Quantification of Entanglement Entropy in Helium by the Schmidt–Slater Decomposition Method

Abstract

In this work, we present an investigation on the spatial entanglement entropies in the helium atom by using highly correlated Hylleraas functions to represent the S-wave states. Singletspin 1sns1 S e states (with n = 1 to 6) and triplet-spin 1sns 3 S e states (with n = 2 to 6) are investigated. As a measure on the spatial entanglement, von Neumann entropy and linear entropy are calculated. Furthermore, we apply the Schmidt-Slater decomposition method on the two-electron wave functions, and obtain eigenvalues of the one-particle reduced density matrix, from which the linear entropy and von Neumann entropy can be determined.

The von Neumann entanglement entropy for Wigner-crystal states in one dimensional N-particle systems

“…Moreover, in order to gain some insight into how the parameter d influences the entanglement, the variation of S g→∞ for d = 3 is also shown in this figure. We can observe how S converges to the values predicted by the HA as g is increased, which confirms both the correctness of our theoretical derivations and the effectiveness of (29). As can be seen, for all cases considered, the vN entropy S shows a monotonic increase with N .…”

“…Our numerical results, together with the ones of [12] are summarized in Table I, where, for the sake of completeness, we also give the optimal values of α. It is apparent from the results that the ansatz (29) gives a reasonable estimate of the linear entropy, which allows us to expect that it also provides good approximations to the true values of the occupancies.…”

“…Such a form of f was recently suggested in [42] and its applicability to the case of charged particles has been demonstrated. We recall at this point that in the limit as g → 0, the system (1) forms a TG gas and its exact ground-state wavefunction is given by (29) with f = |x| [1]. In the case of N = 2, the Hamiltonian (1) separates in terms of…”

“…For example, some light has been shed on entanglement both in quantum dot systems [6][7][8][9][10][11][12][13][14][15] and in systems of harmonically interacting particles in a harmonic trap (the so-called Moshinsky atom) [16][17][18][19][20][21][22][23]. Special attention has also been paid to the study of entanglement in the helium atoms and helium ions [24][25][26][27][28][29][30][31][32]. The effect of the confinement on the entanglement in the ground state of the helium atom has also been discussed in the literature [33].…”

Magnetic alteration of entanglement in two-electron quantum dots

“…Lin et al [19] computed the spatial quantum entanglement of helium-like ions by means of spline techniques. Various types of wave functions have been used to increase the numerical accuracy of the entanglement measures of helium-like atoms with a few electrons [20][21][22]. Also, the entanglement properties of bound states in the two-electron Moshinsky model [23] and a solvable model of two-electron parabolic quantum dot [24] were investigated.…”