Fermionic Entanglement in Superconducting Systems

TL;DR

This paper investigates various measures of fermionic entanglement in superconducting systems, revealing their different behaviors across regimes and highlighting the limitations of BCS approximation for certain entanglement measures.

Contribution

It introduces and compares multiple fermionic entanglement measures in superconductors, showing their distinct behaviors and the necessity of particle number projection for some.

Findings

Global entanglement correlates with the BCS gap in strong coupling.

Bipartite entanglement peaks near the superconducting transition.

Four-mode concurrence requires particle number projection for accurate description.

Abstract

We examine distinct measures of fermionic entanglement in the exact ground state of a finite superconducting system. 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, exhibit 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 $k$ of positive quasimomenta and their time reversed partners $\bar{k}$. In contrast, the entanglement associated with the reduced density matrix of four single particle modes $k,\bar{k}$, $k',\bar{k}'$, which can be measured through a properly defined fermionic concurrence, exhibits a different behavior, showing a peak in the vicinity of the superconducting transition for states $k,k'$ close to the fermi level and becoming small in the strong coupling regime. In the latter such reduced state exhibits, instead, a finite mutual information and quantum discord. And while the first measures can be correctly estimated with the BCS approximation, the previous four-level concurrence lies strictly beyond the latter, requiring at least a particle number projected BCS treatment for its description. Formal properties of all previous entanglement measures are as well discussed.