We compute and analyze distinct measures of fermionic entanglement in the exact ground state of finite superconducting systems [1]. It is first shown that global measures such as the one-body entanglement entropy, which represents the minimum relative entropy between the exact ground state and the set of fermionic Gaussian states and is determined by the one-body density matrix, exhibits a close correlation with the BCS gap, saturating in the strong superconducting regime. The same behavior is displayed by the bipartite entanglement between the set of all single-particle states of positive quasimomenta and their time-reversed partners, which is shown to be closely related to the previous entropy. In contrast, the entanglement associated with the reduced density matrix of two pairs of single-particle modes, which can be measured through a properly defined fermionic concurrence computable for such mixed states, exhibits a different behavior, showing a peak in the vicinity of the superconducting transition for states close to the Fermi level and becoming small in the strong coupling regime. In the latter, such mixed reduced state exhibits, instead, a finite mutual information and quantum discord. The evaluation of these measures with the BCS approximation is also examined. While the first measures can be correctly estimated with the latter, the previous concurrence lies strictly beyond the BCS level, requiring at least a particle-number projected BCS treatment for its adequate description. Formal properties of all previous entanglement measures are as well discussed [1,2].

[1] M. Di Tullio, N. Gigena, R. Rossignoli, Phys. Rev. A 97, 062109 (2018).

[2] N. Gigena, R. Rossignoli, Phys. Rev. A 95, 062130 (2017).