Fig. 5: The quantum operations and usage of Alice’s encoding states on two paired locations (1, 2).
There are 8 systems based on Alice's two paired locations. Among them, the two qudits \({\tilde{A}}_{1}\) and \({\tilde{A}}_{2}\) are measured to obtain the overall photon number ka and the relative phase θa of two optical modes A1 and A2. The two qubits \({A}_{1}^{^{\prime\prime} }\) and \({A}_{2}^{^{\prime\prime} }\) are measured to obtain the relative phase, which is the raw key bit in the X-basis. Another two qubits \({A}_{1}^{\prime}\) and \({A}_{2}^{\prime}\) are measured to obtain the encoded intensity in A1 and the relative encoded intensity, which are used for the key mapping on the Z-basis and the basis assignment, respectively.
