The reconstruction of the Wigner function [1] or density matrix of a propagating quantum field represents a cornerstone in quantum optical measurement theory and experiments. In quantum homodyne tomography [1, 2], for example, the signal is combined with a local oscillator in a beam splitter, and the intensities at the output ports are subtracted to produce the measurement of the amplified field quadratures in terms of a histogram. The latter gives access to all quadrature moments, or, equivalently, the Wigner function [1]. In this procedure, it is of utmost importance that field amplifiers are not needed in the optical domain because photodetectors with sufficient efficiency are available at the single-photon level [1]. In contrast, in the 1 − 10 GHz range, which has become highly relevant due to the advent of circuit quantum electrodynamics (QED) [3–11], only theoretical proposals exist for the detection of propagating single microwave photons [12–14]. Consequently, the detection of few-photon microwave signals requires linear amplification. Within well- established “off-the-shelf” technology, cryogenic high electron mobility transistor (HEMT) amplifiers lend themselves to this purpose. They offer flat gain over a broad frequency range, but they obscure the signals by adding random noise [15, 16] of 10–20 photons at 5 GHz. Nevertheless, we prove here that it is still possible to measure all quadrature moments of few-photon propagating microwaves in this situation. Furthermore, we show that our proposed reconstruction method also produces a measurement of all quadrature moments of the detector noise. In this sense, moving from a single amplification chain to a dual-path configuration constitutes a step beyond pure state reconstruction and towards the complete calibration of the measurement device, i.e., detector tomography [17]. We note that so far only state reconstruction of the intra-cavity field has been demonstrated in circuit QED [9]. However, quantum states of propagating microwaves themselves can be valuable in quantum information processing [18] and their full reconstruction represents an important open issue. We first develop a theory for the measurement of all moments of both a propagating quantum microwave signal and the noise added by the detector. The basic idea is illustrated in Fig. 1a. A signal $_{S}$√$_{2 is equally split at low temperatures by means of a four-port}$ 50-50 microwave beam splitter. The outputs are amplified and synchronously digitized. During this process, the amplifiers add the independent noise contributions χ$_{1}$ and χ$_{2}$ to the split signals in the detection chains 1 and 2, respectively. Assuming a 180 ◦ hybrid ring [19] as a beam splitter (cf. appendix), the recorded time traces are C$_{1}$ = G ( S + V + χ$_{1}$ ) and C$_{2}$ = G ( − S + V + χ$_{2}$ ), where G is the gain and V represents the ancilla state present in 2