Long-range photonic device-independent quantum key distribution using SPDC sources and linear optics

Abstract

We address the question of the implementation of long-distance device-independent quantum key distribution (DI QKD) by proposing two experimentally viable schemes. Those schemes only use spontaneous parametric down-conversion (SPDC) sources and linear optics. They achieve favorable key rate scaling proportional to the square root of channel transmittance $η_t$, matching the twin-field protocol advantage. We demonstrate positive asymptotic key rates at detector efficiencies as low as 80\%, bringing DI QKD within the reach of current superconducting detector technology. Our security analysis employs the Entropy Accumulation Theorem to establish rigorous finite-size bounds, achieving finite-key rates at a detector efficiency of 90\%. This work represents a critical milestone toward device-independent security in quantum communication networks, providing experimentalists with practical implementation pathways while maintaining the strongest possible security guarantees against quantum adversaries.

Figures (8)

• Figure 1: Long-range DI-TF QKD based on heralded entanglement distribution. (a) In the 1-photon protocol, Alice and Bob use spontaneous parametric down-conversion (SPDC) sources pumped by pulsed lasers to generate local bipartite multi-photon entanglement -- two-mode squeezed vacuum states (TMSV). The idler modes $a_2$ and $b_2$ travel through lossy channels with transmittance $\sqrt{\eta_t}$ to reach Charlie's central station, which comprises a symmetric beam splitter and two detectors with efficiency $\eta_c$. The events where exactly one of the two detectors registers a photon herald the state in Eq. \ref{['eq:psiout']}. (b) In the 2-photon protocol, Bob employs a single-photon source obtained by heralding one output from an SPDC crystal at a detector of efficiency $\eta_s$ with the other output routed through a beam splitter with reflectivity-to-transmissivity ratio $r_s:t_s$, effectively generating local single-photon entanglement $\ket{\psi_b}$. (c) The heralded state, Eqs. \ref{['eq:psiout']} or \ref{['eq:psiout2ph']}, enters Alice's and Bob's measurement systems $\mathcal{M}_A$ and $\mathcal{M}_B$. Each measurement comprises a displacement operation, implemented by interfering the signal $a_1$ ($b_1$) with a coherent state $\ket{\alpha}$ ($\ket{\beta}$) on a beam splitter with transmissivity $t_{a,b}\approx 1$, followed by a detector with efficiency $\eta_d$. All detectors used discriminate only between 'photons' and 'no-photons' cases (on/off detection).
• Figure 2: Maximal CHSH parameter $S$ as a function of detector efficiency $\eta_d$ computed for: 1-photon protocol (red solid line, with dashed red line including squeezing operations in the measurements Acin2024), 2-photon protocol (blue) and, for reference, a polarization-based protocol Oudot2024 (green).
• Figure 3: Asymptotic raw key rates $r$ (logarithmic scale) as a function of local detection efficiency $\eta_d$ for visibility $v$ at Charlie's, computed for the 1-photon protocol (red), and the 2-photon protocol (blue), both with optimal preprocessing $q_{\text{opt}}$. The solid lines correspond to the key rate computed using the analytical bound in Eq. \ref{['SKR-CHSH+NP']} for $V=1, 0.95, 0.9$, while dashed lines show results from the BFF method (see SM, Section \ref{['LB_Rate-sec']}.B) for $V=1$.
• Figure 4: Comparison of finite-size and asymptotic key rates $R$ versus distance $L$ for $N \in \{10^8, 10^9, 10^{10}, \infty\}$ protocol rounds, assuming repetition rate $\nu=100$ MHz, optimal parametric gain $g$, computed for: (a) local detection efficiency $\eta_d = 93\%$ for the 1-photon protocol (red), the 2-photon protocol (blue), and the polarization-based protocol from Ref. Oudot2024 (green), all for visibility $V=1$, (b) local detection efficiency $\eta_d = 90\%$ for the 2-photon protocol, computed for visibilities $V=0.95$ (dashed lines) and 1 (solid lines).
• Figure S1: Required visibility (a) to violate the CHSH inequality and (b) to achieve a positive SKR as a function of detection efficiency, shown for the 1-photon (red) and 2-photon (blue) protocols. Solid lines correspond to imperfect visibility in all labs ($V_a=V_b=V_c<1$), while the dashed line assumed the perfect visibility in Charlie's lab ($V_c=1$), and in the dotted line correspond to perfect visibility in the local labs ($V_a=V_b=1$)

Theorems & Definitions (2)

• Lemma S2.1
• proof