The paper https://arxiv.org/abs/2407.20184 deals with calculating the so-called filter function capturing the sensitivity of the fidelity of a quantum channel to the power-spectral-density of a noise source within linear response theory. If the noise is captured by some time trace $h(t)$ times an operator $\mathcal{O}(t)$, that is, by an extra term $\mathcal{H}_{\rm noise}=h(t)\mathcal{O}(t)$ in the Hamiltonian, then the filter function given by (Eq. G12)
$$I(f)=\int{\rm d}t\int{\rm d}\tau\;\cos\left(2\pi f(t-\tau)\right)\left<\mathcal{O}_{H}(t)\mathcal{O}_{H}(\tau)\right>_{\rm c}$$
can be used to calculate the infidelity due to the particular noise term via (Eq. G11)
$$1-\mathcal{F}=\int{\rm d}f\;S(f)I(f)$$
Here the subscript $H$ refers to the operator in the Heisenberg picture, the subscript $\rm c$ refers to the connected correlator and $S(f)$ is the PSD of the noise $h(t)$. According to the authors, the expectation value can be evaluated by averaging over the Haar measure, writing the following explicit formula (Eq. G13)
$$\left<\mathcal{O}_{H}(t)\mathcal{O}_{H}(\tau)\right>_{\rm c}=\frac{1}{D}{\rm Tr}\left[\mathcal{O}_{H}(t)\mathcal{O}_{H}(\tau)P\right]-\frac{1}{D(D+1)}\left[{\rm Tr}\left[\mathcal{O}_{H}(t)P\mathcal{O}_{H}(\tau)P\right]+{\rm Tr}\left[\mathcal{O}_{H}(t)P\right]{\rm Tr}\left[\mathcal{O}_{H}(\tau)P\right]\right]$$
where $D$ is the dimension of the Hilbert space and $P$, in their words, "is the projector from the full Hilbert space to the input Hilbert space".
This last step is unclear to me. For one, I don't understand what precisely $P$ is and where it comes from. But even if I ignore it for a moment, different resources I find (e.g. https://quantum-journal.org/papers/q-2024-05-08-1340/, Eq. 50) state that the first moment Haar average is
$$\left<X\right>=\frac{1}{D}{\rm Tr}\left[X\right]$$
Hence for $P=I$ I would expect
$$\left<\mathcal{O}_{H}(t)\mathcal{O}_{H}(\tau)\right>_{\rm c}=\frac{1}{D}{\rm Tr}\left[\mathcal{O}_{H}(t)\mathcal{O}_{H}(\tau)\right]-\frac{1}{D^{2}}{\rm Tr}\left[\mathcal{O}_{H}(t)\right]{\rm Tr}\left[\mathcal{O}_{H}(\tau)\right]$$
in disagreement with the above. Could someone clarify where are the averages in Eq. G13 in this paper coming from and solve this apparent discrepancy? What is the projector $P$ explicitly? Any relevant references on promoting expectation values into Haar measure averages would be super helpful!
Thanks a lot in advance!
