Numerical security analysis for practical quantum key distribution

Numerical security analysis for practical quantum key distribution introduces a versatile numerical finite-key security framework that remains valid in the presence of nonindependent-and-identically-distributed (non-IID) signals and is applicable to a broad range of practical QKD implementations It accommodates most relevant imperfections at both the transmitter and receiver, including non-independent-and-identically-distributed (non-IID) signals arising in high-speed QKD systems due to the limited bandwidth of optical modulators, while requiring only partial characterization of the apparatuses The framework establishes security against coherent attacks while incorporating the most relevant device imperfections at both the transmitter and receiver, requiring only partial characterization of the apparatuses.

Protocol and Security Condition

The analysis focuses on BB84-type prepare-and-measure (P&M) QKD protocols, though it is valid for more general schemes like entanglement-based or decoy-state setups The transmitted states are modeled as partially characterized and may depend not only on the current setting choice but also on previous choices, with a fidelity constraint of Fσi u u−L Cu, ϕi u⟩Cu ≥ 1 − ε iu, ∀i u u−L (1). This allows the transmitted states to be considered in the alternative pure-state form ψε i u u−L⟩Cu = √1 − ε ϕi u⟩Cu + √ε ϕ⊥ i u u−L⟩Cu, where the unknown states satisfy ⟨ϕ⊥ i uϕi u⟩Cu = 0. The joint state of all transmitted systems and ancilla systems is written as Ψε⟩AN1CN1 = X∑iN1O N u=1 q p A i u iu⟩Auψε i u u−L⟩Cu.

Single-round Operator Inequality

The phase-error probability of any given single round of the virtual protocol can be upper bounded with a semidefinite program (SDP) defined in Eq. (9). This SDP involves maximizing Tr(Eˆph ABρAB) subject to constraints on the state ρAB, characterizing operators Tˆk A, and measurement operators Qˆl AB. The dual SDP in Eq. (11) is used to derive an operator inequality involving the phase-error operator and all the aforementioned measurement operators. This inequality, Eq. (12), implies that Tr(P lη∗ l Qˆl AB + Wˆ A ⊗ 1B)ρAB is a valid upper bound for the phase-error probability Tr(Eˆph ABρAB).

Finite-key Security with Imperfect Devices

To prove security in the finite-key regime, the Virtual Estimation Protocol (VEP) is considered, where each round is designated as a “local W-measurement” (LWM) round with probability α. The resulting bound on the number of phase errors Mph is given by Eq. (B23). This bound, MU ph(MQ,l), holds except with probability at most 3ϵpK + ϵpB. The analysis extends to imperfect detectors suffering from a detection-efficiency mismatch by defining the extended upper bound MU ph,δ as in Eq. (B25).

Decoy-state and MDI Protocols

The framework is robust for decoy-state protocols in the presence of general imperfections, including cross correlations between bit, basis, and intensity settings. For MDI-QKD type protocols, a modified primal SDP is considered in Eq. (A17). The resulting probabilistic bound on the phase-error rate is derived using Theorem 2. This leads to the final secret-key length formula in Eq. (A39).

Results and Performance

Simulations show that the numerical method is robust to multiple realistic device imperfections—even when all of them are taken into account simultaneously—while guaranteeing security against coherent attacks. In Fig. 1, the red curves jointly incorporate most relevant imperfections present in a BB84 setup, yet achieve a maximum transmission distance of ∼ 145km, exceeding half of the maximum distance attainable under ideal conditions. The versatility is especially relevant in the context of decoy-state schemes, where no security proof—analytical or numerical—can accommodate all device imperfections considered in this work.

Conclusion

The work introduces a numerical finite-key security framework for QKD protocols against coherent attacks in the presence of realistic device imperfections. The approach is broadly applicable to prepare-and-measure, measurement-deviceindependent, and entanglement-based QKD protocols, while outperforming state-of-the-art analytical proofs in regimes where a comparison is possible.

How it works

The framework accommodates general state-preparation flaws and side-channel information leakage, and remains applicable even when the emitted states are only partially characterized, requiring only lower bounds on their fidelity to suitable reference states. Crucially, it also incorporates pulse correlations—one of the principal outstanding challenges for previous numerical security proofs—as well as detector imperfections such as basis-dependent detection efficiency.

Protocol and Security Condition

The analysis focuses on BB84-type prepare-and-measure (P&M) QKD protocols, though it is valid for more general schemes like entanglement-based or decoy-state setups. The transmitted states are modeled as partially characterized and may depend not only on the current setting choice but also on previous choices, with a fidelity constraint of Fσi u u−L Cu, ϕi u⟩Cu ≥ 1 − ε iu, ∀i u u−L (1). The joint state of all transmitted systems and ancilla systems is written as Ψε⟩AN1CN1 = X∑iN1O N u=1 q p A i u iu⟩Auψε i u u−L⟩Cu.

Single-round Operator Inequality

The phase-error probability of any given single round of the virtual protocol can be upper bounded with a semidefinite program (SDP) defined in Eq. (9). This SDP involves maximizing Tr(Eˆph ABρAB) subject to constraints on the state ρAB, characterizing operators Tˆk A, and measurement operators Qˆl AB. The dual SDP in Eq. (11) is used to derive an operator inequality involving the phase-error operator and all the aforementioned measurement operators.

Decoy-state and MDI Protocols

The framework is robust for decoy-state protocols in the presence of general imperfections, including cross correlations between bit, basis, and intensity settings. For MDI-QKD type protocols, a modified primal SDP is considered in Eq. (A17)<ref:2605.

1. Numerical security analysis for practical quantum key distribution allows AI systems to provide rigorous security certification for real-world QKD setups by proving security against coherent attacks while accommodating most relevant imperfections at both transmitter and receiver.

2. The framework accommodates nonindependent-and-identically-distributed (nonIID) signals arising in high-speed QKD systems due to limited bandwidth, allowing AI to model and secure protocols with realistic signal correlations.

3. AI systems can establish security against general coherent attacks using a versatile numerical finite key security framework that requires only partial characterization of the apparatuses, making it applicable to a broad class of practical QKD setups.

4. The framework incorporates pulse correlations—scenarios where state emitted in one round depends on previous setting choices—addressing a principal outstanding challenge for previous numerical security proofs.

5. AI systems can establish security against detector imperfections such as basis-dependent detection efficiency mismatch by using the extended upper bound MUph delta which is non-decreasing with respect to Mkey.

6. The framework can model and secure decoy-state protocols in the presence of general imperfections, including cross correlations between bit, basis, and intensity settings.

7. AI systems can compute secret-key length for general QKD schemes by using the bound lkey = Nkey h 1 - h eUph(Nkey, Mi, beta test, Nkey) i − lambdaEC − 2 log(1/2 epsilonPA) − log(2/epsilonEV).

8. The framework can provide upper bounds on the number of phase errors in the virtual protocol by using the bound Mph <= MUph delta 1 + Skey Mkey + Skey DeltaS(Skey,Mkey, epsilonpS).

9. AI systems can extend security proofs to imperfect receivers suffering from detection-efficiency mismatch by deriving an extended upper bound MUph delta which depends on δ1 and δ2 parameters.

10. AI systems can derive probabilistic bounds on the number of Z-basis single-photon detections with mu equals zero by using the bound Mkey,1 >= MLkey,1 (Mvec Z) i <= epsilonpS 2.