Experimental setups and the stages of the experimental implementation. (a) The photon pairs with wavelength 810 nm were prepared by spontaneous parametric down-conversion of the 1.5 cm-long type-II PPKTP nonlinear crystal. The dichroic mirror is to reflect photons of 405 nm and transmit the photons of 810 nm. (b) Experiment setups for preparing the state with tripartite quantum correlation. One photon is sent to the upper experiment installation to act as the auxiliary system A. The other one goes into the lower experimental setting where its polarization modes interact with the spatial modes. The partial CN gate and conditional BF quantum channels are experimentally realized. Different angles of the half-wave plates (HWP) H1 are adjusted to simulate the superposition coefficient p in the tripartite state in equations (15) and (16). The angle of H0 is set to zero. The angle of H2 is set to zero for simulating BF channel, while set to π/4 for CN channel. When the angle is set to π/4, the HWP can perform the inversions between the polarization modes H→V and V→H (in the text the horizontal ( H ) and vertical ( V ) modes are denoted as 0 and 1 for simplicity). (c) The angles of H3 and H4 are set to π/4, together with the BD between them, to realize an anti-BD, which has opposite effects of the ordinary BD, i.e., it transmits horizontally polarized photons and reflects the vertical ones. (d1) Setup for the projective measurements performed on the subsystem A. The quarter-wave plates, HWP, and polarizing beam splitters are employed to realize the measurement basis. (d2) Tomography measurements on the polarization modes of the second photon and the coupled spatial modes. The residual densities can be constructed based on the measurement probability of subsystem A. The other devices are interference filters.