Symmetry-Protected Privacy: Beating the Rate-Distance Linear Bound Over a Noisy Channel

There are two main factors limiting the performance of quantum key distribution—channel transmission loss and noise. Previously, a linear bound was believed to put an upper limit on the rate-transmittance performance. Remarkably, the recently proposed twin-field and phase-matching quantum key distribution schemes have been proven to overcome the linear bound. In practice, due to the intractable phase fluctuation of optical signals in transmission, these schemes suffer from large error rates, which renders the experimental realization extremely challenging. Here, we close this gap by proving the security based on a different principle—encoding symmetry. With the symmetry-based security proof technique, we can decouple the privacy from the channel disturbance, and eventually remove the limitation of secure key distribution on bit error rates. As a direct application, we show that the phase-matching scheme can yield positive key rates even with high bit error rates up to 50%. In the simulation, with typical experimental parameters, the key rate is able to break the linear bound with an error rate of 13%. Meanwhile, we provide a simple finite-data size analysis for the phase-matching scheme under this symmetry-based analysis, which can break the bound with a reasonable data size of ${10}^{12}$. Encouraged by high loss and error tolerance, we expect the approach based on symmetry-protected privacy will provide a different insight into the security of quantum key distribution.

Figure 1

Schematic diagram of a generalized discrete-variable MDI QKD framework. Here we make no assumption on Alice’s and Bob’s sources or their encoding unitary operations. The only assumption is the symmetric property of the encoding operation, i.e., $U^2=I$.

Figure 2

Rate-distance performance of PM QKD under different system misalignment error rates $e_d$. For BB84, $e_d=1.5\mathrm{\%}$. The nonsmooth point indicates the places where the key contribution from unaligned groups with $j_s\ne 0$ turns to 0.

Figure 3

Rate-distance performance of PM QKD under the data size $N=1\times {10}^{12},1\times {10}^{13}$ or infinitely large, and misaligned error $e_d=3$ or $6\mathrm{\%}$.

Figure 4

Schematic diagram for a general entanglement-based version of symmetric encoding protocol. When ${\sigma }_{AB}$ is a parity state, then the protocol is perfectly private without any information leakage.

Figure 5

Schematic diagram for the whole protocol I/Ia when the input state is a general mixed parity state ${\sigma }_{AB}$. Here we postselect the rounds when Eve announces effective clicks. Alice and Bob prepare $n$ rounds of state ${|\mathrm{\Psi }⟩}_{PAB}$ in Eq. (A24). They perform the PM QKD encoding by applying $C(U)$ gates on system $A^{\prime },A$ and $B^{\prime },B$, as illustrated in Fig. 4. After that, they send system $A,B$ to Eve, and perform $I_{A^{\prime }{B}^{\prime }}$ or $I_{A^{\prime }}\otimes X_{B^{\prime }}$ operation on each round of system $A^{\prime },B^{\prime }$. They measure them by $Z_{A^{\prime }}\otimes X_{B^{\prime }}$ to get strings ${\kappa }_{\mathrm{rec}},\gamma$. In protocol Ia, after Eve’s announcement, they send system $P$ to Eve. In phase-error-based proof, Eve’s knowledge of ${\kappa }_{\mathrm{rec}}$ can be characterized by Alice’s knowledge of her $X$-basis measurement result $T_{\gamma }$ on $A^{\prime }$.

Figure 6

Realistic PM QKD protocol with extra $0/\pi$-phase randomization. ${\sigma }_{AB}$ is a generic state on optical modes $A$ and $B$. ${\varphi }_a,{\varphi }_b$ are two random phases, which are either $0$ or $\pi$. In practice, the phase can be absorbed into the controlled-$U(\pi )$ operations afterwards.

Figure 7

Comparison of PM QKD performance with the previous analysis [20] and our results. We set the typical intrinsic misalignment error rates to be $5$ and $9\mathrm{\%}$.