  We propose a measurement strategy which can, probabilistically, reproduce the
statistics of any observable for spatially encoded photonic qubits. It
comprises the implementation of a two-outcome positive operator-valued measure
followed by a detection in a fixed transverse position, making the displacement
of the detection system unnecessary, unlike previous methods. This strategy
generalizes a scheme recently demonstrated by one of us and co-workers
restricted to measurement of observables with equatorial eigenvectors only. The
method presented here can be implemented with the current technology of
programmable multipixel liquid-crystal displays. In addition, it can be
straightforwardly extended to high-dimensional qudits and may be a valuable
tool in optical implementations of quantum information protocols with spatial
qubits and qudits.
