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REVIEW 3 major objections 5 minor 52 references

Quantifying randomness with measurement incompatibility

T0 review · 3 major / 5 minor · reviewed 2026-07-10 · glm-5.2

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Pith's one-line read Incompatible measurements bound eavesdroppers and certify randomness

desk verdict Solid theoretical contribution linking incompatibility robustness to randomness generation; the full-rank assumption is the real but acknowledged limitation. read the letter →

arxiv 2607.08697 v1 pith:7FHILSNP submitted 2026-07-09 quant-ph

Sebastian Schlösser , Pauli Jokinen , Martin Plávala , Leevi Leppäjärvi , Leonardo S. V. Santos , Roope Uola This is my paper · ORCID
classification quant-ph PACS 03.67.Dd03.67.Hk03.65.Ta
keywords measurementincompatibilityrandomnessgenerationprepare-and-measuresemidefiniteprogrammingquantumsteeringjointmeasurabilityrobustnessdevice-independentinformation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that measurement incompatibility—the impossibility of jointly measuring two or more quantum observables—is not merely a foundational curiosity but a directly quantifiable resource for generating secret randomness. In a prepare-and-measure scenario where Alice sends quantum states to an untrusted receiver Bob while an eavesdropper Eve intercepts messages, the authors prove that the scenario is secure (Eve cannot perfectly predict Bob's outcomes) if and only if Bob's effective measurements are incompatible, provided the input states have full rank. They then make this qualitative statement quantitative: the generalised robustness of incompatibility, a geometric measure of how much noise must be added to a set of measurements before they become jointly measurable, upper-bounds Eve's guessing probability through a semidefinite program. Any incompatibility witness—a set of trusted test states whose statistics reveal incompatibility—doubles as a randomness certificate, yielding an explicit protocol that works for any finite number of measurement inputs and any incompatible measurement set.

What carries the argument

The load-bearing object is the semidefinite program (SDP) in Eq. (12), which maximises Eve's guessing probability subject to two constraints: (1) Bob's effective measurements must witness at least a threshold amount of incompatibility robustness, and (2) Eve's joint POVMs must be pairwise jointly measurable with Bob's effective POVMs. The incompatibility witness—trusted positive semidefinite operators whose expectation values are bounded for all jointly measurable measurement sets—provides the lower bound on incompatibility that feeds into the SDP, while the pairwise joint measurability constraint encodes the structural limit on Eve's strategies.

What would settle it

A concrete counterexample would be a set of measurements that is genuinely incompatible (positive incompatibility robustness) and a set of full-rank input states for which the SDP in Eq. (12) still yields Eve's guessing probability equal to 1, meaning the certified incompatibility does not translate into certified randomness.

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Extended reading notes

Core claim

The central mechanism is a structural identity between Eve's information-gathering capability and the mathematical structure of joint measurability. Eve's attack is fully characterised by POVMs that are pairwise jointly measurable with Bob's effective measurements: she can perfectly guess Bob's outcome for a given input if and only if her marginal measurement reproduces his. This means incompatibility of Bob's effective measurements is the exact condition preventing perfect prediction. The authors convert this qualitative equivalence into a quantitative bound by using the generalised incompatibility robustness, which measures the distance from a measurement set to the jointly measurable set.

Load-bearing premise

The security proof requires that the input states sent by Alice have full rank, meaning they are supported on the entire Hilbert space. Without this, there exist incompatible measurements that look jointly measurable on the subspace the states actually probe, and Eve could exploit this gap to perfectly guess outcomes despite apparent incompatibility.

Editorial extensions

If this is right

  • Any set of incompatible measurements, regardless of structure or number of inputs, can be converted into a randomness generation protocol by selecting appropriate test states and solving a single SDP.
  • The framework extends to quantum steering: steerability of a state assemblage and randomness certification are tightly linked for any finite number of measurement inputs, improving noise tolerance over prior two-input or star-incompatibility restrictions.
  • An eavesdropper with a quantum memory can be bounded by replacing joint measurability with dimensional simulability, allowing the framework to handle more powerful adversaries.
  • Incompatibility monogamy relations emerge naturally: higher incompatibility robustness means fewer compatible strategies for Eve, suggesting a resource-theoretic trade-off between incompatibility and eavesdropper capability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the full-rank assumption on input states could be relaxed—for instance, by using dimensional witnesses or self-testing techniques—the protocol would become more device-independent and potentially more experimentally practical, since preparing and verifying full-rank states adds overhead.
  • The connection between incompatibility robustness and state discrimination tasks suggests that optimal randomness-generation protocols could be designed by choosing measurement sets that are maximally hard to discriminate, creating a design principle for randomness sources.
  • Extension to continuous-variable systems seems feasible given that incompatibility robustness has infinite-dimensional counterparts, potentially enabling randomness certification from quadrature measurements or homodyne detection.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript establishes a quantitative trade-off between measurement incompatibility and randomness generation in a semi-device-independent prepare-and-measure scenario. The central qualitative result (Observation 1) states that, for full-rank input states, the scenario is secure if and only if Bob's effective POVMs are incompatible. The quantitative result (Eq. 12) provides an SDP that bounds Eve's guessing probability given a lower bound on the generalised incompatibility robustness certified by a witness. The authors translate their framework to quantum steering (Appendix E), obtaining a tight connection between steerability and randomness for any finite number of inputs, and extend to an Eve with a qubit quantum memory via dimensional simulability (Eq. 14). The proofs are mathematically careful, with both directions of Observation 1 established explicitly.

Significance. The paper provides a constructive, measurement-theoretic route to randomness certification that is not restricted to two-input scenarios or star-incompatibility structures, improving noise tolerance over prior steering-based protocols. The SDP formulation is parameter-free in the sense that the incompatibility lower bound is obtained from observed data via standard witness duality, not introduced as a free parameter. The steering equivalence (Appendix E) is a clean change of variables that makes the result directly applicable to steering experiments. The quantum-memory extension via qubit simulability is a natural and welcome generalisation. These are genuine advances for semi-device-independent quantum randomness.

major comments (3)
  1. Section IV, Observation 1 and Eq. (12): The full-rank assumption on input states is load-bearing for the converse direction of Observation 1 (secure implies incompatible). The authors acknowledge this and cite [36,37], but the quantitative SDP in Eq. (12) inherits this dependence: the witness states must be faithful for the security guarantee to hold. The footnote 3 mentions that adding white noise to the witness allows use of the MUB inequality but notes detection strength may drop. It would strengthen the paper to state more precisely how much the detection strength degrades and whether the SDP bound remains non-trivial after this regularization, at least for the two-MUB example.
  2. Section V, Example 3 (hollow triangle): The authors report witnessed robustness values and guessing probabilities for dimensions 2, 3, and 4 using a single state per basis, and a guessing probability of 0.888 for the qubit case using all witness states. However, no detail is given on how the witness was constructed for the three-MUB hollow triangle case or which specific witness operators were used. Since this example is the primary illustration of the paper's advantage over prior two-input or star-incompatibility results, a brief specification of the witness (or a reference to where it can be found) would be appropriate.
  3. Section VI, steering translation: The authors claim a 'tight connection between steerability and randomness generation in a setting using any finite number of measurement inputs.' The mapping in Appendix E is mathematically clean, but the claim of tightness could be stated more precisely. Does tightness mean that the optimal guessing probability in the steering SDP (Eq. E1) equals that of the incompatibility SDP (Eq. 12) for every full-rank state, or does it refer to the qualitative equivalence? Clarifying this would help the reader assess the strength of the steering result.
minor comments (5)
  1. Fig. 2 caption: The panels are labeled (a) and (b) but the in-text references in Section V refer to them in order without always specifying which panel. This is minor but could be made explicit.
  2. Eq. (6): The notation uses a product over tilde-y of p(e_tilde-y | lambda, tilde-y), which is correct for deterministic post-processing but could be confusing on first read. A brief clarifying sentence that this encodes a deterministic strategy vector would help.
  3. Section V, Example 4: The guessing probability of 0.924 for the qubit-memory Eve is stated without specifying the corresponding incompatibility robustness value or whether this is at maximal robustness. Adding the context would make the number more interpretable.
  4. Reference [48] in the note added: The distinction drawn between the authors' requirement (Eve guesses for some full-rank state unknown to her) and the related work's requirement (guessing for a collection of input states) is somewhat subtle. A slightly more explicit comparison in the main text or a footnote would help readers appreciate the difference.
  5. Typographical: 'PREP ARE-AND-MEASURE' and 'INCOMP A TIBILITY' in section headers appear to have spacing artifacts from the source.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity found

full rationale

The paper's central derivation chain is self-contained. Observation 1 (Section IV) is proven directly: the forward direction constructs an explicit Eve strategy from a Naimark dilation of a joint measurement, and the converse uses full-rank states to deduce joint measurability from perfect guessing. The quantitative bound (Eq. 12) takes a witnessed lower bound α on incompatibility robustness as input and maximizes Eve's guessing probability over the set of pairwise jointly measurable strategies. The witness bound (Eqs. 10-11) is a standard SDP duality result from [10] (Uola et al., PRL 2015), which is an independent, externally published result—not a self-citation chain. The steering equivalence (Appendix E) is shown by explicit variable substitution (G_{e,b|y} = σ^{-1/2} σ_{e,b|y} σ^{-1/2}), not by assumption. The quantum-memory extension (Eq. 14) adds qubit-simulability constraints derived from [32] (Ioannou et al., PRL 2022), another independent external citation. The full-rank assumption on input states is acknowledged as necessary and is not smuggled in. No step reduces to its inputs by construction, and no prediction is a renamed fit. The only minor self-citation is to [27] (Uola et al., arXiv:2212.02815) for the sequential measurement characterization, but this is a supporting remark, not load-bearing for the main trade-off. Score 2 reflects this minor self-citation with no impact on the central claim's independence.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities or postulated objects are introduced. The framework uses established mathematical objects (POVMs, instruments, incompatibility witnesses, SDPs). The 'hollow triangle' and 'star-incompatibility' are existing measurement structures from the literature.

free parameters (3)
  • α (witnessed incompatibility robustness lower bound)
    Not a free parameter in the traditional sense; it is a lower bound on incompatibility robustness obtained from experimental data via an incompatibility witness (Eq. 10-11). It serves as input to the SDP (Eq. 12).
  • p(y) (input distribution) = uniform (in examples)
    The probability distribution over Bob's inputs. The authors state one can optimise over it, but use uniform in examples. This is a protocol choice, not a fitted parameter.
  • Choice of witness states ϱ_{b|y} = MUB-based (in examples)
    The specific incompatibility witness used. The framework works with any witness; MUB witnesses are used for illustration. Not a fitted parameter but a design choice.
assumptions (5)
  • domain assumption Full-rank input states are required for the security proof (Observation 1 converse direction).
    Section IV, proof of Observation 1: 'we further assume that the state ϱ has full rank.' The authors note this is crucial because incompatible measurements whose restriction to any subspace is jointly measurable exist [36, 37].
  • domain assumption Eve is classical (no quantum memory) in the main result.
    Section II: 'We assume a classical Eve, meaning that she does not have a quantum memory.' This is relaxed in Example 4 and Section V using qubit simulability, but the main SDP (Eq. 12) assumes classical Eve.
  • domain assumption Pairwise joint measurability of Eve's POVM with Bob's effective measurements fully characterises Eve's capabilities.
    Remark in Section III, citing [25-27]: 'a classical Eve is fully characterised by those POVMs that are pairwise jointly measurable with all Bob's effective measurements.' This is a known result from sequential measurement theory.
  • domain assumption Bob's Hilbert space is uncharacterised (possibly infinite-dimensional).
    Section III: 'Bob measures the system using some uncharacterised POVMs acting onto a possibly infinite-dimensional Hilbert space.' This is standard in semi-device-independent scenarios.
  • standard math SDP duality gives a valid lower bound on incompatibility robustness via witnesses (Eq. 10-11).
    Section V, citing [10]: standard SDP duality theory applied to the incompatibility robustness SDP.

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Pith. "Pith review of Quantifying randomness with measurement incompatibility." pith.science (2026). https://pith.science/paper/7FHILSNP

@misc{pith2026260708697,
  author       = {Pith},
  title        = {Pith review of: Quantifying randomness with measurement incompatibility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FHILSNP}},
  note         = {Machine review of arXiv:2607.08697}
}
read the original abstract

We present a trade-off between the amount of observed measurement incompatibility and the capabilities of a classical Eavesdropper in a prepare-and-measure scenario. The result is based on a qualitative connection between measurement incompatibility and randomness generation together with the utilization of incompatibility witnesses as randomness certificates. This allows one to use a geometric measure of incompatibility, the generalised robustness, to bound Eve's strategies through a semi-definite program, while providing an explicit protocol for generating randomness from any set of incompatible measurements. By translating the result to quantum steering, we find a tight connection between steerability and randomness generation in a setting using any finite number of measurement inputs. We further show how our techniques can be generalised to scenarios where Eve has a quantum memory by using a dimensional generalisation of joint measurability.

Figures

Figures reproduced from arXiv: 2607.08697 by the authors.

Figure 1
Figure 1. FIG. 1. A trusted quantum state enters a black box device [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Eve’s guessing probability as a function of the observed incompatibility robustness [Eq. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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