Fermionic entanglement in superconducting systems

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 partnersk. In contrast, the entanglement associated with the reduced density matrix of four single particle modes k,k, k ′ ,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.

One-body entanglement as a quantum resource in fermionic systems

“…Such E (|9i) will then be one-body entanglement monotones. Examples are the von Neumann entropy of ρ (1) , S(ρ (1) ) = − P ν λ ν log 2 λ ν , a quantity of interest in various fields [53][54][55][56][57], and the one-body entropy [20,25] (18) which represents, for |9i of definite fermion number N, the minimum relative entropy (in the grand canonical ensemble) between ρ = |9ih9| and any fermionic Gaussian state ρ g : S 1 (ρ (1) 9 ) = Min ρ g S(ρ||ρ g ) [25], for S(ρ||ρ 0 ) = −Trρ(log 2 ρ 0 − log 2 ρ) and ρ g ∝ exp(− P k,k 0 α kk 0 c † k c k 0 ) (pair creation and annihilation terms in ρ g are not required for such |9i [25]). It is also the minimum over all SP bases of the sum of all single mode entropies [20]…”

“…Different approaches have been considered, like mode entanglement [3][4][5], where subsystems correspond to a set of single-particle (SP) states in a given basis; extensions based on correlations between observables [6][7][8][9][10]; and entanglement beyond symmetrization [11][12][13][14][15][16][17][18][19][20][21], which is independent of the choice of SP basis. Several studies on the relation between these types of entanglement [5,16,20,[22][23][24][25][26][27][28] and on whether exchange correlations can be associated with entanglement [29][30][31][32][33] have been recently made. There is also a growing interest in quantum chemistry simulations based on optical lattices [34,35], which would benefit from a detailed characterization of fermionic correlations.…”

Fermionic entanglement in the Lipkin model

“…This quantity, anlogous to the bipartite entanglement of formation [63], is invariant under one-body transformations ρ A → e −ic † Oc ρ A e ic † Oc and vanishes iff ρ A is a convex mixture of SD's. Moreover, if |Ψ is a SD, E(ρ A ) = 0 for any subspace H A since, due to the validity of Wick's theorem, ρ A will be a Gaussian state [35].…”

“…For χ < 0 GS separability no longer occurs and C is positive and parallel ∀ v x > 0. The finite fermionic concurrence of the reduced state ρ pq warrants non-zero bipartite entanglement for any bipartition of the four dimensional sp space [35]. In particular, ρ pq will lead to a finite up-down entanglement, which can be quantified through the pertinent negativity [39,67,68].…”

Robust entanglement of an asymmetric quantum dot molecular system in a Josephson junction

“…The entanglement of indistinguishable particles either bosons or fermions should be characterized with their symmetrized or antisymmetrized wave functions respectively [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] . Particularly, the entanglement of fermions in condensed matter systems can be evaluated by two methods: entanglement of modes [8] , [12] , [14] , [15] and entanglement of particles [6] , [7] , [9] , [10] , [11] . In the former, the entanglement of indistinguishable fermions is associated with the shared modes not particles of subsystems in single-particle Hilbert space.…”