are zero for an equally distributed histogram. Hence, to observe an oscillating third central moment a statistical mixture with an asymmetric histogram (cf. inset of Fig. 3c) is required. In the experiment, we again create a train of pulses with 100 poa, but this time 75% of them are shifted by the phase ϕ and 25% by − ϕ . Figure 3c shows the third central moment calculated from the data of a single amplification chain. In contrast to the case of an equally distributed mixture, a clear oscillating signal is visible within the pulse duration for a skewed histogram. However, there is an offset of approximately − 7 × 10 − $^{5}$V$^{3}$. In Fig. 3d, one can see that this offset disappears when also taking into account the data from the second chain and applying the dual-path detection scheme described by Eqs. (1)-(4). Following these equations, we can write down the third central moment of C$_{1}$ as the sum of the third central moments of signal, noise, and ancilla state: Since 〈 ( V −〈 V 〉 ) $^{3}$〉 = 0 due to the the Gaussian statistics of V , it becomes obvious that the offset in Fig. 3c must be G $^{3}$〈 ( χ$_{1}$ −〈 χ$_{1}$ 〉 ) $^{3}$〉 . In this sense, the noise of the detection chain shows non-Gaussian statistics. Although the exact origin of the latter inside the detection chain remains unclear, our experiments provide a first confirmation that the dual-path method is indeed capable of characterizing signal and detector noise moments simultaneously. In conclusion, we present a method based on off-the-shelf technology and signal recov- ery techniques to gain access to arbitrary moments of both weak propagating quantum microwaves and the detector noise. In particular, we allow for the use of linear amplifiers adding 10–20 noise photons to the signals. We find that it is crucial to move from a single to two amplification and detection chains and successfully perform proof-of-principle ex- periments with statistical mixtures of weak pulsed coherent microwaves. We demonstrate sufficient sensitivity for the first two moments and observe indications for a non-Gaussian statistics of the detector noise. The experiments indicate that our dual-path method is a suitable tool for detecting propagating quantum signals such as squeezed states from a Josephson parametric amplifier [23–26], Fock states [9] leaking out of an on-chip resonator [5] or non-classical microwave field states generated in a two-resonator circuit QED setup [7]. 8 〈 ( C$_{1}$ − 〈 C$_{1}$ 〉 ) $^{3}$〉 = G $^{3}$〈 ( S − 〈 S 〉 ) $^{3}$〉 + G $^{3}$〈 ( χ$_{1}$ − 〈 χ$_{1}$ 〉 ) $^{3}$〉 + G $^{3}$〈 ( V − 〈 V 〉 ) $^{3}$〉 .