Figure 1

(a) Schematic diagram of DDI-QKD. LON is the linear optical network, and BSM is the Bell state measurement. (b) Schematic diagram of a specified example of DDI-QKD proposed in ref. 33. BS: 50:50 beam splitter, BSM: Bell state measurement, HWP: half-wave plate, PM: phase modulator, PBS: polarizing beam splitter, and D _i , with \(i\in \{1,2,3,4\}\): single-photon detectors in the BSM setup. In Bob’s laboratory, the areas before and after the red dashed line denote the LON and the BSM setup, respectively. For the ideal case, a click on each detector D _i corresponds to the projection on one of the four Bell states. |ψ _A 〉 _p is one of the four BB84 polarization states prepared by Alice, and |ψ _B 〉 _s are Bob’s encoding bit in spatial mode. Symbols u, l and a, b denote two paths of the first and second BS, respectively. In practice, Alice prepares |ψ _A 〉 _p by generating phase-randomized weak coherent pulses which can be modelled as Poissonian (with parameter u) mixture of photon number states in different BB84 polarization states^{1, 2, 29}.