Entanglement distribution in pure-state quantum networks

We investigate entanglement distribution in pure-state quantum networks. We consider the case when nonmaximally entangled two-qubit pure states are shared by neighboring nodes of the network. For a given pair of nodes, we investigate how to generate the maximal entanglement between them by performing local measurements, assisted by classical communication, on the other nodes. We find optimal measurement protocols for both small and large one-dimensional networks. Quite surprisingly, we prove that Bell measurements are not always the optimal ones to perform in such networks. We generalize then the results to simple small two-dimensional (2D) networks, finding again counterintuitive optimal measurement strategies. Finally, we consider large networks with hierarchical lattice geometries and 2D networks. We prove that perfect entanglement can be established on large distances with probability one in a finite number of steps, provided the initial entanglement shared by neighboring nodes is large enough. We discuss also various protocols of entanglement distribution in 2D networks employing classical and quantum percolation strategies.

Figure 1

Notation and examples of 1D networks: (a) the standard quantum repeater scenario; (b) entanglement swapping; (c) a two-repeater system.

Figure 2

Representation of the function $h\left(p\right)=\min \left\{f\right(p),g(p\left)\right\}$ governing the SCP after Bell measurements in a two-repeater configuration.

Figure 3

SCP for a system of two repeaters, with ${\beta }_0={\gamma }_0=0.7$. Numerical results (dashed line) show that there exists a better strategy than Bell measurements (solid line) for ${\alpha }_0∊[a_1,,a_2]$. The values $a_1$ and $a_2$ are such that $p^{*}\left(a_1\right)=[1-p_{\max }(a_1\left)\right]∕3$ and $p^{*}\left(a_2\right)=p_{\min }\left(a_2\right)$.

Figure 4

Operations on a square to obtain an entangled pair on the diagonal: first two measurements, then distillation of the resulting states $\alpha$ and $\beta$.

Figure 5

Typical plot of the function $h\left(p\right)$ governing the SCP of the square.

Figure 6

SCP for a square made of four states $\phi$. Numerical results (dashed line) show that Bell measurements (solid line) do not lead to the optimal solution for ${\phi }_0>{\phi }^{\star }\approx 0.664$.

Figure 7

The diamond lattice is formed by iterating the following operation: a single bond (two qubits and one entangled state) is replaced by four bonds forming a diamond shape (four pairs of qubits and four entangled states). After $K$ iterations, the nodes $A$, $B$, $C$, $D$ have $2^K$ links, the nodes on the next level $2^{K-1}$ links, etc.

Figure 8

Recursion relating $E$ on the higher level of lattice iteration to $E^{\prime }$ at the lower level of iteration in the diamond lattice. Each iteration consists of the following steps: (i) WCE and (ii) the two resulting two-qubit states are transformed with probability one into a two-qubit state of the same SCP.

Figure 9

(a) Tree configuration; (b) The nodes in the middle perform WCE. This creates two two-qubit states between the neighboring nodes. These states are transformed into a two-qubit state of the same SCP. The process is iterated until a perfect singlet is established between the two ends of the tree.

Figure 10

(a) “Centipede” with its “legs” and “spine.” (b) Recursive measurement scheme; note that the method can be equally well applied also in higher dimensions.

Figure 11

Recursion relation for the centipede lattice. Only when the entanglement $E_0$ is larger than a threshold $E_{th}$, a trivial stable point at $E=1$ appears.

Figure 12

Each node is connected by two copies of the same two-qubit state $\mid \phi ⟩$. The nodes marked in (a) perform the measurement optimal according to the SCP. A triangular lattice (b) is obtained where the SCP is the same as for the state $\mid \phi ⟩$. Classical entanglement percolation is now possible in the new lattice.

Figure 13

The triangular lattice consists of two different entangled states $\mid \phi ⟩$ and $\mid \tilde{\phi }⟩$ for the solid and dashed lines, respectively. The less entangled states $\mid \tilde{\phi }⟩$ are discarded and some of the nodes perform the optimal measurement according to the SCP. A new triangular lattice is obtained, governed by the SCP of $\mid \phi ⟩$.

Figure 14

(a) Measurements necessary to double the square lattice: the marked nodes apply the optimal one-repeater transformation along the vertical and horizontal directions. (b) Resulting pairs of disjoint square lattices with lattice constant doubled; we want to establish perfect entanglement between any two neighboring points $A$, $A^{\prime }$ versus $B$ and $B^{\prime }$. $A$ and $A^{\prime }$ ($B$ and $B^{\prime }$) are neighbors but belong to different lattices.

Figure 15

“Tree view” of the labels $m$ and their corresponding probabilities after the first two measurements. For symmetry, we choose the root of this tree corresponding to $n=-1$.