Asymptotic Security of Continuous-Variable Quantum Key Distribution with a Discrete Modulation

We establish a lower bound on the asymptotic secret key rate of continuous-variable quantum key distribution with a discrete modulation of coherent states. The bound is valid against collective attacks and is obtained by formulating the problem as a semidefinite program. We illustrate our general approach with the quadrature-phase-shift-keying modulation scheme and show that distances over 100 km are achievable for realistic values of noise. We also discuss the application to more complex quadrature-amplitude-modulation schemes. This result opens the way to establishing the full security of continuous-variable protocols with a discrete modulation, and thereby to the large-scale deployment of these protocols for quantum key distribution.

Popular Summary

Quantum key distribution allows two parties to establish a secret key for encrypting information under the sole assumption that any attack must respect the laws of quantum mechanics. For deploying quantum key distribution at large scale, it must conform as much as possible to telecom standards, which currently involve discrete constellations of coherent states and coherent detection. This is exactly the point of “continuous-variable” (CV) protocols. However, establishing their security is significantly harder than for qubit-based protocols that rely on single-photon detectors. In this paper, we take a major step towards a full security proof for CV protocols.

Up until now, security proofs for CV protocols were restricted to idealized protocols that rely on the exchange of quantum states chosen from a continuous distribution. In practice, such protocols are impossible to implement. By contrast, we consider protocols that rely on discrete modulation, meaning that the states are chosen from a finite set.

In quantum key distribution, the secret key rate (a measure of the number of secret bits) depends only on the amount of information leaked to the adversary. Using techniques from convex optimization, we prove that this secret key rate can be inferred by solving a semidefinite program, which can be done efficiently. This, in turn, provides the minimum number of bits guaranteed to remain secure.

For realistic estimates of noise, we show that it is possible to distribute secret keys along fiber-optic links more than 100 km long, well on par with the performance of qubit-based protocols.

Figure 1

Description of the QPSK protocol with a constellation of four coherent states and the partition of phase space in four quadrants.

Figure 2

Secret key rate versus distance, for a Gaussian channel with transmittance $T={10}^{-0.02d}$ and excess noise $\xi =0.002$. Here, $d$ is the distance between Alice and Bob in km. The value of $\alpha$ is 0.35. The reconciliation efficiency $\beta$ is set to 0.95. The top curve corresponds to the performance of the protocol [39] with a Gaussian modulation, the lower curve to the performance of the four-state protocol while assuming a linear channel (as in Ref. [26]), and the crosses correspond to the lower bound given by our SDP.

Figure 3

Secret key rate versus distance, for a Gaussian channel with transmittance $T={10}^{-0.02d}$ and excess noise $\xi =0.005$. Other parameters are the same as in Fig. 2.

Figure 4

Secret key rate versus distance, for a Gaussian channel with transmittance $T={10}^{-0.02d}$ and excess noise $\xi =0.01$. Other parameters are the same as in Fig. 2.

Figure 5

Secret key rate versus $\alpha$, for a distance of 50 km and excess noise of $\xi =0.002$. Other parameters are the same as in Fig. 2.