Response to defects in multipartite and bipartite entanglement of isotropic quantum spin networks

Quantum networks are an integral component in performing efficient computation and communication tasks that are not accessible using classical systems. A key aspect in designing an effective and scalable quantum network is generating entanglement between its nodes, which is robust against defects in the network. We consider an isotropic quantum network of spin-1/2 particles with a finite fraction of defects, where the corresponding wave function of the network is rotationally invariant under the action of local unitaries. By using quantum information-theoretic concepts like strong subadditivity of von Neumann entropy and approximate quantum telecloning, we prove analytically that in the presence of defects, caused by loss of a finite fraction of spins, the network, composed of a fixed numbers of lattice sites, sustains genuine multisite entanglement and at the same time may exhibit finite moderate-range bipartite entanglement, in contrast to the network with no defects.

Figure 1

A two-dimensional square bipartite lattice, with sublattice A (blue circles) and B (yellow circles). The square bipartite lattice can be generalized to a three-dimensional cubic bipartite lattice.

Figure 2

A two-dimensional bipartite (a) honeycomb lattice and (b) dimer lattice of regular polygons, with sublattice A (blue circles) and B (yellow circles).

Figure 3

Two examples of possible coverings in a two-dimensional bipartite honeycomb lattice, with defects (red circles) in the network. The defects may be (a) clustered or (b) distributed in the lattice, as long as the bipartite state remains intact. The nodes in sublattice A (blue circles) and B (yellow circles) are occupied by spin-1/2 particles.

Figure 4

Maximum permissible bipartite entanglement, $E\left(\rho \right)$. The plot shows the upper bound on BE due to quantum telecloning, quantified using LN, for ${\rho }_{{\text{AB}}_i}^{\left(2\right)}$, as a function of $1-p_3^{\prime }$. $E\left(\rho \right)$ decreases as $M$ increases. Interestingly, in the absence of defects ($p_3^{\prime }$ = 1) no BE is present as $M\to \infty$. The vertical axis is in ebits, while the horizontal one is dimensionless.

Figure 5

Genuine multisite entanglement in an isotropic lattice. Each node either contains a spin, which is a part of a dimer, or a defect. The spatially different but equivalent sets of $x$ (black solid line) and $x^{\prime }$ (black dashed line) nodes contain a common $y$ node (red circle).