Imperfect detectors for adversarial tasks with applications to quantum key distribution

Abstract

Security analyses in quantum key distribution (QKD) and other adversarial quantum tasks often assume perfect device models. However, real-world implementations often deviate from these models. Thus, it is important to develop security proofs that account for such deviations from ideality. In this work, we extend the idea of squashing maps to develop a general framework for analysing imperfect threshold detectors, treating uncharacterised device parameters such as dark counts and detection efficiencies as adversarially controlled within some ranges. This approach enables a rigorous worst-case analysis with exactly characterised devices, ensuring security proofs remain valid under realistic conditions. Our results strengthen the connection between theoretical security and practical implementations by introducing a flexible framework for integrating detector imperfections into adversarial quantum protocols.

Figures (12)

• Figure 1: A set of equivalences (as quantum-to-classical measurement channels) of detection setups to consider dark counts as part of the noise channel. Here, ${ \mathcal{P}_{}}_{\text{sq}}'$ represents the classical post-processing carried out in the protocol, $\mathcal{P}^{{\mathbf{d_B}}}_{}$ is the dark count post-processing, ${ \mathcal{P}_{}}_{\text{sq}}$ is the post-processing required for the existence of the squashing map $\Lambda$, and ${ \mathcal{P}_{}}_{\text{dc}}$ is the post-processing fixed by \ref{['eq:PPswap']}. In the figure, olive post-processing maps represent free choices that can be made to make equivalences (ii) and (iv) hold. Orange post-processing maps represent stochastic processes fixed by equivalences (i) and (iii).
• Figure 2: A set of equivalences (as quantum-to-classical measurement channels) of detection setups to consider dark counts and loss as part of the noise channel, using the flag-state squasher.
• Figure 3: Constructive description of each block of the noise channel. Each line corresponds to a subspace of the input Hilbert space associated with the block-diagonal decomposition of the POVM elements. The dashed lines refer to classical states. The full noise channel is obtained by stacking each of these blocks together. $\mathcal{H}_F$ refers to the flag space, which is common for all the blocks $\mathcal{H}_m$.
• Figure 4: A set of equivalences (as quantum-to-classical measurement channels) of the active basis-choice BB84 detection setups to consider dark counts and loss as part of the noise channel, and to reduce the analysis to a finite-dimensional POVM.
• Figure 5: A set of equivalences (as quantum-to-classical measurement channels) of detection setups to use \ref{['thm:FSSFineGrained', 'thm:lossNoiseChannel']} with the postselection technique. Here, ${ \mathcal{P}_{}}_{\text{cg}}'$ represents the classical post-processing carried out in the protocol, $\mathcal{P}^{{\mathbf{d_B}}}_{}$ is the dark count post-processing, ${ \mathcal{P}_{}}_{\text{cg}}$ is the coarse-graining required for the WPFSS $\Lambda_{\text{WPFSS}}$, and ${ \mathcal{P}_{}}_{\text{dc}}$ is the post-processing fixed by \ref{['eq:WPFSSPPSwap']}. In the figure, olive post-processing maps represent free choices that can be made to make equivalences (ii) and (v) hold. Orange post-processing maps represent stochastic processes fixed by equivalences (i) and (iii).

Theorems & Definitions (9)

• Definition 1: Threshold detection setup with independent dark counts
• Theorem 1: Dark count noise channel
• Theorem 2: Loss noise channel
• Theorem 3: Generic noise channel
• proof
• Theorem 3: Dark count noise channel
• proof
• Theorem 3: Loss noise channel
• proof