Figure 5: Reconstructed density matrices without Bob’s corrections.

Each of the four panels shown above corresponds to the state that Bob receives for each of Alice’s four measurement outcomes. For each state that Charles prepared, Alice measured in a basis that is mutually unbiased in the basis in which Charles applied phases. Each of Alice’s measurement outcomes projected Bob’s photon into one of four orthogonal states. By performing a two-qubit polarization and orbital angular momentum tomography in coincidence with Alice’s measurement, Bob reconstructed the density matrix of each of these states. (a–d) Each of these four density matrices (for clarity we display only the real parts of these matrices) are in the horizontal and vertical polarization basis (H and V) and the right and left orbital angular momentum basis (r and l). The major limiting factor of fidelity in these reconstructions is the crosstalk between orbital angular momentum modes indicated by the differing values of the diagonal elements. After these matrices were reconstructed, Bob numerically performed the corrective transformations to obtain the final reconstructed states shown in Fig. 4.