quantum systems. In the most recent years, many attempts have been made towards understanding entanglement properties of real two-electron systems, i.e., the helium atoms and helium-like ions [4][5][6][7][8][9][10][11][12][13][14][15][16]. Related studies of entanglement in the context of two-electron model systems such as systems of electrons confined by a spherical harmonic potential without and with an on-centre Coulomb impurity and the helium atom immersed in weakly coupled Debye plasmas are also available in the literature, [17], [18] and [19] respectively.…”