Quantum information aspects on bulk and nano interacting Fermi system: A spin-space density matrix approach

Abstract

In this paper, we investigate quantum correlation of an interacting Fermi system, which is a nodal superconductor (d-wave superconductor) at zero temperature, via quantum entanglement of two electron spins forming Cooper pairs (Werner state), tripartite and quantum discord. The energy gap depends on the angle between the electron momentum and the nodal axis; and at zero temperature we use an approximation in which the energy gap is considered as the linear function of the angle. After calculating single-electron Green's functions, the two-electron space-spin density matrix, which has X-state form, is obtained. The dependence of quantum correlation to the relative distance of electrons spins of Cooper pair and energy gap is investigated. One of the results is, for d-wave case, concurrence (as a measure of entanglement), quantum discord and tripartite are sensitive to the change of magnitude of gap. Another result is both concurrence and discord oscillate. Then, we consider three-dimensional rectangular nanosuperconducting grain in the weak coupling frame. The nano-size effect is entered via gap fluctuation. The dependence of quantum correlation to length of superconductor and lower bound of robustness of tripartite entanglement are determined. Moreover, we show that quantum correlation of d-wave nano-size superconducting grain strongly depends on length of grain (in contrast to swave case). In general, it is found that the length of grain lower, the effect of nano-size on quantum correlation higher. Quantum tripartite for nano-scale d-wave superconductor is better than for bulk d-wave superconductor. However, we find out both bulk and nano-size s-wave superconductors have the same tripartite. Furthermore, entanglement length and quantum correlation length are investigated and it is shown that there is a length of superconductor in which discord becomes zero. Also, for a given fixed length of superconductor, both a peak in discord and a peak in concurrence occur simultaneously.

Tuning of Quantum Entanglement of a Superconductor by Transition-Metal and Rare-Earth Impurity Effect and the Role of Potential Scattering on Quantum Phase Transition

“…The entanglement emerges in two ways; first, due to the particle statistics, so that the non-interacting system shows entanglement, and second, due to the external or internal interaction, which is found in the few-or many-body systems, leads to the increase or decrease of the bipartite or tripartite or even multipartite entanglement 9 . We deal with an interacting system namely a superconductor.…”

“…The factor is related to the interactions and also, bipartite entanglement in the non-interacting part of the state can be modified by Feenberg factor. Second, by using the Bogoliubov transformation, the Hamiltonian of an interacting system can be diagonalized as a noninteracting system but with different energy 9 . Of course, the existence of impurity in superconductors will cause to product an additional interaction, which can be considered as a small perturbation on superconductors, accompanied by new Green's functions.…”

“…where is the Fermi velocity and ( ) ( ⃗) was given in Refs. [8][9]. For s-wave superconductors, the angular dependence doesn't exist in Green's Functions.…”

“…8 (a), at a fixed collision time, (= 10), by increasing the value of , concurrence decreases. For comparison, we bring the curve of concurrence without considering any impurity 8,9 . The curves of concurrence in the presence of the non-magnetic impurity with finite case and non-impurity case are overlapped, however, the curve of concurrence for infinite Debye frequency case is less than other cases (at a fixed value of ).…”

“…It should be noted that the entanglement of many-body systems is extraordinarily complicated in comparison with other systems, especially, in the ground state permeate multipartite entanglement at zero temperature. The generation and manipulation of the bipartite and multipartite entangled states were investigated by many-body Hamiltonians [7][8][9][10][11][12] . Other investigations on many-body systems such as the entanglement of electron spins are as follows.…”