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The impact of constellation cardinality on discrete unidimensional CVQKD protocols

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Abstract

We introduce a new unidimensional CV-QKD protocol based on classical capacity achieving constellations and inspect how its cardinality translates into the secret key rate. The protocol requires simple modulation and detection procedures with intensity modulation on a single quadrature and homodyne detection. The secret key rate is computed for several constellation shapes, sizes and detection quantum efficiency for a pure loss quantum channel. We show that the capacity achieving constellations with increasing cardinality in a classical context preserve the same pattern in a QKD context by approximating the performance of Gaussian modulation. We also verify that the non-unit quantum efficiency detection effect is more severe on discrete modulation, diminishing the protocol performance and limiting its link range to no more than 100 km.

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Notes

  1. We call attention the difference between \({\tilde{V}}_m\) and \(V_m\): the first refers to the modulation variance (or average energy), while the second corresponds to the quadrature operator \(\hat{q}\) variance.

  2. Here, we assume the channel to be symmetric on its symplectic invariants, that is, the transmittance and excess noise are equal for both quadratures. The original paper presented a general case where the channel symmetry is initially discarded, forcing sporadic measurements on the non-modulated quadrature.

  3. Here, we consider the operators relations in shot noise units, that is, \(\hat{q}= \hat{a}^{\dagger }+\hat{a}\) and \(\hat{p}= i(\hat{a}^{\dagger }- \hat{a})\), which is slightly different from the development found in [7] but causes no harm as its it will mainly differ on the normalization factor for the output current expected value. Check [12] for a good comparison between the canonical operators relations in different unit systems.

  4. Note that it is a valid bosonic annihilation operator as it preserves the canonical commutation relation \([\hat{\varLambda },\hat{\varLambda }^\dagger ] = 1.\)

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Acknowledgements

This work was performed in the Institute for Studies in Quantum Information and Computation (iQuanta) at the Federal University of Campina Grande (UFCG), Paraíba, Brazil, and supported in part by the National Council for Scientific and Technological Development (CNPq) under research Grant No. 305918/2019-2 and the Coordination of Superior Level Staff Improvement (CAPES/PROEX).

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  1. Department of Electrical Engineering, Federal University of Campina Grande, Campina Grande, Brazil

    Micael A. Dias & Francisco M. de Assis

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  1. Micael A. Dias
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Dias, M.A., de Assis, F.M. The impact of constellation cardinality on discrete unidimensional CVQKD protocols. Quantum Inf Process 20, 284 (2021). https://doi.org/10.1007/s11128-021-03222-w

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