Mismatched-basis statistics enable quantum key distribution with uncharacterized qubit sources

In the postprocessing of quantum key distribution, the raw key bits from the mismatched-basis measurements, where two parties use different bases, are normally discarded. Here, we propose a postprocessing method that exploits measurement statistics from mismatched-basis cases and prove that incorporating these statistics enables uncharacterized qubit sources to be used in the measurement-device-independent quantum key distribution protocol and the Bennett-Brassard 1984 protocol, which is otherwise impossible.

Figure 1

A schematic diagram for the MDIQKD protocol. BSM is the Bell-state measurement, which is an untrusted device and may be controlled by Eve; $|{\phi }_0\rangle ,|{\phi }_1\rangle ,|{\phi }_2\rangle ,|{\phi }_3\rangle$ $\left(|{\phi }_0^{\prime }\rangle ,|{\phi }_1^{\prime }\rangle ,|{\phi }_2^{\prime }\rangle ,|{\phi }_3^{\prime }\rangle \right)$ represent Alice (Bob)'s four encoding states.

Figure 2

The security proof of MDIQKD (left) can be applied to BB84 (right) when the MU is merged into Bob. We impose that the MU honestly announces the $|{\varphi }^{+}\rangle =\left(\right|00\rangle +|11\rangle )/\sqrt{2}$ outcome when it performs the BSM. The two von Neumann (projective) measurements of Bob on the right correspond to the two BB84 bases and each projects onto two orthogonal qubit states. The relationship between the states in MDIQKD and BB84 is that $|{\phi }_y^{\prime }\rangle$ is the complex conjugate of $|{\overline{\phi }}_y^{\prime }\rangle$ for $y=0,1,2,3$.

Figure 3

Secure-key rate $R$ (unit: per pulse of basis 0) vs total transmission loss (dB) of channel from Alice or Bob to MU when single-photon sources are equipped: We set $d={10}^{-5}$ per pulse. The solid line represents MDIQKD with perfectly trustworthy BB84 senders' devices, MU without optics misalignment, which can distinguish one Bell state $|{\varphi }^{+}{\rangle }_{CD}$, and no Eve's attack; the dashed line is for the protocol proposed here, with perfect BB84 senders' devices (but not trusted by Alice and Bob), MU without optics misalignment, which can distinguish one Bell states $|{\varphi }^{+}{\rangle }_{CD}$, and no Eve's attack; the dotted line is for the qubit-MDIQKD protocol proposed in Ref. [42], with perfect BB84 senders' devices (but not trusted by Alice and Bob), MU without optics misalignment, which can distinguish two Bell states $|{\varphi }^{+}{\rangle }_{CD}$ and $|{\psi }^{+}{\rangle }_{CD}$, and no Eve's attack.

Figure 4

Secure-key rate $R$ (unit: per pulse of basis 0) vs total transmission loss (dB) of channel from Alice or Bob to MU when practical coherent sources are equipped: We set $d={10}^{-5}$ per pulse. The solid line represents MDIQKD with trustworthy coherent source with mean photon number $\mu =0.5$, MU without optics misalignment, which can distinguish one Bell state $|{\varphi }^{+}{\rangle }_{CD}$, and no Eve's attack; the dashed line is for the protocol proposed here, with coherent source with mean photon number $\mu =0.5$ (its encoding is not trusted by Alice and Bob but its photon statistics is trustworthy), MU without optics misalignment, which can distinguish one Bell states $|{\varphi }^{+}{\rangle }_{CD}$, and no Eve's attack; the dotted line is for the qubit-MDIQKD protocol proposed in Ref. [42], with coherent source with mean photon number $\mu =0.5$ (its encoding is not trusted by Alice and Bob but its photon statistics is trustworthy), MU without optics misalignment, which can distinguish two Bell states $|{\varphi }^{+}{\rangle }_{CD}$ and $|{\psi }^{+}{\rangle }_{CD}$, and no Eve's attack. Infinite decoy states are employed here.

Figure 5

Secure-key rate $R$ (unit: per pulse of basis 0) vs total transmission loss (dB) of channel from Alice to Bob: We set $d={10}^{-5}$ per pulse. The solid line represents original BB84 protocol with perfect trustworthy BB84 senders' devices and measurement device; the dashed line is for the protocol proposed here, in which perfect BB84 senders' devices (but not trusted by Alice and Bob) are equipped while Bob's measurement devices are also uncharacterized 2-dimensional projections.

Figure 6

Secure-key rate $R$ (unit: per pulse of basis 0) vs total transmission loss (dB) of channel from Alice to Bob: We set $d={10}^{-5}$ per pulse. The solid line represents the proposed BB84 protocol with infinite decoy states; Other lines are all for the three-decoy-state protocol (i.e., vacuum states + weak decoy states +signal states) and the mean photon numbers for decoy states and signal states are 0.1 and 0.5 respectively. The dashed line, dotted line, dashed-dotted line, and dashed-dotted-dotted line are for $N=\infty ,{10}^{10},{10}^8,{10}^6$ respectively.

Figure 7

Secure-key rate $R$ (unit: per pulse of basis 0) vs total transmission loss (dB) of channel from Alice to Bob: We set $d={10}^{-5}$ per pulse. The solid line represents the proposed BB84 protocol without encoding misalignment; the dashed line is for the case in which $a=b=c=3^{\circ }$, the dotted line is for the case in which $a=b=c=6^{\circ }$, and the dashed-dotted line is for the case in which $a=b=c=9^{\circ }$.