Feasibility of entanglement-based QKD protocols with SPDC and QD sources

TL;DR

The study addresses the feasibility of entanglement-based QKD, including DI-QKD and entanglement-based BB84, using realistic SPDC and quantum dot sources under imperfect detectors. It employs a photodetection-theory framework to compute the Bell parameter $S$, QBER $Q$, and Devetak-Winter key rates, incorporating binning strategies and practical imperfections. The results show SPDC sources generally fail to produce secure DI-QKD under standard detection and binning due to vacuum and multiphoton emissions, whereas QD sources remain viable for both DI-QKD and BB84 even with fine-structure splitting, provided detectors are highly efficient. The work highlights the critical role of realistic detector modeling and source-specific imperfections and points to future directions, such as heralding, decoy-state methods, and advanced security proofs, to enable SPDC-based QKD under practical conditions.

Abstract

We theoretically analyze the feasibility of entanglement-based quantum key distribution (QKD) protocols considering widely used spontaneous parametric down-conversion (SPDC) and novel quantum dot (QD) sources. We account for multiphoton emission in SPDC sources and fine-structure splitting (FSS) in QD. In addition, we incorporate imperfect detection, including dark counts and limited efficiency. For SPDC sources, we confirm that the presence of vacuum and multiphoton pairs renders them unsuitable for secure device-independent (DI) QKD implementations under standard detection strategies. Conversely, in the case of QD sources, accounting for the effects of FSS, results in reduced performance of protocols. Our findings are crucial for the practical implementation of entanglement-based QKD protocols using realistic sources and detectors.

Figures (5)

• Figure 1: (a) Generic scheme of a DI-QKD protocol, adapted from Acin2007PRLPironio_2009. DI-QKD is designed to operate without assumptions about the internal working of measurement devices, which are treated as black boxes with multiple inputs and outputs. These devices, along with the source of entangled photons, are considered untrusted and potentially controlled by an eavesdropper, making DI-QKD immune to eavesdropping attacks targeting device imperfections. (b) Realistic setup for implementation of entanglement-based BB84 and DI-QKD using polarization analyzers on each side (A and B). The analyzers consist of a half-wave plate (HWP), polarizing beam splitters (PBS), and two detectors (denoted $D_R$ and $D_T$, respectively, for the light reflected or transmitted by a PBS).
• Figure 2: The Bell parameter \ref{['bell-p']} is plotted as a function of the squeezing parameter $\xi$ for SPDC source with the standard binning in Sec. \ref{['section-5']} (orange line) and with alternative binning Vivoli2015 (violet line). The inset illustrates the dependence of QBER on $\xi$ with the same colour code. In the case of ideal pair generation, the optimal measurements $\theta_{A}^{(1)} = \pi/8$, $\theta_{A}^{(2)} = 3\pi/8$, $\theta_{B}^{(1)} = 0$, and $\theta_{B}^{(2)} = \pi/4$ lead to the maximum CHSH Bell inequality violation \ref{['bell']}. However, for SPDC sources, vacuum and multiphoton events suppress the violation under the current binning strategy. Results with an alternative binning strategy Vivoli2015 yield Bell inequality violations up to $S = 2.30083$ under optimized measurement angles ($\theta_{A}^{(1)} = 0.661$, $\theta_{A}^{(2)} = 1.248$, $\theta_{B}^{(1)} = 2.525$, $\theta_{B}^{(2)} = 3.112$) and a squeezing parameter of $\xi = 0.755$. Both main plot and inset correspond to DI-QKD protocol analysis.
• Figure 3: The Bell parameter \ref{['bell-p']} is plotted as a function of the detection efficiency $\eta$ for a QD source without (solid line) and with (dashed line) the effect of FSS. The results demonstrate that the Bell-parameter violation remains sufficient for secure DI-QKD, even in the presence of FSS and imperfect detection, under a dark count rate of $\nu=10^{-3}$ and an initial state survival probability of $p=0.9$ after depolarisation in the channel. The horizontal blue line indicates the critical minimum Bell parameter required for the DI-QKD secure protocol with the given parameters; values above this threshold (upper half) correspond to the secure DI-QKD regime. For comparison, we also plot the scenario in which polarisation effects in the channel are neglected ($p = 1$, gray lines), which reduces the minimum detection efficiency required for secure DI-QKD.
• Figure 4: The DI-QKD QBER (red line) and the entanglement-based BB$84$ QBER (black line) versus the detection efficiency $\eta$ for the QD source with and without the effect of FSS (both overlap, indicating that the impact of FSS completely cancels out in QBER for both types of protocols). The analysis assumes imperfect detection with a dark-count probability of $\nu=10^{-3}$ and an initial photon-pair survival probability $p=0.9$ after depolarization in the quantum channel. Horizontal green and blue lines mark the maximal tolerable QBER for secure BB84 and secure DI-QKD, respectively; values below these lines correspond to the secure operating regime. For comparison, we also plot the case without channel depolarisation, which lowers the detection-efficiency requirement for DI-QKD. Note that for BB84 the QBER values remain well below the corresponding threshold across the plotted range; in the absence of depolarisation the protocol becomes fully secure, as the QD state \ref{['qd-state']} approaches an ideal Bell state.
• Figure 5: The extractable Devetak–Winter key rates against collective attacks for DI-QKD (blue) and BB84 (green) are plotted versus the detection efficiency $\eta$ for a QD source. For DI-QKD, the curve including FSS is shown dashed; for BB84, FSS has no impact on security and a single solid curve suffices. The analysis includes imperfect detection with a dark-count probability $\nu=10^{-3}$ and an initial-state survival probability $p=0.9$ after depolarisation in the channel. The plot highlights that DI-QKD requires highly efficient detectors, posing an experimental challenge. For comparison, semi-transparent lines depict the scenario without channel depolarisation, which shifts the DI-QKD security threshold to lower detection efficiencies. Note that BB84’s security exhibits only a weak dependence on detection efficiency and is shown primarily for comparison.