Long-range entanglement in quantum dots with Fermi-Hubbard physics

The range of entanglement in quantum dots, under the effect of the Coulomb interaction and the system's size, is investigated. As ququart systems, naturally described by the Fermi-Hubbard model, we show that quantum dots supply long-range entanglement up to the third neighbor. In conjunction with that and using the lower bound of concurrence, we show that the Coulomb interaction can be adjusted to create and increase entanglement between distant parties as well. A rigorous description of the pairs is given in terms of a local half-filled state associated with each pair with an electron number $N=2$ and a spin $S=0$. A thorough study of this state provides a proper explanation related to the pairwise entanglement, namely, its amount and its behavior under the effect of the Coulomb interaction together with the system's size. In addition, we show that the confinement state of quantum dots is genuinely four-partite entangled with a maximum amount for the smallest size $L=4$.

Figure 1

Lower bound of concurrence (2), associated with the pairs (a) ${\rho }_{12}$, (b) ${\rho }_{23}$, (c) ${\rho }_{13}$, and (d) ${\rho }_{14}$, as a function of $U$ for various system sizes up to $L=12$.

Figure 2

Local entanglement (a) $E\left({\rho }_1\right)=E\left({\rho }_L\right)$ and $E\left({\rho }_i\right)$ with $1<i<L$ [Eq. (7)] at the ends and the middle of the chain, respectively, as a function of $U$ for system size (a) $L=4$ and (b) $L=12$. The inset in (a) shows the effect of $U$ on the single- and double-occupation probabilities $s_i$ and $d_i$, respectively, at the end sites ($i=1,4$) and the middle sites ($i=2,3$).

Figure 3

(a) Probabilities corresponding to the 16 basis states for each pair at $U=0$ with $L=4$. The basis states with $N=2$ and $S=0$ are marked by red squares, those with $N=\{1,3\}$ and $S=\pm \frac{1}{2}$ by green triangles, and those with $N=\{0,4\}$ and $S=0$ and with $N=2$ and $S=\pm 1$ by blue circles. Also shown are the probabilities $P_k$ associated with the $4^2$ eigenstates $|{\psi }_k\rangle$ of (b) ${\rho }_{12}$, (c) ${\rho }_{23}={\rho }_{14}$, and (d) ${\rho }_{13}$. The $P_{\text{LHFS}}$ associated with the LHFS is marked by a red dashed line.

Figure 4

Probability $P_k$ associated with the LHFS as a function of $U$ with various system sizes $L$ for (a) ${\rho }_{12}$ and (b) ${\rho }_{23}$.

Figure 5

(a) Representative scheme showing the direct correlation bonds between each pair and the remaining sites for $L=4$. (b) Entanglement $E\left({\rho }_{ij}\right)$ externally shared by the different pairs for $L=4$.

Figure 6

The LBC associated with the LHFS (red solid and dotted lines) and with the global state ${\rho }_{ij}$ with a frozen LHFS (blue solid and dashed lines) for (a) ${\rho }_{12}$, (b) ${\rho }_{23}$ and ${\rho }_{14}$, and (c) ${\rho }_{13}$.