Long-distance entanglement generation in two-dimensional networks

We consider two-dimensional networks composed of nodes initially linked by two-qubit mixed states. In these networks we develop a global error correction scheme that can generate distance-independent entanglement from arbitrary network geometries using rank-2 states. By using this method and combining it with the concept of percolation, we also show that the generation of long-distance entanglement is possible with rank-3 states. Entanglement percolation and global error correction have different advantages depending on the given situation. To reveal the trade-off between them we consider their application to networks containing pure states. In doing so we find a range of pure-state schemes, each of which has applications in particular circumstances: For instance, we can identify a protocol for creating perfect entanglement between two distant nodes. However, this protocol cannot generate a singlet between any two nodes. In contrast, we can also construct schemes for creating entanglement between any nodes, but the corresponding entanglement fidelity is lower.

Figure 1

(a) A quantum network composed of nodes (large open circles) that contain qubits (filled circles). Each qubit is initially entangled with another [thin (black) lines] and these two-qubit states form the edges that are contained within each bond [thick (gray) lines]. (b) A $5\times 5$-node square network with its dual network. Individual edges and qubits are not shown. However, dual nodes are presented as small open (red) circles connected by dashed lines, including the network exterior (dashed square with small circle). (c) A $5\times 5$-node triangular network with its honeycomb dual network.

Figure 2

One possible pure-state network ${\rho }_{\{a_i\},\{b_i\}}$ from the ensemble given by Eq. (1). Each node (open circles) contains qubits (filled circles) that are part of two-qubit entangled states (thin lines) and edges are located in bonds [thick (gray lines]. For $a=0$ the edge line is solid, and for $a=1$ it is dashed. $Z\otimes Z$ measurement operations are carried out within each node [(black) ovals] around one plaquette. This allows a value of $L_{\mathrm{loop}}$ to be associated with each plaquette. The dual network [dashed (red) lines] is shown with these values contained within each dual node [dashed (red) circles]. The dual vertex representing the exterior is represented by a dashed (red) box.

Figure 3

(a) A single-edged network is shown where three edge errors have occurred [dashed (black) lines]. The local $Z\otimes Z$ measurement outcomes are displayed inside boxes, with $\pm$ referring to a $(-1){}^{m_j}=\pm 1$ eigenstate. Around each plaquette the values of $L_{\mathrm{loop}}$ are then calculated and shown on the dual-network nodes [dashed (red) circles linked by dotted (red) lines]. Values of $1$ are defects and are matched together along a path [(blue) arrows]. (b) Edge errors are assumed to lie across the matching path and $X$ gates are applied to one qubit within each of these edges [(black) arrows]. This “flips” ($+\leftrightarrow -$) some of the local measurement outcomes. Each qubit can then be put into two groups (labeled 0 and 1) using the measurement outcomes (a minus sign means the groups need to be different for the two qubits), and qubits that are in a network edge are put in the same group. To assign the values we pick an initial qubit and value at random and then assign values in all remaining qubits. (c) $X$ gates are applied to one group [nodes containing (white) crosses], and for each pure ensemble state a GHZ state is created with patches [shaded (red) area] of bit-flip errors. These occur where the actual edge error path differed from the assumed path.

Figure 4

Regular square network formed by single-edged bonds. The dual lattice is shown by (red) dashes and includes the error syndromes on the dual vertices [small filled (red) circles; $L_{\mathrm{loop}}=1$, filled; $L_{\mathrm{loop}}=0$, outlined]. The edge errors causing these plaquette errors are shown by thick (blue) bars. The error syndromes are matched together [(black) arrows], and this can be used to give the most likely edge error locations for the recorded measurement syndrome. The shaded (gray) areas between the assumed error locations and the actual edge errors identify groups of nodes. For this particular pure state in the ensemble we will obtain a GHZ state with $X$ errors present on the qubits within these nodes. In practice we do not know what the initial state was and hence do not know the location and size of the shaded (gray) patches, but for the regions to become larger, more errors are required and hence are less likely.

Figure 5

Here a $20\times 20$ network has had its bonds removed with a probability of $0.05$ and then $X$ errors introduced with a probability of $0.05$. We apply our global error correction procedure, and eventually this generates a GHZ state with patches of nodes that exhibit unwanted, extra bit flips [filled (red) circles]. If we try to link a node from within a patch to outside of them (dashed line), the final-state two-qubit state exhibits a bit-flip error. Examples are also given of qubit pairings that would not exhibit errors [bold (black) lines].

Figure 6

Fidelities are plotted from simulations of global error correction when run on complete square and triangular networks to produce a two-qubit entangled state between two randomly chosen qubits. In the square case (solid lines) we see a lower threshold than in the triangular network (dashed lines). Also included are the approximate threshold values (dotted vertical lines) calculated in Sec. 3. We used networks of $10\times 10$ nodes and $20\times 20$ nodes, with the larger networks presented by thicker lines; the threshold becomes more pronounced when the network gets larger.

Figure 7

(a) $p_x^{\prime }$ coefficient for the binary state created using a PCM. (b) The probability, $P_c$, of generating a binary state from two initial states, i.e., $m=2$, of the type given by Eq. (9).

Figure 8

(a) The probability of a node existing in the largest cluster, $\varphi$. (b) Based on a $25\times 25$ square network, the fidelity $F$ between two randomly chosen nodes in the largest cluster of connected nodes is shown, depending on the bond conversion probability $P_c$ and $X$-error probability $p_x^{\prime }$. The boundary for values that allow long-distance entanglement distribution in infinite networks is shown as the thick solid (black) line. Pure states can be probabilistically transformed into binary states with parameters along the dashed lines. These lines are for the pure states $\alpha =0.75,0.76,\dots ,0.85$, from lowest to highest $p_x^{\prime }$ values, respectively.

Figure 9

Starting from pure states we can create a range of binary states (dashed lines in Fig. 8). For each of these the largest cluster contains a different proportion $\varphi$ of total nodes, and any two of these can be linked using the global error correction, to leave a two-qubit entangled state that has fidelity $F$. The different values of $F$ and $\varphi$ are given for each initial pure state. Shown are plots corresponding to 11 initial pure states with $\alpha =0.75,0.76,0.77,\dots ,0.85$ (top to bottom).