The impact of constellation cardinality on discrete unidimensional CVQKD protocols

Abstract

Consider the problem of using a finite set of coherent states to distribute secret keys over a quantum channel. It is known that computing the exact secret key rate in this scenario is intractable due to the infinite dimensionality of the Hilbert spaces and usually one computes a lower bound using a Gaussian equivalent bipartite state in the entangled based version of the protocol, which leads to underestimating the actual protocol capability of generating secret keys for the sake of security. Here, we define the QKD protocol's non-Gaussianity, a function quantifying the amount of secret key rate lost due to assuming a Gaussian model when a non-Gaussian modulation was used, and develop relevant properties for it. We show that if the set of coherent states is induced by a random variable approaching the AWGN channel capacity, then the protocol's non-Gaussianity vanishes, meaning that there is no loss of secret key rate due to the use of a Gaussian model for computing bound on the secret key rate. The numerical results show that by using a 256-QAM with Gauss-Hermite shaping, the loss of secret key rate quickly falls below 10 −5 as the distance increases.

Amplitude-Phase Modulated CVQKD Protocol

“…Several constellation shapes of unidimensional (UD) constellations were analyzed in [18] and it was shown that the ones applying some kind of shaping have a better convergence towards Gaussian modulation performance as its cardinalities increase than the ones with no geometric and/or probabilistic shaping. This capability of approximating Gaussian modulation by shaping the constellation is also translated to QKD protocols, as it was shown in [19] where the authors used the UD constellations in [18] and compared with an UD Gaussian modulation [20], and they concluded that capacity archiving UD constellations also reaches the performance of Gaussian modulated UD protocol. The authors also presented optimal four state constellations obtained by exhaustive search on its geometry and probability to each transmittance in the interval 0 < τ < 1.…”

“…Four types of unidimensional constellations where examined in [18] for different signal-to-noise values and increasing cardinalities. The performance of such constellations in a QKD context where investigated in [19] and they also presented optimal four-state unidimensional constellation. The search for these optimal four-state constellations was made exhaustively in the following way.…”

“…With that in mind, here we propose new eight and sixteen-states amplitude-phase shifted constellations based CVQKD protocol. Our constellations are based on the optimal UD four-state constellations presented in [19], where we used the four-state constellation amplitudes and probabilities while applied several rotations to obtain a proper eight-state constellation. We analyze the protocol performance in reverse reconciliation and assume a noiseless quantum channel.…”

Theoretical development of discrete-modulated continuous-variable quantum key distribution

“…The first binarymodulated scheme (Silberhorn et al, 2002), which sends states with a Gaussian distribution but interprets a state as logical 0 or one according to the displacement direction (Heid and Lütkenhaus, 2007), is the early form of discrete modulation. Later, schemes with different signal states or different probability distributions evolved (Zhao et al, 2020a;Dias and de Assis, 2021;Kaur et al, 2021).…”