Entanglement-based continuous-variable quantum key distribution with multimode states and detectors

Secure quantum key distribution with multimode Gaussian entangled states and multimode homodyne detectors is proposed. In general the multimode character of both the sources of entanglement and the homodyne detectors can cause a security break even for a perfect channel when trusted parties are unaware of the detection structure. Taking into account the multimode structure and potential leakage of information from a homodyne detector reduces the loss of security to some extent. We suggest the symmetrization of the multimode sources of entanglement as an efficient method allowing us to fully recover the security irrespectively to multimode structure of the homodyne detectors. Further, we demonstrate that by increasing the number of the fluctuating but similar source modes the multimode protocol stabilizes the security of the quantum key distribution. The result opens the pathway towards quantum key distribution with multimode sources and detectors.

Figure 1

Scheme of the multimode entanglement-based CV QKD with Gaussian entangled multimode source. The multimode beams emitted by the source are measured by two trusted parties Alice and Bob after the untrusted lossy and noisy channel. The channel is under control of an eavesdropper and is parametrized by transmittance $T$ and excess noise $\epsilon$. Measurement is modeled as performed by the single-mode homodyne detectors (H) precessed by the linear-optical modes coupling (LOC) and resulting in outcomes of the form (1). (a) Untrusted homodyne scenario, corresponding to the leakage of the auxiliary modes to the environment, where they can be potentially measured by an eavesdropper. (b) Trusted homodyne scenario, where auxiliary mode outputs of the LOC are discarded within the trusted stations.

Figure 2

Multimode CV QKD scheme with a single-mode source $V$ and $N$-mode detection in the general case (a) and the equivalent scheme in the case of balanced multimode detection (b). The particular case of the purely lossy channel (i.e., the excess noise $\epsilon =0$) is considered.

Figure 3

Multimode CV QKD scheme with two-mode source and detection in the purification scheme, where modes are coupled and then measured by a single-mode homodyne detector.

Figure 4

Lower bound on secure key rate in the case of collective attacks vs distance in fiber with standard attenuation $-0.2$ dB/km for multimode entanglement-based CV QKD with two-mode states upon trusted, given in green (dark gray), and untrusted, given in orange (light gray), homodyne detection at $V_1=3$. ${\lambda }_1^2=0.5$ and $V_2=1$ (dotted lines), ${\lambda }_1^2=0.5$ and $V_2=1.1$ (dashed lines), ${\lambda }_1^2=0.95$ and $V_2=1$ (solid lines). Black solid thick line represents the benchmark set by single-mode CV QKD protocol with source variance $V=3$. In all the cases channel noise is $\epsilon =5\%$ SNU and postprocessing efficiency $\beta =95\%$.

Figure 5

Security region against collective attacks for the two-mode CV QKD in terms of mode variances $V_1$ and $V_2$ in case of untrusted (left) and trusted (right) multimode homodyne detection. In both the cases channel noise $\epsilon =0,1,2,3,4,5\%$ SNU (lines from left to right) is present, mode coupling in detection is balanced. Channel transmittance $T=3\times {10}^{-2}$, postprocessing is perfect ($\beta =1$). The area of parameters above each of the lines is secure upon given conditions.

Figure 6

Lower bound on secure key rate in the case of collective attacks vs distance in fiber with standard attenuation $-0.2$ dB/km for multimode entanglement-based CV QKD upon three different situations given in Table 1: 3 mode (solid line), 2 mode (dashed line), and 1 mode (dotted line); the postprocessing efficiency is $\beta =95\%$.

Figure 7

Numerical simulation of lower bound on the secure key rate in case of collective attacks for multimode CV QKD with varying mode variance $V\sim \mathcal{N}(3,0.75)$ for number of modes $N=5$ (light gray) and $N=100$ (black) vs the time flow of the protocol runs. The single-mode CV QKD key rate for constant source variance $V=3$ is given in blue (dark gray) dashed line for the reference. In all the cases channel transmittance $T=3\times {10}^{-2}$ (corresponding to approximately 76 km of telecom fiber), channel noise $\epsilon =5\%$ SNU, and postprocessing efficiency $\beta =95\%$.