# Quantum Calibration of Photon-Number-Resolving Detectors Based on Multi-Pixel Photon Counters
## 2 Materials and Methods
### Quantum Detector Tomography (QDT)  
To characterize MPPC detector, we need to determine its corresponding POVM. Different from the partial calibration of PNRDs or the establishment of a complex model based on multiple assumptions, the POVM elements can be directly obtained and applied to describe quantum detectors under very few assumptions by means of QDT [35]. If a series of quantum states \(\{\rho\}\) put into the under-test quantum detector is already known, \(\mathrm{P}_{j}\) is the probability of obtaining \(j\) clicks. According to the input quantum state \(\rho\) and the output \(j\), the probability \(\mathrm{P}_{j}\) can be calculated by:  
\[\mathrm{P}_{j}=Tr\big{[}\Pi^{j}\rho\big{]}, \tag{1}\]  
where a set of semipositive Hermitian operators \(\big{\{}\Pi^{j}\big{\}}\) are the POVM elements of the detector, and \(Tr\) corresponds to the trace. Equation (1) connects the quantum characteristics of the input signal with the classical output of the detector. By measuring a set of known probe states, we are able to characterize an unknown detector and find its POVM elements. POVM elements are complete and non-negative as:  
\[\sum_{j}\Pi^{j}=1,\Pi^{j}\geq 0. \tag{2}\]  
In the experiment, coherent state \(|\alpha\rangle\) obtained from an attenuated laser is chosen as the alternative to photon number state \(|n\rangle\) to be the input state, since the pure photon number state \(|n\rangle\) is difficult to prepare. The relationship between the distribution of the coherent state \(|\alpha\rangle\) and the photon number state \(|n\rangle\) is in accordance with the Poisson distribution:  
\[|\alpha\rangle=\sum_{0}^{\infty}e^{-\frac{|n|^{2}}{2}}.\frac{d^{n}}{\sqrt{n }}|n\rangle. \tag{3}\]  
Different average incident photon number of coherent states can be obtained by varying the attenuation. The range of input quantum states shall include photon number states that saturate the output of the detector so that the POVM space of the detector can be fully covered. Elements except for the diagonal ones of POVM elements can be assumed as zero since MPPC is phase independent. The equation of the POVM elements is:  
\[\Pi^{j}=\sum_{n=0}^{M}\theta_{n}^{j}|n\rangle\langle n|, \tag{4}\]  
Figure 1: **(a)** Structure diagram of MPPC. AMP, amplifier; OSC, oscilloscope. **(b)** Output waveform from MPPC captured by an oscilloscope. The effective photosensitive area of MPPC (C13366-1350 GA Hamamatsu) is \(1.3\times 1.3\) mm\({}^{2}\) with 667 pixels. **(b)** Output waveform displayed by an oscilloscope. The output pulse duration is about 40 ns. where \(M\) is the number of photons when the detector is in the saturation state, and \(\theta_{n}^{j}\) is the diagonal element of POVM elements \(\left\{\Pi^{j}\right\}\). Therefore, the probability that the number of incident photons is \(i\) and that the number of photons detected by the detector is \(j\) can be calculated as:  
\[ \begin{array}{l}P_{i,j}=Tr\left[\Pi^{j}\rho_{i}\right]\\ =Tr\left[e^{-|\alpha_{i}|^{2}}\sum_{n=0}^{M}\frac{\alpha_{i}^{2}\alpha_{i}^{ \alpha_{i}^{\alpha_{i}^{\alpha_{i}^{\alpha_{i}^{\alpha_{i}^{\alpha_{i}^{\alpha_{i} }}}}}}}{n!}|n\rangle\langle n|\cdot\sum_{m=0}^{M}\Pi_{m}^{j}|m\rangle\langle m |\right]\\ =Tr\left[e^{-|\alpha_{i}|^{2}}\sum_{n=0}^{M}\frac{|\alpha_{i}|^{2}n}{n!}\Pi_{n }^{j}|n\rangle\langle n|\right]\\ =e^{-|\alpha_{i}|^{2}}\sum_{n=0}^{M}\frac{|\alpha_{i}|^{2}}{n!}\Pi_{n}^{j}. \end{array}  \tag{5}\]  
Here, if A\({}_{i,n}\) is the density matrix of the input photon number states and \(\Pi_{n,j}\) are the POVM elements of MPPC to be measured, then:  
\[\mathrm{A}_{i,n}=e^{-|\alpha_{i}|^{2}}\frac{|\alpha_{i}|^{2}}{n!}, \tag{6}\]  
\[\Pi_{n,j}=\Pi_{n}^{j}, \tag{7}\]  
and the matrix of probability can be defined as:  
\[\mathrm{P}_{i,j}=\mathrm{A}_{i,n}\cdot\Pi_{n,j}. \tag{8}\]  
According to the already known A\({}_{i,n}\) and the measurement of the corresponding P\({}_{i,j}\), POVM elements of the detector can be derived from the utilization of the maximum-likelihood estimation (MLE) [39]. There are only two main preconditions for QDT.