Closing the detection loophole in tripartite Bell tests using the W state

We study the problem of closing the detection loophole in three-qubit Bell tests, the experimentally most relevant case beyond the usual bipartite scenario, and show that the minimal detection efficiencies required can be considerably lowered compared to the two-qubit case. The lowest reported detection efficiency thresholds for two and three qubits so far are $\sim 66.7\mathrm{\%}$ and $60\mathrm{\%}$, respectively. Using the three-qubit W state and a 3-setting Bell inequality, we beat these thresholds and with an 8-setting Bell inequality we reach $50.13\mathrm{\%}$. We also investigate generic three-qubit states which allow us to attain a detection efficiency of $50\mathrm{\%}$ in a 4-setting Bell test. We conjecture that the limit of $50\mathrm{\%}$ is unbeatable using three-qubit states and any number of measurements.