Re-examination of the role of displacement and photon catalysis operation in continuous variable measurement device-independent quantum key distribution

We investigate the benefits of using m-photon catalysed two-mode squeezed coherent (m-PCTMSC) state in continuous variable measurement device-independent quantum key distribution (CV-MDI-QKD). To that end, we derive the Wigner characteristic function of the m-PCTMSC state and show that the 0-PCTMSC state is a Gaussian state and is an inferior choice as compared to the zero photon catalyzed two-mode squeezed vacuum state for CV-MDI-QKD. We carry out the optimization of the secret key rate with respect to all state parameters, namely variance, transmissivity, and displacement. Contrary to many recent proposals, the results show that zero- and single-photon catalysis operation provides only a marginal benefit in improving the maximum transmission distance. Secondly, we find that displacement offers no benefit in improving CV-MDI-QKD.