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arXiv:2606.12284

Multipartite reference-frame-independent quantum cryptographic communication

Donghwa Lee, Kyujin Shin, Hyang-Tag Lim, Yosep Kim, Yong-Su Kim

quant-phSubmitted Jun 10, 2026• Announced Jun 11, 2026• 0 citations
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Abstract

Reference frame mismatch among communication parties introduces errors in quantum cryptographic protocols. As the number of participants increases, aligning reference frames becomes increasingly difficult, complicating multipartite quantum cryptographic implementations. Here, we theoretically and experimentally investigate multipartite reference-frame-independent (RFI) quantum cryptographic communication using Greenberger-Horne-Zeilinger (GHZ) states. We generalize the bipartite RFI security parameter $C$ to an $N$-party parameter $C_N$ and derive the asymptotic secret key rate expressed solely in terms of experimentally accessible quantities. We analyze the key rate under global and local depolarizing noise models and find that increasing the number of parties $N$ enhances robustness against global depolarizing noise while increasing vulnerability to local channel noise. We also present a proof-of-principle experimental demonstration of four-party RFI quantum cryptographic communication using four-photon GHZ states, confirming the reference-frame invariance of both the $C_4$ parameter and the secret key rate under various reference frame rotations.

Comments
8 pages, 4 figures

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Reference frame mismatch among communication parties introduces errors in quantum cryptographic protocols. As the number of participants increases, aligning reference frames becomes increasingly difficult, complicating multipartite quantum cryptographic implementations. Here, we theoretically and experimentally investigate multipartite reference-frame-independent (RFI) quantum cryptographic communication using Greenberger-Horne-Zeilinger (GHZ) states. We generalize the bipartite RFI security parameter CCC to an NNN-party parameter CNC_NCN​ and derive the asymptotic secret key rate expressed solely in terms of experimentally accessible quantities. We analyze the key rate under global and local depolarizing noise models and find that increasing the number of parties NNN enhances robustness against global depolarizing noise while increasing vulnerability to local channel noise. We also present a proof-of-principle experimental demonstration of four-party RFI quantum cryptographic communication using four-photon GHZ states, confirming the reference-frame invariance of both the C4C_4C4​ parameter and the secret key rate under various reference frame rotations.
We present a three-party reference frame independent quantum key distribution protocol which can be implemented without any alignment of reference frames between the sender and the receiver. The protocol exploits entangled states to establish a secret key among three communicating parties. We derive the asymptotic key rate for the proposed protocol against collective attacks and perform a finite-size key security analysis against general attacks in the presence of statistical fluctuations. We investigate the impact of reference frame misalignment on the stability of our protocol, and we obtain a transmission distance of 180 km, 200 km, and 230 km for rotation of reference frames β = π/6, β = π/8 and β = 0, respectively. Remarkably, our results demonstrate that our proposed protocol is not heavily affected by an increase in misalignment of reference frames as the achievable transmission distances are still comparable to the case where there is no misalignment in reference frames (when β = 0). We also simulate the performance of our protocol for a fixed number of signals. Our results demonstrate that the protocol can achieve an effective key generation rate over a transmission distance of about 120 km with realistic 107 finite data signals and approximately achieve 195 km with 109 signals. Moreover, our proposed protocol is robust against noise in the quantum channel and achieves a threshold error rate of 22.7%.
Compared with the traditional protocols of quantum key distribution (QKD), the reference-frame-independent (RFI)--QKD protocol has been generally proved to be very useful and practical, since its experimental implementation can be simplified without the alignment of a reference frame. In most RFI-QKD systems, the encoding states are always taken to be perfect, which, however, is not practical in realizations. In this paper, we consider the security of RFI QKD with source flaws based on the loss-tolerant method proposed by Tamaki et al. [Phys. Rev. A 90, 052314 (2014)]. As the six-state protocol can be realized with four states, we show that the RFI-QKD protocol can also be performed with only four encoding states instead of six encoding states in its standard version. Furthermore, the numerical simulation results show that the source flaws in the key-generation basis (Z(Z(Z basis) will reduce the key rate but are loss tolerant, while the ones in XXX and YYY bases almost have no effect and the key rate remains almost the same even when they are very large. Hence, our method and results will have important significance in practical experiments, especially in earth-to-satellite or chip-to-chip quantum communications.
Reference frame independent quantum key distribution (RFI-QKD) has gained widespread attention due to the unique advantage for practical application, as it circumvents the need for active reference frame alignment within the system. However, in comparison to the standard BB84 protocol, the original 6-state RFI protocol requires a greater number of quantum states to be operated by Alice and Bob, which is an aspect that merits optimization. In this work, we propose a 4-state RFI protocol and illustrate that Alice and Bob each require only four quantum states to perform channel estimation that remains independent of reference frame deviation, which can proficiently reduce the system complexity. Furthermore, through numerical simulations taking the finite-size key effect into consideration, we show that 4-state RFI protocol can achieve a secure key rate and transmission distance on par with the original 6-state RFI protocol. Finally, a experiment over 200 km is inplemented to conducted the feasibility of our scheme. We believe that our protocol can streamline the implementation of RFI-QKD and thereby contribute to the practical advancement of RFI-QKD.
Current quantum secure direct communication (QSDC) protocols guarantee communication security by estimating the error rates of photons in the X and Z bases. This take the reference frame calibration between communicating parties as a necessary prerequisite. However, in mobile communications scenarios, achieving continuous and accurate reference frame calibration poses significant challenges. To address this issue, this paper proposes a reference-frame-independent (RFI) QSDC protocol. This protocol only requires ensuring the calibration accuracy of one direction of the reference frame, while allowing a misalignment angle βββ in the other two directions. To improve the protocol's robustness against reference frame fluctuations, we introduce a βββ-independent parameter C into the security analysis framework and rederive the protocol's security bounds. Additionally, we construct a system model and optimize the pulse intensity of the signal states, enabling the protocol to achieve optimal performance under each level of channel attenuation. At an attenuation of 10 dB (corresponding to a communication distance of 25 km), the secrecy message capacities for $β= 0^{ \circ} $ and $45^{ \circ} $ are 8.765×10−68.765 \times10^{-6}8.765×10−6 bit/pulse and 4.150×10−64.150 \times10^{-6}4.150×10−6 bit/pulse, respectively. Compared with the single-photon-based QSDC, the communication distance of the protocol proposed in this paper is significantly extended. When $β= 0^{ \circ} $ and $45^{ \circ} $, the maximum transmission distances of the RFI QSDC protocol are 27.875 km and 26.750 km, which is about 155.9 % and 149.7 % of that of the single-photon-based QSDC protocol.
Reference-frame-independent measurement-device-independent quantum key distribution (RFI-MDI-QKD) is a novel protocol that eliminates all possible attacks on the detector side and the necessity of reference-frame alignment in source sides. However, its performance may degrade notably due to statistical fluctuations since more parameters, e.g., yields and error rates for mismatched-basis events, must be accumulated to monitor the security. In this work, we find that the original decoy states method estimates these yields overpessimistically since it ignores the potential relations between different bases. Through jointly processing parameters of different bases, the performance of RFI-MDI-QKD is greatly improved in terms of the secret key rate and achievable distance when statistical fluctuations are considered (the statistical fluctuation analysis is the first step to the final analysis of the finite key size effect). Our results pave an avenue towards practical RFI-MDI-QKD.