〈 χ n 1 〉 = + 〈 C n 1 〉 /G n − n ∑ k =1 k ∑ j $_{=0}$( n k )( k j )$_{〈}$ χ n − k 1 〉〈 S k − $^{j}$〉〈 V $^{j}$〉 , (2) 〈 χ n 2 〉 = + 〈 C n 2 〉 /G n − n ∑ k =1 k ∑ j $_{=0}$( n k )( k j ) ( − 1) k − $^{j}$〈 χ n − k 2 〉〈 S k − $^{j}$〉〈 V $^{j}$〉 . (3) G 〈 S 〉 = 〈 C$_{1}$ 〉 = −〈 C$_{2}$ 〉 G $^{2}$〈 S $^{2}$〉 = − 〈 C$_{1}$C$_{2}$ 〉 + G $^{2}$〈 V $^{2}$〉 G $^{3}$〈 S $^{3}$〉 = − 〈 C 2 $_{1}$C$_{2}$ 〉 − 〈 C$_{1}$ 〉 ( 〈 C 2 1 〉 + 〈 C$_{1}$C$_{2}$ 〉 − 3 G $^{2}$〈 V $^{2}$〉 ) G $^{4}$〈 S $^{4}$〉 = − G $^{4}$〈 V $^{4}$〉 − 6 G $^{2}$〈 V $^{2}$〉〈 C$_{1}$C$_{2}$ 〉 + 6 G $^{4}$〈 V $^{2}$〉2 + 〈 C$_{1}$C$_{2}$ 〉 2 + 〈 C 2 $_{1}$C 2 2 〉 − 〈 C 2 1 〉〈 C 2 2 〉 〈 χ$_{1}$ 〉 ≡ 0 G $^{2}$〈 χ 2 1 〉 = 〈 C 2 1 〉 + 〈 C$_{1}$C$_{2}$ 〉 − 2 G $^{2}$〈 V $^{2}$〉 G $^{3}$〈 χ 3 1 〉 = 〈 C 3 1 〉 + 〈 C 2 $_{1}$C$_{2}$ 〉 − 2 〈 C$_{1}$ 〉 ( 〈 C 2 1 〉 + 〈 C$_{1}$C$_{2}$ 〉 ) 4 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: 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.