# Conditional Measurements with Silicon Photomultipliers
## 2 Materials and Methods
### Theoretical Description
#### 2.1.1 Detected-Event Statistics of a TWB in the Presence of the OCT  
A TWB is a multi-mode entangled state of light generated through a nonlinear process known as parametric down conversion [22], which is investigated in the specific context of the photon counting described in [23]. Under the assumption that the energy is equally distributed among the \(\mu\) modes, a TWB state can be written as the tensor product of \(\mu\) single-mode squeezed states [4; 24], i.e.,  
\[\hat{\Lambda}=\bigotimes_{j=1}^{\mu}|\lambda\rangle\rangle_{jj}\langle\langle\lambda| \tag{1}\]  
where  
\[|\lambda\rangle\rangle=\sqrt{1-\lambda^{2}}\sum_{n}\lambda^{n}|n\rangle|n\rangle, \tag{2}\]  
being \(n\) the number of photons, and  
\[\lambda^{2}\equiv\frac{N}{N+\mu} \tag{3}\]  
with \(N\) as the mean number of photons in each beam. The conditioning measurement is performed on one of the two parties of the TWB state, typically named as the idler, so that the corresponding state of the other beam, which is called the signa, is ideally reduced to the same outcome, accordingly with [PERSON]'s rule [1]. In the absence of the OCT effects, the POVM describing a direct measure of the photon-number operator \(\hat{n}\) over multi-mode radiation reads [4]  
\[\hat{\Pi}_{m}(\eta,\mu)=\bigotimes_{j=1}^{\mu}\sum_{l_{j}}\delta_{m,\gamma} \hat{\Omega}_{l_{j}}(\eta) \tag{4}\]where \(m\) is the number of detected photons and \(\gamma\equiv\sum_{j=1}^{\mu}l_{j}\), being \(l_{j}\) the contribution of mode \(j\) to the number of detected photons. The detection is assumed to be affected by a limited quantum efficiency \(\eta\) and  
\[\hat{\Omega}_{l}(\eta)=\left(\frac{\eta}{1-\eta}\right)^{l}\sum_{n=l}^{\infty} \binom{n}{l}(1-\eta)^{n}|n\rangle\langle n| \tag{5}\]  
is the single-mode photon counting POVM. The effect of the OCT is typically described [12; 13; 14; 19] by the probability \(\varepsilon\) that an avalanche triggers another single spurious avalanche from a different cell. Assuming first-order OCT events, the number of fired spurious cells cannot be larger than the number of detected photons, which implies that, for the detected event \(k\) in the presence of the OCT, we have \(m\leq k\leq 2m\Rightarrow k/2\leq m\leq k\). Note that this assumption on the OCT model is quite strong. In principle, one should consider that a primary avalanche may be related to more than one OCT event [25; 26; 27]. Indeed, it may happen that more than one of the carriers in the primary avalanche triggers a secondary one, or that a secondary avalanche triggers a tertiary one as well, and the tertiary a quaternary and so on. However, here we develop a first-order OCT model since the class of SiPMs employed in the experiment is characterized by a very low cross-talk probability. Therefore, considering higher orders would be useless. Indeed, in a previous paper of ours [14] we have shown that the cross-talk probability associated to a cascade model can be assimilated to that limited to first order as long as a larger effective value of OCT is considered. Given this picture, we generalize the POVM in Equation (4) as follows  
\[\hat{\Omega}_{k}(\eta,\varepsilon,\mu)=\bigotimes_{j=1}^{\mu}\sum_{l_{j}}\delta _{k,\gamma}\hat{\Omega}_{l_{j}}(\eta,\varepsilon) \tag{6}\]  
with  
\[\hat{\Omega}_{l}(\eta,\varepsilon)=\left(\frac{\varepsilon}{1-\varepsilon} \right)^{l}\sum_{t=\lceil l/2\rceil}^{l}\binom{t}{l-t}\left(\frac{(1- \varepsilon)^{2}}{\varepsilon}\right)^{t}\left(\frac{\eta}{1-\eta}\right)^{t} \sum_{n=t}^{\infty}\binom{n}{t}(1-\eta)^{n}|n\rangle\langle n| \tag{7}\]  
where \(\lceil\cdot\rceil\) is the ceiling function. It can be shown that the operator in Equation (6) is a POVM, i.e., \(\hat{\Omega}_{k}\geq 0\) and \(\sum_{k}\hat{\Omega}_{k}=\hat{\mathbb{I}}\).