a quasiprobability) for the quantum noise of interest. For truly classical noise, Eq.  reduces to classical MGF in Eq. . The moments and corresponding quasiprobability defined here are intrinsic to the noisy system, and can be used to predict the outcomes of a wide class of schemes designed to measure this noise [@Nazarov2003; @Hofer2017]. They also have a direct role in Keldysh non-equilibrium field theory: they characterize the fluctuations of the “classical" field associated with the operator $\hat{\xi}(t)$. At each order, the Keldysh-ordered correlation function describes the intrinsic fluctuations of the system [@Kamenev2011book]. In contrast, the remaining independent correlation functions at the same order describe how the system responds to external fields which couple to $\hat{\xi}(t)$ (either higher-order response coefficieints, or generalized noise susceptibilities; see Supplemental Material (SM) for details [@SI]). We stress again that at each order, the Keldysh-ordered correlation function is precisely the correlation function “seen" by the qubit.

It follows that the Keldysh-ordered cumulants $C^{(k)}(\vec{t}_{k}) \equiv \langle \langle \hat{\xi}\left(  t_1\right) \cdots 
\hat \xi \left(  t_k\right) \rangle\rangle_\mathcal{K}$ of the noise can be generated from $\chi[F(t);t_f] \equiv \ln \Lambda[ F(t); t_f ] $ via $$\label{eq:ncdef}
    \chi[F(t);t_f]  =  \sum_{\ell=1}^\infty\!\frac{(-i)^\ell }{  \ell !}\prod_{j=1}^\ell
    \left[ \int_0^{t_f}\!\! dt_j F(t_j) \right]\!C^{(\ell)}(\vec{t}_{\ell}),$$ where we define 