The use of beam and fiber splitters in quantum random number generators based on vacuum fluctuations

Quantum random number generators based on vacuum fluctuations produce truly random numbers that can be used for applications are requiring a high degree of randomness. A beam splitter with two inputs and two outputs is normally used in these systems. In this paper, mathematical descriptions were obtained for the use of such beam splitter and fiber $\mathrm{Y}$-splitter in quantum random number generation systems with homodyne detection. We derived equations which allowed us to estimate the impact of the scheme parameters' imperfection upon measurement results. We also obtained mathematical expressions, demonstrating the equivalence of quantum descriptions for a $\mathrm{Y}$-splitter and a beam splitter with two inputs.