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arxiv: 2501.05890 · v2 · submitted 2025-01-10 · 🪐 quant-ph

High-dimensional quantum key distribution rates for multiple measurement bases

Nikolai Wyderka , Giovanni Chesi , Hermann Kampermann , Chiara Macchiavello , Dagmar Bru{\ss} This is my paper

Pith reviewed 2026-05-23 05:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords high-dimensional QKDmutually unbiased basesfinite-key analysissecret key rateBBM92 protocolquantum cryptographyasymptotic key rate
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The pith

For small numbers of rounds in high-dimensional QKD, three mutually unbiased bases yield higher key rates than d+1 bases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a BBM92-like quantum key distribution protocol using systems of dimension d greater than 2 and varying numbers of mutually unbiased bases. It derives an analytic formula for the asymptotic secret key rate when all d+1 possible bases are used. The analysis then shifts to finite-key scenarios, where the key rate is optimized against different types of attacks for any number of bases. A key result is that when the total number of protocol rounds is limited, employing only three bases can produce a higher key rate than using the full set of d+1 bases. This finding matters because practical QKD implementations often operate with finite data, so the choice of bases affects achievable security and throughput.

Core claim

In a high-dimensional BBM92-like QKD protocol, an analytic expression for the asymptotic key rate is derived when d+1 MUBs are used. For the finite-key case, optimization against collective and coherent attacks reveals that for sufficiently small number of rounds, the highest key rate is achieved by using three MUBs rather than d+1, for generic d.

What carries the argument

Finite-key rate optimization against collective and coherent attacks, parameterized by the number of mutually unbiased bases employed.

If this is right

  • An analytic expression exists for the asymptotic key rate when the protocol uses the maximum d+1 MUBs.
  • Varying the number of MUBs changes the achieved key rate in both the asymptotic and finite-key regimes.
  • Below a threshold on the number of rounds, three MUBs give the maximum finite key rate for any prime-power dimension d.
  • The optimization result applies uniformly across all admissible numbers of MUBs and generic system dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hardware that generates only a few MUBs may suffice for short sessions, lowering implementation cost without sacrificing rate.
  • The crossover point where three MUBs cease to be optimal could be used to set session-length thresholds in deployed systems.
  • Similar finite-size effects might appear when the same optimization is applied to other high-dimensional protocols or attack models.

Load-bearing premise

The finite-key optimization against collective and coherent attacks correctly identifies the reported optimum for generic d and all MUB counts.

What would settle it

An independent recalculation of the finite-key rates that shows d+1 MUBs always produce strictly higher rates than three MUBs for every round count would falsify the central claim.

Figures

Figures reproduced from arXiv: 2501.05890 by Chiara Macchiavello, Dagmar Bru{\ss}, Giovanni Chesi, Hermann Kampermann, Nikolai Wyderka.

Figure 1
Figure 1. Figure 1: Asymptotic key rates for five-dimensional systems and all [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: , where we plot, in the symmetric case, the maximum tolerable error rate Qmax, namely the error rate at which the key rate becomes zero, as a function of the dimension d of the system. There, we plot the same quantity also for the case m = d + 1, which shows again the enhancement that one gets by exploiting more than two MUBs, as displayed in Figs. 1 and 2 for the particular cases d = 5 and d = 47. Finally… view at source ↗
Figure 5
Figure 5. Figure 5: Asymptotic key rates for d-dimensional systems and two MUBs as a function of the two error rates QX and QZ . The analyti￾cal expression is given in Eq. (18). [19, 32, 41, 42], as for the collective attacks. This approxima￾tion can be achieved by using the so-called postselection tech￾nique [33]. Alternatively, one can exploit suitable entropic uncertainty relations (EURs), but their generalization to more … view at source ↗
Figure 6
Figure 6. Figure 6: Top: Finite key rates secure against collective (solid lines, Eq. (27)) and coherent (dot-dashed lines, Eq. (30)) attacks for [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

We investigate the advantages of high-dimensional encoding for a quantum key distribution protocol. In particular, we address a BBM92-like protocol where the dimension of the systems can be larger than two and more than two mutually unbiased bases (MUBs) can be employed. Indeed, it is known that, for a system whose dimension $d$ is a prime or the power of a prime, up to $d+1$ MUBs can be found. We derive an analytic expression for the asymptotic key rate when $d+1$ MUBs are exploited and show the effects of using different numbers of MUBs on the performance of the protocol. Then, we move to the non-asymptotic case and optimize the finite key rate against collective and coherent attacks for generic dimension of the systems and all possible numbers of MUBs. In the finite-key scenario, we find that, if the number of rounds is small enough, the highest key rate is obtained by exploiting three MUBs, instead of $d+1$ as one may expect.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes a high-dimensional BBM92-like QKD protocol that employs up to d+1 mutually unbiased bases (MUBs) for prime-power dimensions d. It derives a closed-form asymptotic key rate expression when all d+1 MUBs are used, examines the effect of using fewer bases, and performs a numerical optimization of the finite-key rate against collective and coherent attacks for arbitrary d and all admissible numbers of MUBs. The central finite-key result is that, for sufficiently small total round counts, the optimal rate is achieved with three MUBs rather than d+1.

Significance. If the finite-key optimization is reproducible, the work supplies concrete, dimension-dependent guidance on the number of measurement bases that maximizes the extractable key in the short-block-length regime, which is directly relevant to near-term high-dimensional QKD implementations. The analytic asymptotic formula is a clear strength that can be checked independently.

major comments (2)
  1. [finite-key optimization] Finite-key optimization section: the numerical maximization of the key rate against coherent attacks for generic d and varying MUB counts is presented only as a result; the explicit objective function, the precise finite-key correction terms (smoothing or concentration bounds), the SDP constraints that encode the MUB overlap structure, and the solver tolerances are not stated. Without these, the reported crossover to three MUBs for small N cannot be independently verified and is therefore load-bearing for the headline claim.
  2. [asymptotic analysis] Asymptotic rate derivation: while an analytic expression is stated for the d+1-MUB case, the manuscript does not show how the expression reduces when the number of MUBs is lowered to three; the dependence on the number of bases therefore remains implicit and prevents direct comparison with the finite-key optimum.
minor comments (2)
  1. Notation for the number of MUBs should be introduced once and used consistently (e.g., m versus k).
  2. Figure captions should explicitly state the dimension d and the total number of rounds N used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the recognition of the significance of our finite-key results for high-dimensional QKD. Below, we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [finite-key optimization] Finite-key optimization section: the numerical maximization of the key rate against coherent attacks for generic d and varying MUB counts is presented only as a result; the explicit objective function, the precise finite-key correction terms (smoothing or concentration bounds), the SDP constraints that encode the MUB overlap structure, and the solver tolerances are not stated. Without these, the reported crossover to three MUBs for small N cannot be independently verified and is therefore load-bearing for the headline claim.

    Authors: We agree that providing these details is crucial for reproducibility of the numerical results. In the revised manuscript, we will include the explicit form of the objective function, specify the finite-key correction terms with the exact smoothing and concentration bounds employed, detail the SDP constraints reflecting the MUB overlap structure, and report the numerical solver tolerances. These additions will enable independent verification of the reported crossover to three MUBs for small total round counts. revision: yes

  2. Referee: [asymptotic analysis] Asymptotic rate derivation: while an analytic expression is stated for the d+1-MUB case, the manuscript does not show how the expression reduces when the number of MUBs is lowered to three; the dependence on the number of bases therefore remains implicit and prevents direct comparison with the finite-key optimum.

    Authors: The analytic expression is presented for the d+1 MUB case because this configuration yields the highest asymptotic rate. The effects of using fewer MUBs are demonstrated through numerical evaluation in the manuscript. However, to make the dependence explicit and facilitate direct comparison, we will add in the revision a derivation showing the reduction of the expression for the three-MUB case by appropriately modifying the entropy terms corresponding to the reduced number of bases. revision: yes

Circularity Check

0 steps flagged

No circularity: analytic derivation and numerical optimization are independent of their outputs

full rationale

The abstract states an analytic expression is derived for the asymptotic key rate with d+1 MUBs and that finite-key rates are optimized numerically against collective/coherent attacks for generic d and varying MUB counts. No equation is presented that defines a rate or optimum in terms of itself, renames a fitted parameter as a prediction, or reduces the central claim to a self-citation chain. The reported preference for three MUBs at small round counts is framed as the output of an external optimization procedure rather than a definitional identity. Standard QKD formulas and known MUB existence results supply the inputs; the optimization step is treated as a computational result, not a tautology. This satisfies the default expectation of a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The claims rest on standard QKD security definitions for collective and coherent attacks and on the existence of d+1 MUBs for prime-power dimensions; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • standard math Existence of d+1 mutually unbiased bases when d is prime or prime power
    Invoked in the abstract to justify using up to d+1 MUBs.
  • domain assumption Validity of BBM92-like protocol extension to dimension d and multiple MUBs
    The protocol is described as BBM92-like without further justification in the abstract.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Security of deterministic key distribution with higher-dimensional systems

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    Higher-dimensional two-way QKD protocols using mutually unbiased bases and Heisenberg-Weyl operators yield secret keys for stronger individual attacks and improved robustness to collective eavesdropping via entropic u...

  2. Quantum Uncertainty and Entropy

    quant-ph 2026-04 unverdicted novelty 1.0

    A review of quantum uncertainty relations that covers foundational and practical applications of both variance- and entropy-based uncertainties.

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