Demonstration of Local Expansion Toward Large-Scale Entangled Webs

We demonstrate an optical gate that increases the size of polarization-based $W$ states by accessing only one of the qubits. Using this gate, we have generated three-photon and four-photon $W$ states with fidelities $0.836\pm 0.042$ and $0.784\pm 0.028$, respectively. We also confirmed the existence of pairwise entanglement in every pair of qubits, including the one that was left untouched by the gate. The gate is applicable to any size of $W$ states and hence is a universal tool for expanding entanglement.

Figure 1

(a) Concept of local state expansion. (b) Experimental setup. BBO: $\beta$-barium borate crystals (thickness $1.5\mathrm{mm}+1.5\mathrm{mm}$). HWP: half-wave plate. QWP: quarter-wave plate. UV: ultraviolet pulse. BS: beam splitter. IF: narrowband interference filter (wavelength 790 nm; bandwidth 2.7 nm). Q: quartz crystals. C: BBO (thickness 1.65 mm) used to compensate walk-off effects. Although the ideal operation of the gate requires a two-photon Fock state in mode 2, in the experiments the two photons of mode 2 are provided by a weak coherent pulse (WCP). All the detectors ($D_{0,4,5,6}$) are silicon avalanche photodiodes placed after single-mode optical fibers.

Figure 2

Observed two-photon interference. The best fit to the experimental data is represented by the solid Gaussian curve (coherence length $l_c\simeq 144\mu \mathrm{m}$ and visibility is 0.85).

Figure 3

Results of the expansion of $|W_1⟩$ to a three-photon polarization-entangled $W$ state $|W_3⟩$. (a) The real part of the reconstructed density matrix of the final state ${\rho }_{456}$. (b) The real parts of the reconstructed reduced density matrices ${\rho }_{45}$, ${\rho }_{46}$, and ${\rho }_{56}$.

Figure 4

Results of the expansion of $|W_2⟩$ to a four-photon polarization-entangled $W$ state $|W_4⟩$. (a) The real part of the reconstructed density matrix of the final state ${\sigma }_{0456}$. (b) The real parts of the reconstructed reduced density matrices ${\sigma }_{45}$, ${\sigma }_{46}$, and ${\sigma }_{56}$, and (c) those of ${\sigma }_{04}$, ${\sigma }_{05}$, and ${\sigma }_{06}$.