# Conditional Measurements with Silicon Photomultipliers
## 1 Introduction  
Given an entangled state, a conditional measurement, which is a scheme exploiting the reduction postulate [1], is a well-known option for the generation and manipulation of nonclassical and non-Gaussian states [2, 3]. Remarkably, optical states have proven to be suitable for this task [3, 4, 5, 6, 7, 8], especially in sight of Quantum Information protocols [9, 10, 11].  
Here we focus on the detection of conditional states of light in the discrete-variable regime via photon-number-resolving (PNR) detectors. In particular, a novel class of PNR detectors, known as Silicon Photomultipiers (SiPMs), has recently experienced a remarkable technological improvement [12] and attracted attention for Quantum Optics applications [13, 14, 15]. Due to both their outstanding PNR capability and to their compactness and robustness, SiPMs may now be considered for discrete-variable Quantum-Information protocols [16]. Motivated by these points, we have recently tested a pair of SiPMs for the detection of nonclassical states of light [17, 18]. Specifically, in [17] we generated a mesoscopic multi-mode twin-beam (TWB) state via type-I parametric down-conversion and post-selected one of the entangled beams by measuring the photon-number observable on the other one. We succeeded in assessing the nonclassicality of the detected conditional states.  
However, as far as we know, the conditioning protocol via SiPMs on a TWB still lacks a full theoretical description. Indeed, the existing model of the effects of detection [3, 4] does not include the influence of the major drawback of the SiPMs, i.e., the Optical Cross-Talk (OCT) [12, 13, 19, 20]. The OCT is a process intrinsically connected to the very pixel structure of these devices. Being each pixel a single-photon avalanche diode, there is a chance that the avalanche triggered by a photon emits a secondary photon, which may fire a supplementary cell, resulting in a spurious count. Thus, the OCT influences the output statistics and may conceal the nonclassicality of the detected state.  
Here we extend the model presented in [4] by including the effects of the OCT and provide a comparison with our experimental results. In Section 2 we define the positiveoperator-valued measure (POVM) describing photon counting affected by a limited quantum efficiency and by the OCT, and provide the tools needed for retrieving the statistics of the unconditioned and conditional states. Finally, we address the nonclassicality of the conditional states as sub-Poissonianity and recall the definition of the Fano factor.  
In Section 3 we show our results. Firstly, we provide an analytic closed formula for the statistics of a multi-mode thermal state affected by the OCT. In a previous work of ours [21], we have already shown that in the single-mode case such a distribution is expressed in terms of the Fibonacci polynomials. In the present paper, we derive the distribution of the conditional state and consider the limit case of a TWB with an infinite number of modes. We also include the effect of the imbalance between the quantum efficiencies of the detectors. We show the effects of the OCT on the first moment of the statistics and, finally, we provide the Fano factor of the output conditioned distribution.  
In Section 4 the theoretical predictions of the developed model are compared with the data from our experiment. In the same Section we also discuss how the OCT affects the light statistics and especially the consequences for the nonclassicality of the conditional states.  
In Section 5 we draw our conclusions and suggest further improvements to our model.