Entanglement purification protocols for all graph states

We present multiparty entanglement purification protocols that are capable of purifying arbitrary graph states directly. We develop recurrence and breeding protocols and compare our methods with strategies based on bipartite entanglement purification in static and communication scenarios. We find that direct multiparty purification is of advantage with respect to achievable yields and minimal required fidelity in static scenarios, and with respect to obtainable fidelity in the case of noisy operations in both scenarios.

Figure 1

(Color online) A three-colorable graph $G$ and the three corresponding two-colorable sub-graphs $g_1$, $g_2$, and $g_3$. $g_1$ corresponds to the red color (vertices 2, 4, and 6), $g_2$ to the green (vertex 7), and $g_3$ to the blue (vertices 1,3, and 5).

Figure 2

(Color online) Effect of a CNOT gate on a product of two graph states. The final state is associated with a graph composed of the two initial graphs with additional edges between the control qubit $b_1$ and all neighbors of the target qubit $b_2$.

Figure 3

(Color online) Yield as function of fidelity for the breeding protocol applied to the five-qubit ring state.

Figure 4

(Color online) A three-colorable graph $G$ with colors $A_1=\left\{7\right\}$, $A_2=\{2,4,6\}$, and $A_3=\{1,3,5\}$. ${\tilde{g}}_1$ and ${\tilde{g}}_2$ are two-colorable subgraphs which can be obtained from $G$ and which give $G$ when they are merged together. We use the method described in Sec. 5 to construct the subgraphs. For example, to obtain ${\tilde{g}}_1$, we choose $A_1$ as the first color and erase all edges between $A_2$ and $A_3$ using the procedure described in Sec. 4A2. The subgraph ${\tilde{g}}_2$ is obtained by erasing the vertex $A_1$ via a ${\sigma }_z$ measurement, and no further processing is required since this graph is already two-colorable.

Figure 5

(Color online) Local noise equivalent (LNE) corresponding to the maximal reachable fidelity and to the minimal required fidelity as function of the amount of local noise $p_l$ for the five-qubit ring state in the communication scenario. Given a state of fidelity $F$, the LNE is the level of local depolarizing noise which has to be applied to the perfect state to obtain fidelity $F$. The red solid line and the green dashed line stand for the LNE of the maximal reachable fidelity and the minimal required fidelity respectively for the MEPP strategy. The same values are plotted for the BEPP strategy. The blue small-dashed line is the LNE of a ring state teleported using four purified pairs and the pink dotted line is the LNE corresponding to the minimal required fidelity of a pair obtained by sending one of its qubits through a depolarizing channel. Note that the maximal reachable fidelity is the same in a communication and a static scenario. We also plotted the maximal reachable fidelity for the alternative strategy where a five-qubit cluster state and a Bell pair are purified before being connected. This value is given by the light-blue dot-dashed line.

Figure 6

(Color online) Maximal reachable fidelity and minimal required fidelity as function of the amount of local noise for the five-qubit ring state in the static scenario. The red solid line and the green dashed line stand for the maximal reachable fidelity and the minimal required fidelity respectively, for the MEPP strategy, while the blue small-dashed line and the pink dotted line stand for the same quantities in the BEPP scenario. The value of $F_{\mathit{min}}$ plotted for the BEPP scenario is the minimum value of fidelity of the ring state, so that the pairs we extract from the ensemble of rings can be purified.