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Optimal key rates for quantum key distribution with partial source characterization

Margarida Pereira, Guillermo Currás-Lorenzo, Mateus Araújo

TL;DR

This work extends conic-optimization methods to scenarios where only partial information about the emitted states is known, covering both prepare-and-measure and measurement-device-independent protocols, and demonstrates that the method outperforms state-of-the-art analytical and numerical approaches under realistic source imperfections.

Abstract

Numerical security proofs based on conic optimization are known to deliver optimal secret-key rates, but so far they have mostly assumed that the emitted states are fully characterized. In practice, this assumption is unrealistic, since real devices inevitably suffer from imperfections and side channels that are extremely difficult to model in detail. Here, we extend conic-optimization methods to scenarios where only partial information about the emitted states is known, covering both prepare-and-measure and measurement-device-independent protocols. We demonstrate that our method outperforms state-of-the-art analytical and numerical approaches under realistic source imperfections, especially for protocols that use non-qubit encodings. These results advance numerical-based proofs towards a standard, implementation-ready framework for evaluating quantum key distribution protocols in the presence of source imperfections.

Optimal key rates for quantum key distribution with partial source characterization

TL;DR

This work extends conic-optimization methods to scenarios where only partial information about the emitted states is known, covering both prepare-and-measure and measurement-device-independent protocols, and demonstrates that the method outperforms state-of-the-art analytical and numerical approaches under realistic source imperfections.

Abstract

Numerical security proofs based on conic optimization are known to deliver optimal secret-key rates, but so far they have mostly assumed that the emitted states are fully characterized. In practice, this assumption is unrealistic, since real devices inevitably suffer from imperfections and side channels that are extremely difficult to model in detail. Here, we extend conic-optimization methods to scenarios where only partial information about the emitted states is known, covering both prepare-and-measure and measurement-device-independent protocols. We demonstrate that our method outperforms state-of-the-art analytical and numerical approaches under realistic source imperfections, especially for protocols that use non-qubit encodings. These results advance numerical-based proofs towards a standard, implementation-ready framework for evaluating quantum key distribution protocols in the presence of source imperfections.
Paper Structure (10 sections, 1 theorem, 61 equations, 3 figures)

This paper contains 10 sections, 1 theorem, 61 equations, 3 figures.

Table of Contents

  1. Justification for Eq. (9)
  2. Handling linearly dependent reference states
  3. Optimization problem for MDI-type protocols
  4. Full state characterization
  5. Partial state characterization
  6. Protocol modeling
  7. BB84 protocol
  8. Coherent-light-based MDI protocol
  9. Reconstructing solution
  10. Additional simulations

Key Result

Lemma 1

Let $\{\rho_j\}_j$ with $j \in \{0,1,...,n-1\}$ be a set of states satisfying for a fixed set of pure states $\{\ket{\phi_j}\}_j$. Then there exist a CPTP map $\Phi$ and pure states $\{\ket{\psi_j}\}_j$ of the form where $\braket{\phi_j^\perp}{\phi_j} = 0$, such that As a consequence, for any prepare-and-measure QKD protocol, if security can be established for all pure state preparations $\{\ke

Figures (3)

  • Figure 1: Secret-key rate (logarithmic scale) as a function of distance for the BB84 protocol with $\delta = 0.14$ and different values of $\epsilon$. The solid lines correspond to our conic optimization approach while the dashed lines correspond to the phase-error estimation method of curras-lorenzoNumericalSecurity2025. The curve obtained for $\epsilon = 0$ when using the latter analysis is omitted as its visually indistinguishable from that obtained when using our approach.
  • Figure 2: Secret-key rate rate (logarithmic scale) as a function of distance for the coherent-light-based MDI-type protocol introduced in navarretePracticalQuantum2021, with $\delta = 0.14$ and different values of $\epsilon$. The parameter $\alpha$ has been optimized for each distance point, taking values in the range $[0.03, 0.5]$. All computations were performed with double floats for increased precision and stability.
  • Figure 3: Secret-key rate (logarithmic scale) as a function of distance for the coherent-light-based MDI protocol with $\delta = 0$, $p_d = {10^{-4}}$ and different values of $\epsilon$ (which accounts for both Alice's and Bob's source imperfections). The solid lines correspond to our conic optimization approach while the dashed lines correspond to the phase-estimation method of curras-lorenzoNumericalSecurity2025. The parameter $\alpha$ has been optimized for each distance point, taking values roughly in the range $[0.1, 0.4]$. All computations were performed with double floats for increased precision and stability.

Theorems & Definitions (2)

  • Lemma 1
  • proof