Numerical security analysis for quantum key distribution with partial state characterization

TL;DR

This work develops a semidefinite-programming (SDP) based security proof for quantum key distribution that requires only partial information about emitted states, addressing practical imperfections and side channels. It extends to both prepare-and-measure and measurement-device-independent QKD, including decoy-state scenarios, by proving upper bounds on the phase-error rate $e_{ m ph}$ through a Gram-matrix formulation, thereby yielding a lower bound on the asymptotic key rate $R = Y_Z [1-h(e_{ m ph}) - f h(e_Z)]$. Numerical results show limited gains for BB84-like four-state protocols but substantial improvements for non-qubit encoding schemes, such as coherent-light MDI-QKD, and demonstrate applicability to Trojan-horse attacks in decoy-state protocols. The approach bridges theory and practice by enabling rapid evaluation across diverse imperfections, albeit with open questions on finite-size security and extension to general attacks using GEAT or PST techniques.

Abstract

Numerical security proofs offer a versatile approach for evaluating the secret-key generation rate of quantum key distribution (QKD) protocols. However, existing methods typically require perfect source characterization, which is unrealistic in practice due to the presence of inevitable encoding imperfections and side channels. In this paper, we introduce a novel security proof technique based on semidefinite programming that can evaluate the secret-key rate for both prepare-and-measure and measurement-device-independent QKD protocols when only partial information about the emitted states is available, significantly improving the applicability and practical relevance compared to existing numerical techniques. We demonstrate that our method can outperform current analytical approaches addressing partial state characterization in terms of achievable secret-key rates, particularly for protocols with non-qubit encoding spaces. This represents a significant step towards bridging the gap between theoretical security proofs and practical QKD implementations.

Figures (4)

• Figure 1: Asymptotic secret-key rates for the BB84 protocol with an imperfect source in its (a) standard four-state version and (b) alternative three-state version boileauUnconditionalSecurity2005tamakiLosstolerantQuantum2014 using the numerical analysis presented in this work (solid lines) compared with the analytical results in curras-lorenzoSecurityFramework2023 (dashed lines). We consider $\delta = 0.063$honjoDifferentialphaseshiftQuantum2004xuExperimentalQuantum2015 and several values of $\epsilon$, which correspond to the magnitudes of the characterized SPFs and the uncharacterized imperfections, respectively.
• Figure 2: Illustration of the coherent-light-based MDI-type setup introduced in navarretePracticalQuantum2021. In every round, each of Alice and Bob (ideally) prepares a state in the set $\{\ket*{\sqrt \mu}, \ket*{-\sqrt \mu}, \ket{\rm vac}\}$ and sends it to the untrusted middle node Charlie through a quantum channel. Charlie measures the incoming signals by interfering them with a 50:50 beamsplitter, followed by two threshold single-photon detectors $D_c$ and $D_d$, which are associated with constructive and destructive interference, respectively; and announces the outcomes.
• Figure 3: Asymptotic secret-key rates for the coherent-light-based MDI protocol navarretePracticalQuantum2021 illustrated in \ref{['fig:coherentMDI']} using the numerical analysis presented in this work (solid lines) compared with the analytical results in curras-lorenzoSecurityFramework2023 (dashed lines). We consider several values of $\xi \coloneqq 1-(1-\epsilon)^2 \approx 2 \epsilon$ and optimize over the value of the coherent-light intensity $\mu$ for each distance value.
• Figure 4: Asymptotic secret-key rates as a function of the maximum intensity ($I_{\rm max}$) of the back-reflected light for a decoy-state BB84 protocol under a Trojan-horse attack (THA) when using our analysis (solid lines), compared with the analytical results in navarreteImprovedFiniteKey2022 (dashed lines). We assume that Alice uses three intensity settings $\{\mu_0,\mu_1,\mu_2\}$, where for simplicity we set $\mu_1 = 0.02$, $\mu_2 =0$, and optimize over the value of $\mu_0$ for each distance value.

Theorems & Definitions (2)

• Lemma : curras-lorenzoSecurityFramework2023
• proof