Singlet generation in mixed-state quantum networks

We study the generation of singlets in quantum networks with nodes initially sharing a finite number of partially entangled bipartite mixed states. We prove that singlets between arbitrary nodes in such networks can be created if and only if the initial states connecting the nodes have a particular form. We then generalize the method of entanglement percolation, previously developed for pure states, to mixed states of this form. As part of this, we find and compare different distillation protocols necessary to convert groups of mixed states shared between neighboring nodes of the network into singlets. In addition, we discuss protocols that only rely on local rules for the efficient connection of two remote nodes in the network via entanglement swapping. Further improvements of the success probability of singlet generation are developed by using particular forms of “quantum preprocessing” on the network. This includes generalized forms of entanglement swapping and we show how such strategies can be embedded in regular and hierarchical quantum networks.

Figure 1

Mixed-state quantum network. Qubits in a node (circles) may be connected by bonds (thick lines), that is, they share mixed entangled states [“edges” (solid black lines)] of qubits (black dots) with other nodes. When two “paths” of states of the form (1) connect A and B, a singlet (dashed line) can be created with finite probability. This is proven by partitioning the nodes into two groups with one containing A (shaded region) and the other B. For it to be possible to generate a singlet between A and B, these groups must be linked by at least two states of the form (1) for all possible partitions.

Figure 2

Basic arrangement for entanglement swapping. Entanglement swapping involves a measurement in the Bell basis at node $C_2$ and classical communication between the nodes followed by local unitaries which causes $C_1$ and $C_3$ to become entangled.

Figure 3

Illustration of classical entanglement percolation in a square network. Pairs of qubits (black dots) in neighboring nodes (circles) are in identical, pure, partially entangled states (solid black lines). The percolation scheme involves these entangled states being converted into singlets (dashed lines) with probability $p$. If $p$ exceeds the percolation threshold, these form large clusters and we can obtain a singlet between any two qubits within a cluster by performing swapping operations.

Figure 4

Triangular network. This is a simple 2D arrangement of PMSs in which CEP is possible.

Figure 5

The purification setup consists of $n$ PMSs (solid lines) shared between two nodes A and B. The aim is to distill these states into a singlet.

Figure 6

Singlet conversion probability for $n$-edged bonds using the recycling scheme for $\alpha =1/2$ and $n=2,(3),4,6,8,10,12,14,16$ (bottom to top). The $n=3$ line (dashed) corresponds to the DSS scheme. The percolation thresholds for triangular (T), square (S), and honeycomb (H) lattices are given by the horizontal lines.

Figure 7

The recycling scheme consists of splitting the states into pairs that are then purified. If no singlets are successfully produced some of the states may still have been transformed into PMSs, and given enough of these the process can be repeated.

Figure 8

Success probabilities for the recycling schemes that split the states into pairs (dashed lines) and sets of three (solid lines). Shown are the SCPs for 6, 9, and 12 initial states (bottom to top) for $\alpha =1/2$.

Figure 9

Procedure to join a node’s singlets onto a GHZ state. Here the GHZ state is represented by a dotted box. Additional qubits (black dots) that are a part of the GHZ state are linked to the dotted box by a dotted line. A C-NOT gate and measurement (both are represented by a shaded oval) are performed between the qubit already in the GHZ state and those that are part of a singlet (dashed lines). Each measurement outcome needs to be sent (gray arrow) to the other singlet qubit to perform a local unitary (shaded square). This extends the GHZ state to include qubits connected by edges to the node being attached. Once this has occurred for each qubit in a singlet the process is repeated by sending out a signal to repeat the step at each node that was linked by a singlet.

Figure 10

After the singlets are formed we can repeatedly extend a GHZ state from node A. This procedure uses the operation shown in Fig. 9 to add qubits to the GHZ state. The black squares depict qubits that are part of the GHZ state. Arrows represent a message to add nodes to the GHZ state along singlet paths. Each node keeps a record of the node from which it received this message, symbolized here by a white dot. When a node cannot extend the GHZ state any further (highlighted by a dashed outline) it measures its qubits in the $X$ basis (open squares) and sends this information back toward A (thin arrows) along the route recorded. Certain nodes are selected beforehand not to perform the measurement (here A and B) and these will form the resulting GHZ state. At each node the incoming data can be combined and sent back along one path if the routes branch. Once this data returns to A a phase operation can be performed on the qubit there to correct for any errors and the final GHZ state (here a singlet between A and B) will remain.

Figure 11

Three methods can be used to generate a singlet between two nodes A and C via an intermediate node B in an arrangement with two edges per bond. The thick black lines indicate pure but not maximally entangled states.

Figure 12

Comparison of the three methods described in the text for creating a singlet between nodes A and C in the setup shown in Fig. 11 for pure states. Shown are the success probabilities if the bonds are made up initially of the states $|\alpha \rangle$ and $|1/2\rangle$ for CEP (solid line), direct swapping (dotted line), and hybrid swapping (dashed line).

Figure 13

Success probability to generate a singlet between the end nodes of the swapping setup shown in Fig. 11 for the classical scheme (solid line), direct swapping (dotted line), and the hybrid scheme (dashed line). Each bond initially contains the states $\rho (\alpha ,\lambda )$ and $\rho (1/2,\lambda )$. We have indicated the percolation threshold of a face-centered cubic network.

Figure 14

Illustration of entanglement percolation in a 3D network. The circles represent nodes containing qubits and the lines represent bonds containing pairs of two-qubit entangled states (the edges are not shown). The 3D network can be transformed into a face-centered cubic network by performing the swapping operations (see Fig. 9) over the smaller nodes. For some bond parameters the hybrid scheme allows percolation to occur where classical percolation fails.

Figure 15

Application of the hybrid scheme in a square network. This involves transforming the PMSs into pure states probabilistically and then applying a suitable pure-state procedure (see text). In the case shown, all of the conversions are successful. When this happens a swapping operation can be performed and the resulting states distilled into a singlet.

Figure 16

Comparison of singlet conversion probabilities for the different strategies in the square configuration, that is, ${\mathrm{p̃}}_{\mathrm{CEP}}$ (solid line) and $p_{\mathrm{sq}}$ (dashed line) for $\lambda =\nu =0.98,\beta =0.5$. We have also indicated the percolation threshold for a triangular network.

Figure 17

By applying the square protocol on the shaded regions a triangular network of randomly distributed singlets (dashed lines in the right figure) is recovered. In the left figure the nodes (circles) are linked by bonds (solid lines) each containing two edges (not shown).

Figure 18

The first three iterations of a diamond lattice. We aim to create a singlet between nodes A and B in each case.

Figure 19

First three iterations of the tree lattice. Each iteration is given by repeating the previous lattice twice and linking the pair of previous endpoints at new endpoints. We aim to create a singlet between nodes A and B.

Figure 20

Probability of succeeding in generating a singlet between the endpoints of a diamond lattice for the 2nd and 3rd iterations (dashed lines). These give higher probabilities than the classical protocol (solid lines). The bonds contain two edges with parameters $\lambda =\nu =0.9,\beta =0.5$.

Figure 21

Probability of succeeding in generating a singlet between the endpoints of the 1st, 2nd, and 3rd iterations of the tree lattice (dashed lines). These also outperform the classical protocol (solid lines). The bonds contain two edges with parameters $\lambda =\nu =0.9,\beta =0.5$.