Robust and scalable scheme to generate large-scale entanglement webs

We propose a robust and scalable scheme to generate an $N$-qubit $W$ state among separated quantum nodes (cavity-QED systems) by using linear optics and postselections. The present scheme inherits the robustness of the Barrett-Kok scheme [S. D. Barrett and P. Kok, Phys. Rev. A 71, 060310(R) (2005)]. The scalability is also ensured in the sense that an arbitrarily large $N$-qubit $W$ state can be generated with a quasipolynomial overhead $~$$2^{O[(\log {}_{2}N){}^2]}$. The process to breed the $W$ states, which we introduce to achieve the scalability, is quite simple and efficient and can be applied for other physical systems.

Figure 1

Three-level atoms are embedded in cavities. The two long-lived states $|0\rangle$ and $|1\rangle$ are used as a qubit, and the transition between the states $|1\rangle \leftrightarrow |e\rangle$ is coupled to the cavity mode. The output modes are mixed with four 50:50 BSs and measured by photodetectors ${\mathrm{D}}_i$.

Figure 2

Economical breeding. Two $|W_N\rangle$’s, which are depicted symbolically as circles connected with lines, are converted to a $|W_{2(N-1)}\rangle$ probabilistically. Even if the conversion fails, the two $|W_{N-1}\rangle$’s are left and can be recycled.

Figure 3

The overheads $R_N$ for the concatenated entangling (red $ˆ$) and the breeding (blue $\times$), respectively, are plotted as functions of the number of qubits $N$, where ${\eta }_d{\eta }_l=0.5,0.7,1$ from top to bottom.