  Quantum state reconstruction involves measurement devices that are usually
described by idealized models, but not known in full detail in experiments. For
weak propagating microwaves, the detection process requires linear amplifiers
which obscure the signal with random noise. Here, we introduce a theory which
nevertheless allows one to use these devices for measuring all quadrature
moments of propagating quantum microwaves based on cross-correlations from a
dual-path amplification setup. Simultaneously, the detector noise properties
are determined, allowing for tomography. We demonstrate the feasibility of our
novel concept by proof-of-principle experiments with classical mixtures of weak
coherent microwaves.
