  Quantum states and measurements exhibit wave-like --- continuous, or
particle-like --- discrete, character. Hybrid discrete-continuous photonic
systems are key to investigating fundamental quantum phenomena, generating
superpositions of macroscopic states, and form essential resources for
quantum-enhanced applications, e.g. entanglement distillation and quantum
computation, as well as highly efficient optical telecommunications. Realizing
the full potential of these hybrid systems requires quantum-optical
measurements sensitive to complementary observables such as field quadrature
amplitude and photon number. However, a thorough understanding of the practical
performance of an optical detector interpolating between these two regions is
absent. Here, we report the implementation of full quantum detector tomography,
enabling the characterization of the simultaneous wave and photon-number
sensitivities of quantum-optical detectors. This yields the largest
parametrization to-date in quantum tomography experiments, requiring the
development of novel theoretical tools. Our results reveal the role of
coherence in quantum measurements and demonstrate the tunability of hybrid
quantum-optical detectors.
