One of the most intriguing physics processes that remain untested is the pure photon electron-positron pair production via quantum-vacuum fluctuations described by the nonlinear Breit-Wheeler theory. These fluctuations generate virtual pairs that can be turned into observable particles by applying strong electric fields above the Schwinger critical limit of \num{1.3d18}~V/m~\cite{Schwinger.1951, Ritus.1985}. Despite the advent of high-intense lasers, the critical limit is still far beyond achievable. However, such fields can be achieved on the rest frame of the real particles after the collision of a high-energy $\gamma$-ray photons with the laser beam. To diagnose the created pairs, this thesis describes the design of a particle detection system capable of successfully probing the single leptons created from strong-field quantum electrodynamics (SF-QED) interactions at the upcoming SF-QED experiments E-320 at FACET-II and FOR2783 at CALA. The designed detection system is composed of tracking layers made of LYSO:Ce scintillating screens and a Cherenkov calorimeter that, having their signals combined, can identify a positive event with a confidence level above 99%. At the E-320 experiment, electron beams generated by the FACET-II linear accelerator with an energy of 13~GeV collide with an intense laser beam of $\anot \approx 10$, and nonlinear Breit-Wheeler pairs are produced in the nonperturbative full quantum regime of SF-QED interaction ($\chie > 1$ and $\anot > 1$). About 100 electron-positron pairs per shot are expected to be created. According to Monte-Carlo simulations of the experimental layout, the detection system will be placed on a region permeated by a shower of x-rays and few-MeV $\gamma$-photons, however, a signal-to-noise ratio of $\SNRsig \approx 18$ on the detectors is achieved.