Entanglement in fermionic systems

The anticommuting properties of fermionic operators, together with the presence of parity conservation, affect the concept of entanglement in a composite fermionic system. Hence different points of view can give rise to different reasonable definitions of separable and entangled states. Here we analyze these possibilities and the relationship between the different classes of separable states. The behavior of the various classes when taking multiple copies of a state is also studied, showing that some of the differences vanish in the asymptotic regime. In particular, in the case of only two fermionic modes all the classes become equivalent in this limit. We illustrate the differences and relations by providing a complete characterization of all the sets defined for systems of two fermionic modes. The results are applied to Gibbs states of infinite chains of fermions whose interaction corresponds to a $XY$ Hamiltonian with transverse magnetic field.

Figure 1

Scheme of the construction of the different sets.

Figure 2

(Color online) Regions of parameters that correspond to separable reduced density matrices of two neighboring fermions according to the different criteria. The different curves correspond to values of the parameters for which one of the conditions of separability (see Table ) is satisfied with equality. For a fixed value of $\lambda$ [$\lambda =0.5$ for (a), $\lambda =0.95$ for (b)], the area at the bottom corresponds to values of $\beta ,\gamma$ for which the reduced density matrix is in $\mathcal{S}{2}_{\pi }^{\prime }$. In both cases, the region of $\mathcal{S}{2}_{\pi }$ lies on the horizontal axis. Notice that for $\lambda =0.95$ there is a small range of values of $\gamma \lesssim 0.2$ for which the entanglement shows up when increasing the temperature, as illustrated quantitatively by the figure below.

Figure 3

(Color online) Entanglement of formation of the reduced density matrix of two neighboring fermions with respect to the sets $\mathcal{S}{2}_{\pi }$ (dots) and $\mathcal{S}{2}_{\pi }^{\prime }$ (crosses), for fixed values of $\lambda$ and $\gamma$, as a function of the inverse temperature.