## Appendix 

The ancilla state is assumed to be well known and, hence, its quadrature moments can not be detected by our method. This is not a restriction in practice because vacuum or thermal states at gigahertz frequencies can be prepared reliably in an experiment by controlling the temperature [20, 21]. Furthermore, it is more relevant to characterize quantum states such as Fock states, where part of the Wigner function is negative. 

Beam splitter types for the dual-path method. Although in this work we always assume a 180 ◦ hybrid ring as beam splitter, other choices are possible, as long as they provide enough isolation between the output ports. However, one has to keep in mind that all lossless, matched and reciprocal beam splitters must be four port devices [19]. Sometimes, as in the case of the Wilkinson power divider [19, 21], the fourth port may be hidden internally and therefore complicate the analytical treatment of the ancilla state. Nevertheless, the equivalent of Eqs. (1)–(3) of the supplemented paper can in principle be calculated for any beam splitter. 

Offset correction. The raw data is divided into segments of 4128 traces equivalent to 0.5 s measurement time. For each of these segments, the time average is subtracted from each data point before any other manipulation. Effectively, this procedure acts as a high-pass filter eliminating slow drifts in the data. 

The calibration of the input line can be performed in situ in our setup. To this end, the 50 Ω load at the ancilla port of the hybrid ring is temperature controlled using a heater and a thermometer. Except for small deviations, we can assume the hybrid ring, ancilla load, and effective 50 Ω load at the signal port to have the same temperature. Both loads inject thermal voltage fluctuations into the hybrid ring. The auto-variance of these fluctuations follows the well-known Planck function. Knowing the bandwidth of our setup from measurements with a spectrum analyzer (51 MHz), the gains of the amplification and detection chains are inferred from a numerical fit of the Planck function to the auto-variance data [21]. We find gains of approximately 110 dB for both chains, where small residual gain asymmetries have been absorbed in a compensation factor already. Together with the 0 . 5 dB loss of the hybrid ring and the total transmission from source to digitizer, we can extract an input line attenuation of 94 dB. From this, we determine the signal power P$_{in}$ at the input of the hybrid ring. The corresponding number of photons on average is the pulse energy P$_{in}$T$_{pulse}$ divided by 

