  We propose methods for the preparation and entanglement detection of
multi-qubit GHZ states in circuit quantum electrodynamics. Using quantum
trajectory simulations appropriate for the situation of a weak continuous
measurement, we show that the joint dispersive readout of several qubits can be
utilized for the probabilistic production of high-fidelity GHZ states. When
employing a nonlinear filter on the recorded homodyne signal, the selected
states are found to exhibit values of the Bell-Mermin operator exceeding 2
under realistic conditions. We discuss the potential of the dispersive readout
to demonstrate a violation of the Mermin bound, and present a measurement
scheme avoiding the necessity for full detector tomography.
