Surpassing the currently achievable distance of quantum key distribution based on sending-or-not-sending approach

TL;DR

SNS-PM-QKD introduces a long-distance QKD protocol that fuses the sending-or-not-sending idea with phase-matching QKD to improve phase-mismatch tolerance and extend transmission distance. The authors provide an asymptotic security proof under collective attacks using a Lin-Lutkenhaus-based framework and derive key-rate expressions that incorporate phase-randomization and postselection. Numerical simulations show that SNS-PM-QKD, especially with phase randomization and AOPP, surpasses PM-QKD and SNS-TF-QKD benchmarks, achieving multi-hundred-kilometer gains and even exceeding reported experimental distances for SNS-TF-QKD. The work establishes a path toward ultra-long-distance QKD with robust phase handling and outlines directions for finite-key analysis and experimental realization. The results suggest practical impact for wide-area quantum networks by leveraging the square-root scaling characteristics of TF-QKD-family protocols while enhancing error tolerance through postselection and phase-randomized strategies.

Abstract

Protocols based on the sending-or-not-sending (SNS) principle have been intensively studied in recent years and have been shown to enable the longest transmission distances in quantum key distribution (QKD). In this work, we propose a sending-or-not-sending phase-matching QKD protocol (SNS-PM-QKD) that improves tolerance to phase mismatch, thereby extending the achievable transmission distance. We present a security analysis of SNS-PM-QKD in the asymptotic (infinite-key) regime under collective attacks. The performance of the proposed protocol is compared with that of standard phase-matching QKD, theoretical SNS-type twin-field QKD protocols (SNS-TF-QKD), and an experimental SNS-TF-QKD operated over transmission distances of up to 1002km. Our results show that SNS-PM-QKD achieves greater transmission distances than these existing protocols, highlighting its potential for long-distance quantum communication.

Figures (11)

• Figure 1: Schematic setup of SNS-PM-QKD. Alice (Bob) independently chooses to send or not to send her (his) signal and secondary coherent states to a third party Charlie. He performs a joint (beam splitter) measurement on the coherent states coming from the couplers, C$_{\text{s}}$ and C$_{\text{r}}$. The couplers output: either 1) the interference of Alice's and Bob's signal states (output of C$_{\text{s}}$) and interference of Alice's and Bob's secondary (reference) states (output of C$_{\text{r}}$)— sending-sendingss scenario; or 2)Alice's signal state (output of C$_{\text{s}}$) and Alice's secondary state (output of C$_{\text{r}}$)— sending-not sending (sns) scenario; or 3)Bob's signal state (output of C$_{\text{s}}$) and Bob's secondary state (output of C$_{\text{r}}$)— not sending-sending (sns) scenario; or 4) nothing (neither C$_{\text{s}}$ nor C$_{\text{r}}$ outputs a coherent state)— not sending-not sending (nn) scenario. Next Charlie publicly announces the measurement outcome. BS— balanced beam splitter; D— single-photon detector; C— coupler.
• Figure 2: Schematic setup of SNS-PM-QKD with phase randomization. Alice (Bob) independently chooses to send or not to send her (his) signal and secondary coherent states to a third party, Charlie. He performs a joint (beam splitter) measurement on the coherent states coming from the couplers, C$_{\text{s}}$ and C$_{\text{r}}$. The couplers output: either 1) the interference of Alice's and Bob's signal states (output of C$_{\text{s}}$) and interference of Alice's and Bob's secondary (reference) states (output of C$_{\text{r}}$)— sending-sendingss scenario; or 2)Alice's signal state (output of C$_{\text{s}}$) and Alice's secondary state (output of C$_{\text{r}}$)— sending-not sending (sns) scenario; or 3)Bob's signal state (output of C$_{\text{s}}$) and Bob's secondary state (output of C$_{\text{r}}$)— not sending-sending (sns) scenario; or 4) nothing (neither C$_{\text{s}}$ nor C$_{\text{r}}$ outputs a coherent state)— not sending-not sending (nn) scenario. In the above schematic, the secondary states of Alice and Bob swap positions by means of mirrors M and phase shifts PH of $\pi$ [rad]. Note that the role of a PH is to cancel the phase shift induced into a coherent state upon its reflection off a mirror. After a conclusive measurement, Charlie publicly announces the corresponding outcome. BS— balanced beam splitter; D— single-photon detector; C$_{\text{x}}$— coupler (x$=$ r— coupler for secondary (reference) states; x$=$ s— coupler for signal states); M— mirror; PH— phase shift of $\pi$ [rad].
• Figure 3: Model of a POVM attack on SNS-PM-QKD schematic setup. The attack is in accordance with Ref. lin-lutkenhaus. Eve performs a POVM $F^{\varkappa}$ on the coherent states coming from the couplers' outputs. Eve publicly announces the measurement (POVM) outcome $\varkappa\in\{+,-,?,d\}$ ($+$ indicates a click at detector $D_+$ (see Fig. \ref{['setup']}); $-$ indicates a click at detector $D_-$ (see Fig. \ref{['setup']}); $?$ indicates the absence of a click at any detector; $d$ indicates the presence of clicks at both detectors). $\delta$---phase mismatch; $\eta_t$---quantum channel transmittance; C$_{\text{x}}$---coupler (x $=$ r---coupler of reference states; x $=$ s---coupler of signal states).
• Figure 4: Model of a double-POVM attack on SNS-PM-QKD schematic setup. Eve performs a POVM $F^{\varkappa'}$ on Alice's coherent states and a POVM $F^{\varkappa"}$ on Bob's coherent states. Based on outcomes $\varkappa'$ and $\varkappa"$ of these POVMs, Eve decides on the value of $\tilde{\varkappa}$, the actual outcome announcement. Eve aims to replicate the outcome announcement $\varkappa$ of the POVM attack (Fig. \ref{['model1']}).
• Figure 5: Error rate $e^{\text{loss-only}}_{\text{distinguish}}$ in a double-POVM attack given a loss-only operation of SNS-PM-QKD. The above curve is evaluated for protocol parameters: channel attenuation $\alpha=0.2$dB, detection efficiency $\eta_{\text{det}}=1$, dark count rate $p_{\text{dark}}=10^{-11}$, mode (intensity) mismatch $V=0.95$, phase mismatch $\delta=\frac{\pi}{60}$, coherent-state intensity $\mu=0.1$, and sending probability function $\epsilon(L)$ of the form shown in Fig. \ref{['epsilon-profile']} with $\epsilon_0=0.05$, $\epsilon_{\text{max}}=0.45$, $L_{\text{max}}=950$km.