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We note that our conjectures of equal gain in both chains and a perfect 180 ◦ hybrid ring are not a restriction in practice. In the derivation of the above formulas, the mutual statistical independence of S , V , χ$_{1}$ and χ$_{2}$ is crucial because it implies 〈 S $^{β}$V $^{γ}$χδ $_{1}$χ ϵ 2 〉 = 〈 S $^{β}$〉〈 V $^{γ}$〉〈 χ δ 1 〉〈 χ ϵ 2 〉 for β, γ, δ, ϵ ∈ N$_{0}$ . The latter formula also shows that Eqs. (1)-(3) are suitable for quantum signals, where S , V , χ$_{1}$ , and χ$_{2}$ have to be interpreted as operators. 

Explicit expressions for moments of signal and detector noise can be calculated in the spirit of Eqs. (1)–(3). We assume 〈 V 2 j $^{+1}$〉 = 0 ( j ∈ N ) for the ancilla as, e.g., in the case of Gaussian statistics. We note that for the initial correlation 〈 C ℓ $_{1}$C m 2 〉 other choices than ℓ = n − 1 and m = 1 are possible, as long as ℓ + m = n and ℓ, m ∈ N . Typically, balanced products with ℓ ≈ m result in simpler expressions because of higher symmetry. Starting from 〈 C$_{1}$C$_{2}$ 〉 , 〈 C 2 $_{1}$C$_{2}$ 〉 , 〈 C 2 $_{1}$C 2 2 〉 we obtain: 

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