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Maximal intrinsic randomness of noisy quantum measurements

T0 review · 1 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Noisy quantum measurements now have exact private-randomness formulas.

desk verdict Closed-form guessing probabilities for two classes of noisy measurements, proved by matching decompositions and dual certificates; the inherited quantum-to-classical reduction is the soft spot, but the paper is honest and solid. read the letter →

arxiv 2506.22294 v2 pith:IVDON66V submitted 2025-06-27 quant-ph

Fionnuala Curran , Morteza Moradi , Gabriel Senno , Magdalena Stobinska , Antonio Acín This is my paper · ORCID
classification quant-ph MSC 81P1581P4594A17 PACS 03.65.Ta03.67.-a
keywords quantumrandomnessguessingprobabilitynoisymeasurementsmin-entropyPOVMqubitwhitenoisesideinformation
topics The Measurement Problem
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

Given a noisy quantum measurement, how much private randomness can it generate when an adversary may hold side information about the noise? The paper solves this optimization exactly for two families of practical measurements: every qubit measurement with two outcomes, and every rank-one projective measurement mixed with white noise in any dimension. The answer is a closed-form expression for the adversary's optimal guessing probability, which converts directly into the conditional min-entropy that bounds how many private random bits can be extracted. A further result is a warning about attribution: when the same observed statistics can arise with noise on the state, on the measurement, or on both, putting noise on both gives the adversary noticeably more guessing power.

What carries the argument

The machinery is the guessing-probability semidefinite program: for a pure input state $|\varphi\rangle$, Eve's optimal strategy is a convex decomposition of the measurement into sub-POVMs, and her guessing probability is the sum of probabilities that her label matches Alice's outcome. The paper uses a known reduction from the quantum side-information problem to this classical decomposition, sets it up as a semidefinite program with a dual, and then exhibits optimal primal and dual solutions. The key object is the 'square-root decomposition': Eve splits $M_1$ into the rank-one term $\sqrt{M_1}|\varphi\rangle\langle\varphi|\sqrt{M_1}$ plus a remainder, and symmetrically for $M_2$; the generalized $d$-dimensional version splits each $M_x$ around $|\varphi\rangle$ and is proven optimal for the unbiased state via explicit dual variables. The matching lower and upper bounds from these decompositions close the min-max problem and produce the exact formulas.

What would settle it

For the qubit POVM with $M_1 = \mathrm{diag}(0.8, 0.1)$, solve the semidefinite program (4) or perform a dense search over input states and decompositions: any computed guessing probability below $1 - 0.9 + \frac{1}{2}(\sqrt{0.8} + \sqrt{0.1})^2$ would falsify Theorem 1. For arbitrary dimension, take the $d=3$ noisy projective measurement at $\varepsilon = 0.3$, numerically minimize equation (4) over $|\varphi\rangle$, and compare the result with $\frac{1}{3}(\operatorname{tr}\sqrt{M_1})^2$.

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Extended reading notes

Core claim

The central discovery is a pair of exact formulas. For any two-outcome qubit POVM (positive operator-valued measure) $M = \{M_1, M_2\}$ with $\operatorname{tr} M_1 \leq \operatorname{tr} M_2$, the guessing probability minimized over all input states is $P^*_{\mathrm{guess}}(M) = 1 - \operatorname{tr} M_1 + \frac{1}{2}(\operatorname{tr} \sqrt{M_1})^2$. For the noisy projective measurement $M_d$ obtained by mixing a rank-one projective measurement with white noise in dimension $d$, $P^*_{\mathrm{guess}}(M_d) = \frac{1}{d}(\operatorname{tr} \sqrt{M_1})^2$. Since the conditional min-entropy is $H^*_{\min} = -\log P^*_{\mathrm{guess}}$, these give the maximal intrinsic randomness of the measurement. The optimizing input state is unbiased to the measurement basis; for the noisy projective measurement this also produces a uniform outcome distribution, while for a general two-outcome qubit POVM uniform outcomes occur only when $\operatorname{tr} M_1 = \operatorname{tr} M_2$. The paper also shows that for fixed observed statistics, a realization with noise on both state and measurement can be guessed much more successfully than either single-noise realization, with perfect guessing reached at noise $\varepsilon^* = 1 - 1/\sqrt{2}$ in the qubit case.

Load-bearing premise

The whole calculation assumes that when Alice feeds in a pure state, a quantum eavesdropper with entanglement is no more powerful than a classical eavesdropper who knows which sub-measurement is being performed; the paper cites this equivalence rather than proving it, and if it failed the closed-form guessing probabilities would not describe the true adversarial scenario.

Editorial extensions

If this is right

  • Any noisy channel applied to a rank-one qubit projective measurement, such as depolarizing, phase-flip, or amplitude-damping noise, produces a two-outcome qubit POVM whose maximal intrinsic randomness is now computable in closed form.
  • For a noisy projective measurement in any dimension, the maximal intrinsic randomness equals that of the corresponding noisy pure state with the same noise parameter, so the noise can be attributed entirely to the state or entirely to the measurement without changing the min-entropy.
  • Coarse-graining a $d$-dimensional noisy projective measurement into two outcomes gives no more randomness than the qubit version; the adversary's best attack is to inflate her optimal qubit attack rather than coarse-grain her optimal $d$-dimensional attack.
  • When noise is shared between the state and the measurement, Eve's guessing probability is larger for every total-noise level, and she reaches perfect guessing already at $\varepsilon^* = 1 - 1/\sqrt{2}$ (equivalently $\delta = 1/2$), whereas single-device noise reaches perfection only at maximal noise.
  • Upper bounds are given for the conditional von Neumann and max-entropies of the noisy projective measurement under the unbiased state; the paper leaves open whether the unbiased state is optimal for all conditional entropies and whether the square-root decomposition saturates those bounds.

Reading between the lines

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

  • An implicit consequence for device-dependent quantum random number generators is a certification threshold: once the shared noise level passes $\delta = 1/2$ in the qubit scenario, the measured statistics certify zero private randomness even though the outcomes remain maximally random statistically.
  • Going beyond the paper, the square-root form of Eve's optimal decompositions suggests a conjecture that for arbitrary POVMs the optimal attack is some coherent square-root splitting of each element; three-outcome qutrit POVMs would be a direct numerical test bed.
  • The paper leaves open whether the unbiased state maximizes conditional von Neumann and max-entropies; a testable extension is to compute those quantities for $d > 2$ and see whether the equality between measurement-side and state-side randomness survives beyond min-entropy.
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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

1 major / 5 minor

Summary. The paper studies the maximal intrinsic randomness of a quantum measurement, defined as the minimum over input states of the guessing probability of an eavesdropper who holds optimal side information. The main results are exact closed-form formulas for this quantity for two classes of measurements: any two-outcome qubit POVM (Theorem 1, P*_guess(M) = 1 - tr M1 + (1/2)(tr sqrt(M1))^2) and d-dimensional noisy projective measurements (Theorem 2, P*_guess(Md) = (1/d)(tr sqrt(M1))^2). Lower bounds are obtained from explicit POVM decompositions; matching upper bounds are obtained from feasible dual variables and a permutation-symmetry argument. The paper also presents a Corollary upper bound for arbitrary two-outcome POVMs, upper bounds for the conditional von Neumann and max-entropies, a coarse-graining analysis, and a study of the scenario where noise is shared between the state and the measurement.

Significance. If correct, these are the first exact solutions for the intrinsic randomness of noisy measurements, complementing the corresponding state-side results of Ref. [10]. The formulas are simple, parameter-free, and correctly reproduce the known limits (projective measurements, maximally mixed POVMs). The constructive proof technique, matching explicit decompositions with dual certificates, is elegant and likely to be reusable. The main caveat is that the identification of the classical decomposition SDP (4) with the quantum guessing probability (2) for pure input states is imported from Ref. [13] rather than proved; this is a legitimate citation, but it is the load-bearing external input to the advertised quantum-scenario interpretation. The auxiliary results (Corollary 1, the shared-noise bound, and the entropy upper bounds) are honestly labeled as not fully tight, which strengthens the credibility of the central claims.

major comments (1)
  1. [Main text, Eq. (4) and Appendix A] Theorems 1 and 2 solve the classical guessing problem (4), and the equality of this quantity with the quantum guessing probability (2) for pure input states is taken from Ref. [13] (stated near Eq. (4) and in Appendix A). This is a published theorem covering exactly the finite-dimensional pure-state setting, so I consider the reliance legitimate; nonetheless, because this bridge is load-bearing for the central claim, an explicit statement of its hypotheses (or a short derivation) would make the paper more self-contained. This does not affect my assessment of the internal correctness of the classical proofs.
minor comments (5)
  1. [Eq. (4) and Appendix A17] The normalization constraint in the displayed SDP is garbled: 'Σ_x dK_{x,j} = 1' and the incomplete 'Σ_x tr K_{x,j}' do not form a correct condition. The intended constraint is Σ_x K_{x,j} = (1/d) Σ_x tr(K_{x,j}) 1, as used in the dual derivation (A24). Please correct the typesetting.
  2. [Main text Lemmas 1-4 and Appendix B/C] The lemma numbering is inconsistent between the main text and the appendix: the qubit lower bound is Lemma 1 in the main text but is proved as 'Lemma 2' in Appendix B1, and the qubit upper bound is Lemma 2 in the main text but 'Lemma 3' in Appendix B3. The same offset occurs in Appendix C, where the noisy-projective lower and upper bounds are proved as Lemma 4 and Lemma 5 rather than Lemma 3 and Lemma 4. Please renumber for consistency.
  3. [Discussion] The sentence 'an isotopic state would have the same max- and min-entropy as an isotropic measurement' contains a typo: 'isotopic' should be 'isotropic'.
  4. [Appendix C2, Eq. (C29)] The definition of Y_x is garbled in the current rendering: the term 'Mx − 1 2' should be the matrix M_x^{-1/2} with the exponent as a superscript, not a separate constant. Please ensure the typesetting is unambiguous.
  5. [Main text, after Eq. (23)] The phrase 'This begs the question' is a misuse of the idiom; 'raises the question' would be more appropriate. This is a minor stylistic issue.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: Theorems 1 and 2 are exact SDP solutions obtained by matching primal and dual bounds; the only imported ingredient is the published pure-state quantum-to-classical guessing reduction of [13], which is a parameter-free external bridge rather than a fitted input.

full rationale

The derivation chain in this paper is not circular. Theorem 1 is proved by a lower bound from an explicit valid decomposition (Lemma 1, Appendix B1/B2) and a matching upper bound from a feasible dual/vector solution (Lemma 2, Appendix B3), with the two bounds exactly equal. Theorem 2 is proved the same way: a constructive decomposition gives the lower bound (Lemma 3, Appendix C1), and feasible dual variables, or an independent permutation-symmetry argument, give the matching upper bound (Lemma 4, Appendix C2/C3). No parameter is fitted to the target formula, and no 'prediction' is the value of an input; the closed forms are derived, not assumed. The one load-bearing imported premise is the reduction from the quantum guessing problem (2) to the classical decomposition SDP (4)/(A23), stated as 'Happily, it is known [13] that when Alice chooses a pure state, the quantum guessing probability (2) is equal to the guessing probability of a classical Eve.' This is a genuine external input to the paper's formal results, and it is the right place to look for hidden regularity conditions. However, [13] is a published, parameter-free theorem whose stated assumptions (finite dimension, pure input state) do not include the specific formulas of Theorems 1 and 2, so it is independent support rather than a circular self-citation. Overlapping authorship with [13] and [10] is present, but the cited results are not being used as proxies for the present claims. The paper also openly labels the genuinely open items: Corollary 1's bound is numerically but not analytically saturated, the shared-noise comparison is a lower bound in a classical joint-decomposition model, and the von Neumann and max-entropy results are upper bounds. These limitations do not touch the validity of Theorems 1 and 2 and do not signal circularity. Overall, the central derivation is self-contained once the external [13] bridge is accepted; the score reflects only the minor caveat that this bridge deserves an independent check.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theorems rest on standard quantum measurement formalism and on the adversarial-model reduction of refs [12,13]; the latter is the only premise the paper imports rather than proves. The concavity lemma that justifies pure optimal states is proven in Appendix A, and the SDP duality used for upper bounds is standard with strict feasibility checked in Appendix A2. No free parameters are fitted: the noise parameter epsilon and the reparameterization delta = epsilon(2 - epsilon) are inputs and definitions, and the constants chosen in the dual constructions are proof scaffolding that do not appear in the final formulas. No new physical entities are postulated.

assumptions (6)
  • domain assumption POVM formalism and Born rule for outcome probabilities
    Setting section: a measurement is a set of positive operators Mx summing to identity, with p(x) = tr(rho Mx). This is the standard quantum measurement model.
  • domain assumption Naimark dilation theorem: any POVM is the marginal of a projective measurement on a larger system with an auxiliary state
    Eq. (1) and ref [5]; the adversarial model parameterizes all realizations of M through such dilations.
  • domain assumption For a pure Alice state, the quantum guessing probability with entangled side information equals the classical guessing probability over convex POVM decompositions
    Main text after eq. (2), attributed to refs [12,13]; this is the load-bearing premise that lets Theorems 1 and 2 solve the classical SDP (4).
  • standard math Strong duality for the guessing-probability SDP holds when both primal and dual are strictly feasible
    Appendix A2, citing the conic duality theorem of Nesterov and Nemirovskii [23]; justifies using the dual (A30) for full-rank POVMs.
  • standard math Data-processing inequality for von Neumann entropy
    Appendix D, eq. (D19), citing Nielsen and Chuang [16]; used to lower bound the conditional entropy from the explicit dilation.
  • standard math Permutation averaging preserves optimality for states unbiased to the measurement basis
    Appendix C3; the symmetrized decomposition (C62) is valid and gives the same guessing probability for |psi>.

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Pith. "Pith review of Maximal intrinsic randomness of noisy quantum measurements." pith.science (2026). https://pith.science/paper/IVDON66V

@misc{pith2026250622294,
  author       = {Pith},
  title        = {Pith review of: Maximal intrinsic randomness of noisy quantum measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IVDON66V}},
  note         = {Machine review of arXiv:2506.22294}
}
read the original abstract

Quantum physics exhibits an intrinsic and private form of randomness with no classical counterpart. Any setup for quantum randomness generation involves measurements acting on quantum states. In this work, we consider the following question: Given a quantum measurement, how much randomness can be generated from it? In real life, measurements are noisy and thus contain an additional, extrinsic form of randomness due to ignorance. This extrinsic randomness is not private since, in an adversarial model, it takes the form of quantum side information held by an eavesdropper who can use it to predict the measurement outcomes. Randomness of measurements is then quantified by the guessing probability of this eavesdropper, when minimized over all possible input states. This optimization is in general hard to compute, but we solve it here for any two-outcome qubit measurement and for projective measurements in arbitrary dimension mixed with white noise. We also construct, for a given measured probability distribution, different realizations with (i) a noisy state and noiseless measurement (ii) a noiseless state and noisy measurement and (iii) a noisy state and measurement, and we show that the latter gives an eavesdropper significantly higher guessing power.

Figures

Figures reproduced from arXiv: 2506.22294 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 1
Figure 1. Figure 1: FIG. 1: Vector visualizations of the SDP constraints for a qubit two-outcome measurement. [PITH_FULL_IMAGE:figures/full_fig_p014_1.png]
Figure 2
Figure 2. Figure 2: FIG. 2: Conditional entropies arising from the noisy qubit measurement [PITH_FULL_IMAGE:figures/full_fig_p029_2.png]
Figure 3
Figure 3. Figure 3: FIG. 3: A joint decomposition of a noisy state and measurement pair with noise parameter [PITH_FULL_IMAGE:figures/full_fig_p033_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4: Eve’s decompositions conditioned on the random variable [PITH_FULL_IMAGE:figures/full_fig_p034_4.png]

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Forward citations

Cited by 1 Pith paper

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

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Reference graph

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    Proof of Corollary 1 Corollary 1. Let M = {Mx}x be any POVM with two outcomes, with λmin (M1) + λmax (M1) ≤ λmin (M2) + λmax (M2), where λmax (.) and λmin (.) denote the largest and smallest eigenvalues of the argument. Then P ∗ guess M ≤ 1 − 1 2 p λmax (M1) − p λmin (M1) 2 . ...

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    F ull proof of Lemma 4 Lemma 4. Let Md be the noisy projective measurement in dimensiond. The following lower bound holds, P ∗ guess Md ≥ 1 d tr p M1 2 . (C1) Proof. We start by expressing Alice’s chosen state |ϕ⟩ in the eigenbasis {|x⟩} of the POVM as |ϕ⟩ = X x ⟨x|ϕ⟩ |x⟩ , (C...

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    Proof of Lemma 5 based on dual variables Lemma 5. Let Md be the noisy projective measurement in dimensiond, and let|ψ⟩ = 1√ d (1, ...,1)T . Then Pguess |ψ⟩ , M ≤ 1 d tr p M1 2 . (C23) Proof. We prove Lemma 5 by finding feasible dual variables {Yx} and {Gj} that satisfy the con...

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    We prove that the decomposition {Kj} from (C4) in Appendix C 2 is optimal for the unbiased state |ψ⟩

    Proof of Lemma 5 based on permutations Proof. We prove that the decomposition {Kj} from (C4) in Appendix C 2 is optimal for the unbiased state |ψ⟩. First, note that because all of the elements of |ψ⟩ are real when expressed in the computational basis, we are free to assume tha...

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