Multimode states in decoy-based quantum-key-distribution protocols

Every security analysis of quantum-key distribution (QKD) relies on a faithful modeling of the employed quantum states. Many photon sources, such as for instance a parametric down-conversion (PDC) source, require a multimode description but are usually only considered in a single-mode representation. In general, the important claim in decoy-based QKD protocols for indistinguishability between signal and decoy states does not hold for all sources. We derive bounds on the single-photon transmission probability and error rate for multimode states and apply these bounds to the output state of a PDC source. We observe two opposing effects on the secure key rate. First, the multimode structure of the state gives rise to a new attack that decreases the key rate. Second, more contributing modes change the photon number distribution from a thermal toward a Poissonian distribution, which increases the key rate.

Figure 1

(Color online) Eve blocks all photons in the second mode. This results in different single-photon yields for signal and decoy state.

Figure 2

(Color online) The photon number distributions for different pump widths $\sigma$.

Figure 3

(Color online) The mode occupation probabilities $m_1$ and $m_2$ for a single-photon event in dependence of the pump intensity.

Figure 4

(Color online) Solid (red) line: lower bound on the secure key rate for the KTP crystal with a pump width of 4 nm. Dashed (green) line: lower bound on secure key rate for the same photon number distribution, but with all photons in the same spectral mode. The inset shows a zoom of the 10–11 dB region. The weak decoy mean photon number is 0.1 in both cases, and the signal mean photon number is optimized to result in the largest possible rate.

Figure 5

(Color online) Lower bound on the secure key rate for different pump widths and therefore different mode contributions. The asymptotic cases are the rates for thermal and Poissonian distribution. The inset shows the optimized mean photon number of the signal state. The weak decoy mean photon number is fixed to 0.1 for all scenarios.