Measure of tripartite entanglement in bosonic and fermionic systems

We describe an efficient theoretical criterion suitable for the evaluation of the tripartite entanglement of any mixed three-boson or three-fermion state, based on the notion of the entanglement of particles for bipartite systems of identical particles. Our approach allows one to quantify the accessible number of quantum correlations in the systems without any violation of the local particle number superselection rule. A generalization of the tripartite negativity is here applied to some correlated systems including the continuous-time quantum walks of identical particles (for both bosons and fermions) and compared with other criteria recently proposed in the literature. Our results show the dependence of the entanglement dynamics upon the quantum statistics: The bosonic bunching results in a low number of quantum correlations while Fermi-Dirac statistics allows for higher values of the entanglement.

Figure 1

The tripartite entanglement ${\epsilon }_G$ (left panel) and ${\epsilon }_T$ (right panel) as a function of the phases $\alpha$ and $\beta$ ranging from 0 to $\pi$. Specifically, the quantum correlations exhibited by the three-fermion state $|\Phi \rangle$ of Eq. (11) among the three parties $A=\{1,2\}$, $B=\{3,4\}$, and $C=\{5,6\}$ are here considered.

Figure 2

Entanglement as a function of the “time” $\tau$ for the fermionic (blue line) and bosonic (green line) systems when two different partitions are considered: $A=\{1,2\}$, $B=\{3,4\}$, $C=\{5,6\}$ (left panel) and $A^{\prime }=\{1,4\}$, $B^{\prime }=\{2,5\}$, $C^{\prime }=\{3,6\}$ (right panel).

Figure 3

Single- and two-particle density probability of the bosonic and fermionic systems initially described by the state $|\Psi \rangle$ of Eq. (15) for $\tau =8.7$. (a) The single-particle density distribution ${\rho }_r$ which is identical for fermions and bosons. (b) The correlation matrix ${\Gamma }_{rs}$ for bosons computed from Eq. (16). Bosons tend to localize in the the same or different sites of the first half of the lattice. (c) The correlation matrix ${\Gamma }_{rs}$ for fermions computed from Eq. (17). Fermions are found with high probability in the neighboring sites of the first half of the lattice. (d) The interparticle distance probability for bosons. They tend to appear at the same site or at neighboring sites. (e) The interparticle distance probability for fermions. They are more likely to be separated by a small number (1–2) of sites.