A preprint reports that changing which party's measurement device is characterized can change the calculated secret-key rate in one-sided quantum key distribution. Across representative protocols, the modeled rate varied substantially with that choice, but no single assignment was best in every case. In a CHSH protocol, the Alice-trusted calculation matched the fully device-independent result, while the Bob-trusted assignment produced a higher rate than that comparison.
An asymmetric protocol question
The work presents tools, methods and benchmarks for one-sided device-independent quantum key distribution, or DIQKD. It applies them to CHSH-based QKD, lossless and lossy entanglement-based BB84, a qutrit mutually unbiased-bases protocol and an I3322-based protocol. The comparison changes which party's measurement device is characterized: Alice in one case and Bob in the other, with Alice used for raw-key generation.
From entropy to a key rate
For the asymptotic calculation, meaning the large-sample limit, the secret-key rate is formulated as the infimum—the lowest value allowed by the optimization—of the difference between an entropy conditioned on Eve and one conditioned on Bob. The optimization considers states compatible with the statistics used for parameter estimation.
Two routes through the optimization
To carry out that optimization, the authors develop two extensions of the NPA hierarchy using semidefinite-programming relaxations. NPA-AC imposes matrix-algebra constraints, while NPA-MP uses matrix-valued polynomials. Either hierarchy can be combined with the BFF or KS approximation for conditional von Neumann entropy.
For finite-size secret-key bounds, the proof adapts the Generalized Entropy Accumulation Theorem, or GEAT, to general attacks with arbitrary device memory. It reduces finite-key analysis to the same single-round conditional-entropy optimization used for the asymptotic rate.
Under the stated Archimedean condition, the convergence theorem identifies the limit of the semidefinite-programming optima with the optimum of the associated polynomial-optimization problem. Put more simply, the hierarchy approaches the target optimization in the limit, provided that mathematical condition holds.
A small speed-versus-memory trade-off
In the one-sided qubit BB84 benchmark using the KS calculation, NPA-AC took 1.3 seconds and used 245 MiB of memory. NPA-MP took 1.6 seconds and used 201 MiB. That single reported configuration favored NPA-AC on runtime and NPA-MP on memory; broader performance was described as problem-dependent.
Different protocols, different patterns
For lossy BB84 with trusted Alice, the reported asymptotic key rate had O(ηA) scaling, where ηA denotes Alice's detection probability. No fitted coefficient or tabulated scaling estimate was reported alongside that result.
In a separate qutrit mutually unbiased-bases comparison, the figure reported better noise tolerance for the von Neumann-entropy calculation at d = 3 than for the min-entropy comparison at d = 7. The exact thresholds were not reported in the text, and the conclusion comes from the figure.
Taken together, the examples make trust placement a protocol question rather than a fixed rule. In several examples, the key-generating party was favorable, but the CHSH case showed the opposite pattern. The results therefore stop short of a universal ranking between Alice and Bob.
Code and publication status
The paper reports code repositories for the NPA-MP implementation, key-rate calculations and a Python implementation reproducing a subset of the CHSH and BB84 results. Its front matter identifies the work as arXiv:2608.25915v1, dated 26 Aug 2026, and as a preprint.
The acknowledgements also report support from the French National Research Agency through the France 2030 program and LabEx PERSYVAL, as well as the ANR PraQPV project.
Paper data and sources
Original title: Computing key rates for one-sided device-independent quantum key distribution
Authors: Andreas Bluhm, Gereon Koßmann, Martin Sandfuchs et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text