One-shot entanglement generation over long distances in noisy quantum networks

We consider the problem of creating a long-distance entangled state between two stations of a network, where neighboring nodes are connected by noisy quantum channels. We show that any two stations can share an entangled pair if the effective probability for the quantum errors is below a certain threshold, which is achieved by a local encoding of the qubits and a global bit-flip correction. In contrast to the conventional quantum-repeater schemes, we do not need to store the qubits in quantum memory for a long time: our protocol is a one-shot process (i.e., the elementary entangled pairs are used only once) involving one-way classical communication. Furthermore, the overhead of local resources increases only logarithmically with the size of the network, making our proposal favorable to long-distance quantum communication.

Figure 1

(Color online) A quantum network with a station located at each vertex of an $N\times N$ square lattice. The goal is to create a long-distance entangled state between some previously chosen stations $A$ and $B$.

Figure 2

(Color online) Bit-flip error correction: (a) Parity check pattern associated with two erroneous edges, and (b) its plaquette representation. (c) Two of the three possible edge identifications (with the minimal number of errors) for the given syndrome pattern. If the inferred edges do not correspond to the erroneous ones, a loop is created and sites lying inside this region will apply the wrong bit-flip correction.

Figure 3

(Color online) Simulation of the (classical) bit-flip error correction in a square lattice with periodic boundary conditions, so that syndrome plaquettes always appear in pairs. An unknown bit is sent through the lattice and a possible minimal set of erroneous edges leading to the corresponding parity check pattern is found using Edmonds’ algorithm for minimal weight perfect matching [10]. We show here the probability that two random sites infer the same value of the bit in an $N\times N$ lattice with bit-flip error probability ${\epsilon }_b$. The extrapolated function $P_{\infty }$ is plotted with a bold line and the dashed line represents its behavior for small ${\epsilon }_b$: $P_{\infty }\left({\epsilon }_b\right)\approx 1-6{\epsilon }_b^2+O\left({\epsilon }_b^3\right)$.

Figure 4

(Color online) Local resources and operations at each station. At least five qubits are needed: one for the parity check measurement (dashed box) and four for the teleportations (dotted boxes). The choice of the rotations $R$, depending on the outcomes of the Bell measurements, can be tracked classically and thus no classical communication is necessary between the stations during the process; the interpretation of the parity checks can be postponed until its very end.

Figure 5

(Color online) One-dimensional quantum computer embedded in a lattice: (a) two teleportations transport the quantum computer between time $t$ and $t+1$, and (b) two-qubit gates can be applied on neighboring qubits.