Security proof of continuous-variable quantum key distribution using three coherent states

We introduce a ternary quantum key distribution (QKD) protocol and asymptotic security proof based on three coherent states and homodyne detection. Previous work had considered the binary case of two coherent states and here we nontrivially extend this to three. Our motivation is to leverage the practical benefits of both discrete and continuous (Gaussian) encoding schemes creating a best-of-both-worlds approach; namely, the postprocessing of discrete encodings and the hardware benefits of continuous ones. We present a thorough and detailed security proof in the limit of infinite signal states which allows us to lower bound the secret key rate. We calculate this is in the context of collective eavesdropping attacks and reverse reconciliation postprocessing. Finally, we compare the ternary coherent state protocol to other well-known QKD schemes (and fundamental repeaterless limits) in terms of secret key rates and loss.

Figure 1

Phase-space configurations of the ternary coherent state QKD protocol. Note that each subsequent coherent state is ${120}^{\circ }$ from the other one. Alice's role is to continually and randomly choose from these three options and then send them to Bob who performs homodyne detection on the incoming states by randomly alternating between the $Q$ and $P$ quadratures. As is standard, the quantum channel is assumed to be monitored by the eavesdropper, Eve.

Figure 2

Secret key rates as functions of loss $1-\eta$ for several values of the channel excess noise $\delta =(0,0.0004,0.001,0.005,0.01)$ (the pink dots). The black curve is the ultimate achievable bound without an energy constraint for $\delta =0$. The orange curve is an achievable bound for $\delta =0$ taking into account the input energy constraint [6]. All curves are functions of the channel loss.

Figure 3

The red dots depict the optimizing overlaps $r$ for $\delta =0$. The black curve is a boundary $\varepsilon =0$ of (A8e) ($\vartheta =\pi /2$) given by $r^2\frac{3\sqrt{3}}{2}\eta =\pi /2$ [see below Eq. (6)] below which the proofs of monotonicity and concavity exist ($\varepsilon \le 0$).

Figure 4

Properties of some trigonometric functions.

Figure 5

The ternary Shannon entropy Eq. (1) is shown. The blue line depicts $u_1=(1-u_2)/2$ while the green one is the plot of $u_1=-2u_2+1$.