Quantum Information Aspects on Bulk and Nano Interacting Fermi System:   Spin-Space Density Matrix Approach
  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. After calculating single-electron Green 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 nano-superconducting 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 s-wave 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. 
Reza Afzali, Neda Ebrahimian, Bashir Eghbalifar
cond-mat.supr-con quant-ph