vector components $\rho_m\in\{\rho_0,...,\rho_n\}$ are the probabilities to generate $m$ PDC photon pairs, and $0\le m\le n$ is an integer number. For our case of spectrally multi-mode PDC, the photon probabilities obey Poisson statistics and are, thus, given by $$\label{eq:02}
\rho_m=\frac{e^{-\langle m \rangle}\langle m\rangle^{m}}{m!}, \;m\in\mathbf{N_0}.$$ Because the mean photon pair number per optical pulse can be written as $$\label{eq:03}
\langle m\rangle=\sum\limits_{m=0}^\infty m\rho_m,$$ a reasonable photon-triplet generation approach must provide $\langle m\rangle\ll1$ for the primary PDC process. Given that case, the higher-order photon contributions are significantly reduced such that $\sum_{m\ge2}^\infty m\rho_m\ll\rho_1$, meaning that we pump the secondary PDC stage almost exclusively with single photons. Thus, at low pump powers, we will measure mainly genuine photon triplets in the overall process. Reference [@Krapick2015] provides an in-depth theoretical analysis of quantitative measures for state preparation fidelities based on observed experimental parameters.

Long-term stable experimental setup
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![Experimental setup for photon-triplet verification: the tripartite states reside in the two output beams behind the device. A silicon filter blocks the pump and photons at $790.3\,\mathrm{nm}$ in both beams. The lower output comprises the primary idler photons and the upper beam contains secondary photon pairs. We separate the secondary photon pairs quasi-deterministically according to their spectral correlations 