(a) Illustration of the model of a random external classical field acting only on the qubit b, whereas qubit a is isolated. The random dephaser shifts of π, with probability 1/2, the phase (π/2) of the input field. (b), Experimental setup simulating the physical model of panel a. The state preparation process is started by separating the ultraviolet pulses into two paths with the Wollaston prism. The relative amplitude between the two paths can be controlled by a HWP1. HWP2 and HWP3, with angles set to 22.5°, are used to control the polarization of the pump light by rotating |H〉 and |V〉 to and , respectively. Photon pairs are emitted into modes a and b. The birefringence in the beta-barium-borate crystals is compensated by the quartz plates (CP). The photon in mode b is then coupled by a single-mode fiber and directed to the environment part. The Soleil-Babinet compensator (SBC) and quartz plates (QPs) add a relative phase , proportional to their length, between |H〉 and |V〉. HWP4 and HWP7, with angles set to 0° introduce a relative π phase between horizontal and vertical polarizations. The angles of HWP5 and HWP6 are set to be 22.5°, which rotates |H〉 and |V〉 to and , respectively. Quantum state tomography is implemented with the polarizations of the final states analysed by a quarter-wave plate (QWP), a HWP and a polarization beam splitter (PBS) in each arm. The photons are detected by single photon avalanche detectors (D1 and D2) with 3 nm interference filters (IFs) in front of them.