Phase-Matching Quantum Cryptographic Conferencing

Quantum cryptographic conferencing (QCC) holds promise for distributing information-theoretic secure keys among multiple users over a long distance. Limited by the fragility of Greenberger-Horne-Zeilinger (GHZ) states, QCC networks based on directly distributing GHZ states over a long distance still face a big challenge. Another two potential approaches are measurement device-independent QCC and conference-key agreement with single-photon interference, which were proposed on the basis of the postselection of GHZ states and the postselection of the W state, respectively. However, implementations of the former protocol are still heavily constrained by the transmission rate $\eta$ of optical channels and the complexity of the setups for postselecting GHZ states. Meanwhile, the latter protocol cannot be cast as a measurement device-independent prepare-and-measure scheme. Combining the idea of postselecting GHZ states and recently proposed twin-field quantum-key-distribution protocols, we report a QCC protocol based on weak coherent-state interferences named “phase-matching quantum cryptographic conferencing,” which is immune to all detector side-channel attacks. The proposed protocol can improve the key-generation rate from $O({\eta }^N)$ to $O({\eta }^{N-1})$ compared with the measurement device-independent QCC protocols. Meanwhile, it can be easily scaled up to multiple parties due to its simple setup.

Figure 1

Setup for a $N$-party PM-QCC network. ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_N\in [0,2\pi )$ label the random phases for parties $P_1$, $P_2$, and $P_N$, respectively. $k_1$, $k_2$, and $k_N\in \{0,1\}$ label the random bits for parties $P_1$, $P_2$, and $P_N$, respectively. $D_{L_1}(D_{R_1})$ is the left (right) detector of the first measurement branch. $D_{L_2}(D_{R_2})$ is left (right) detector of the second measurement branch. $D_{L_{N-1}}(D_{R_{N-1}})$ is the left (right) detector of the $(N-1)\mathrm{th}$ measurement branch. BS, beam splitter; FC, fiber channel.

Figure 2

(a) The PM-QCC protocol with phase-matching condition ${\varphi }_1={\varphi }_2=\cdots ={\varphi }_N=\varphi$ satisfied. (b) Equivalent PM-QCC protocol after Eve’s splitting with phase-matching condition ${\varphi }_1={\varphi }_2=\cdots {\varphi }_N=\varphi$ satisfied. Eve splits each pulse of parties $P_2,\dots ,P_{N-1}$ into two separated coherent pulses using a beam splitter (BS1) to perform interference measurements. Once a successful detection event is achieved, the encoded phases of $N$ parties are correlated with each other. $p_1,p_{2L},\dots ,p_N$ are the path modes after Eve’s splitting. $D_{L_1}(D_{R_1})$ is the left (right) detector of the first measurement branch. $D_{L_2}(D_{R_2})$ is the left (right) detector of the second measurement branch. $D_{L_{N-1}}(D_{R_{N-1}})$ is the left (right) detector of the $(N-1)\mathrm{th}$ measurement branch.

Figure 3

Key-generation rate $R$ of three-party PM-QCC, three-party PM-QCC* (without phase postselection in signal pulses), and three party MDI-QCC networks versus transmission distance $L$. The simulation results are obtained with parameters from Ref. [17]: dark-count rate $p_d=1\times {10}^{-7}$, loss rate of the channel $\alpha =0.2\mathrm{dB}/\mathrm{km}$, detection efficiency ${\eta }_d=93\mathrm{\%}$, error-correction efficiency $f=1.16$, misalignment error for MDI QCC $e_d^{\mathrm{MDI}}=1.5\mathrm{\%}$, and phase misalignment error for PM QCC* $e_{\delta }=1.5\mathrm{\%}$. The number of phase slices $M$ for PM QCC is optimized at different transmission distances.

Figure 4

Key-generation rate $R$ for three-party PM QCC, three-party PM QCC with four decoy states ($\nu$, $\omega$, $\tau$, and 0), and four-party PM QCC versus transmission distance $L$. Parameters used in the simulation are obtained from Ref. [56]: dark-count rate $p_d=7.2\times {10}^{-8}$, loss rate of the channel $\alpha =0.2\mathrm{dB}/\mathrm{km}$, detection efficiency ${\eta }_d=65\mathrm{\%}$, and error-correction efficiency $f=1.16$.

Figure 5

(a) The entanglement-based version of the $N$-party PM-QCC network. (b) The equivalent entanglement-based version of the $N$-party PM-QCC network with virtual sources after Eve’s splitting. In 5,5, random phases are supposed to be ${\varphi }_1={\varphi }_2=\cdots =\varphi$. Here we omit the virtual qubits for simplicity in (b). $p_1,p_{2L},\dots ,p_N$ are path modes after Eve’s splitting. $D_{L_1}(D_{R_1})$ is the left (right) detector of the first measurement branch. $D_{L_2}(D_{R_2})$ is the left (right) detector of the second measurement branch. $D_{L_{N-1}}(D_{R_{N-1}})$ is the left (right) detector of the $(N-1)\mathrm{th}$ measurement branch. BS, beam splitter. ${\mathrm{C}}_{\pi }$, controlled phase gate.

Figure 6

(a) The three-party PM-QCC network and the two-party PM-QKD protocol reduced from the five-party PM-QCC network. In some rounds of the five-party PM-QCC network only sets $\{P_1,P_2,P_5\}$ and $\{P_4,P_5\}$ have successfully interference, while the interference of $\{P_3,P_4\}$ failed. (b) The entanglement-based PM-QKD protocol for parties $P_4$ and $P_5$. (c) The equivalent entanglement-based PM-QKD protocol for $P_4$ and $P_5$ with a virtual source. BS, beam splitter; FC, fiber channel; ${\mathrm{C}}_{\pi }$, controlled phase gate.