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REVIEW 2 major objections 4 minor 63 references

Separation of measurement uncertainty into quantum and classical parts based on Kirkwood-Dirac quasiprobability and generalized entropy

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that the total uncertainty of any POVM measurement decomposes additively into a genuine quantum part and a classical remainder.

desk verdict Proposition 2 is false — the infimum over all POVMs is 0, not the quantum impurity — so the paper's central advertised result collapses, even though the KD-based decomposition framework has real merit. read the letter →

arxiv 2412.10619 v1 pith:M6RYZQKK submitted 2024-12-13 quant-ph

Agung Budiyono This is my paper · ORCID
classification quant-ph MSC 81P15
keywords measurementuncertaintygenuinequantumKirkwood-Diracquasiprobabilitygeneralizedentropyimpurityweakvaluecontextualitydisturbance
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

The paper claims that the total uncertainty of a measurement described by any positive-operator-valued measure (POVM) can be split additively into a genuinely quantum part and a classical remainder. The quantum part is identified with the maximized Kirkwood-Dirac nonreality or nonclassicality of the state relative to the POVM, while the classical part is whatever uncertainty remains. Two generalized entropies of the measurement outcomes, $S(\{p_a\})=\sum_a \sqrt{p_a(1-p_a)}$ and $T(\{p_a\})=\sum_a \sqrt{p_a}-1$, serve as the total uncertainty, and the decomposition obeys natural requirements, including vanishing classical uncertainty for sharp projective measurements on pure states. If the decomposition is right, the quantum part is directly accessible in experiments through weak-value measurements and signals quantum contextuality, which would give the separation operational meaning.

What carries the argument

The central object is the generalized Kirkwood-Dirac quasiprobability $\Pr_{\mathrm{KD}}(a,b|\rho,M_a,\Pi_b)=\mathrm{Tr}\{\Pi_b M_a\rho\}$, a complex-valued analog of a joint probability that can take nonreal and negative values when $\rho$ and the measurement do not commute. Its nonreality, the $\ell^1$-norm of the imaginary parts, and its nonclassicality, $\sum_{a,b}|\Pr_{\mathrm{KD}}|-1$, each maximized over all rank-1 PVM bases $\{\Pi_b\}$, are the proposed genuine quantum parts of the measurement uncertainty (Eqs. (4a) and (4b)). Because $\Pr_{\mathrm{KD}}(a,b|\rho,M_a,\Pi_b)$ equals the weak value $\langle b|M_a\rho|b\rangle/\langle b|\rho|b\rangle$ times the postselection probability $\langle b|\rho|b\rangle$, the quantum parts can be read off from weak-value data; the total uncertainties are the tight upper bounds given by the $S$ and $T$ entropies of the outcome distribution.

What would settle it

Check, in a low-dimensional Hilbert space (e.g., d=3 or d=4), whether every pair consisting of a pure state basis and a projective measurement basis admits a rank-one projective basis with equal overlap with both; if any pair fails, compute the KD-nonclassicality quantum uncertainty for that pair and compare it with the T-entropy total uncertainty—a strict gap would falsify the equality half of Proposition 1 and hence property QCD1.

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Extended reading notes

Core claim

The paper's central claim is that for any state $\rho$ and any POVM basis $\{M_a\}$, the total measurement uncertainty, quantified by $S(\{p_a\})=\sum_a \sqrt{p_a(1-p_a)}$ or $T(\{p_a\})=\sum_a \sqrt{p_a}-1$ with $p_a=\mathrm{Tr}\{M_a\rho\}$, decomposes additively as $U^{\mathrm{Total}}=U^{\mathrm{Quant}}+U^{\mathrm{Class}}$. The quantum part is the Kirkwood-Dirac nonreality (Eq. (4a)) or the Kirkwood-Dirac nonclassicality (Eq. (4b)) of $\rho$ relative to $\{M_a\}$, each maximized over the reference rank-1 PVM basis, and the classical part is the difference. The decomposition is shown to satisfy a list of natural requirements; in particular, for sharp rank-1 projective measurements on pure states the classical part vanishes, and the infimum of the total uncertainty over all POVMs equals the quantum impurity of the state, $\mathrm{Tr}\{(\rho-\rho^2)^{1/2}\}$ or $\mathrm{Tr}\{\sqrt{\rho}\}-1$, attained by measuring in the state's eigenbasis and hence entirely classical. The paper further claims that a nonvanishing quantum part is necessary and sufficient for a proof of generalized quantum contextuality via weak measurement with postselection, and that the quantum part is experimentally estimable from weak values.

Load-bearing premise

The load-bearing premise is that, for any pure state and any sharp projective measurement, there always exists a third rank-one projective measurement that has equal overlap with both the measurement basis and a basis containing the state; this is stronger than the known existence of mutually unbiased triples and is assumed in the proof without proof or reference.

Editorial extensions

If this is right

  • The genuine quantum part of a POVM measurement's uncertainty can be estimated directly from weak-value measurements with postselection, without full state tomography.
  • A nonzero quantum part is both necessary and sufficient to demonstrate generalized quantum contextuality by weak measurement with postselection.
  • The minimum total uncertainty over all measurements equals the impurity of the state, so classically mixed states have an irreducible but entirely classical uncertainty floor.
  • For pure states under sharp rank-1 projective measurements the classical part of the uncertainty is zero; classical uncertainty enters only through mixed preparations, unsharp POVMs, or both.
  • For rank-1 projective measurements, the KD-nonreality quantum part equals half the total trace distance between the state and its post-measurement state under a nonselective binary measurement, linking quantum uncertainty to measurement disturbance.

Reading between the lines

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

  • The same decomposition strategy could be applied with other generalized entropies; a future derivation might recover the Shannon-entropy-based decomposition as a special case and clarify which physical settings pick out the $S$ and $T$ entropies.
  • If the mutual-unbiasedness assumption behind the equality case fails in some dimension, property QCD1 would fail for those states, so a numerical search in small dimensions for counterexamples would directly test the tightness of the decomposition.
  • Because the quantum part is expressed in terms of weak values, it may serve as a practical nonclassicality resource in tasks such as metrology or quantum-information processing, though the paper does not address resource theory.
  • The equivalence with contextuality suggests the quantum part could be used as a witness for nonclassicality in experiments that already measure weak values, potentially bypassing full state reconstruction.
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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

2 major / 4 minor

Summary. The paper proposes two additive decompositions of the total uncertainty of a POVM measurement on a quantum state into a genuine quantum part and a classical remainder. The quantum parts are defined as the maximized Kirkwood-Dirac (KD) nonreality and KD nonclassicality relative to the POVM and an optimized rank-1 PVM basis, Eqs. (4a)-(4b). The total uncertainties are identified with the S and T entropies of the outcome probabilities, Eqs. (10a)-(10b), and the classical parts are the differences, Eqs. (16a)-(16b). The paper claims these decompositions satisfy a list of plausible requirements QCD1-QCD5, and that the infimum of the total uncertainty over all POVMs equals the quantum impurity quantified by S(ρ) and T(ρ) (Proposition 2, QCD6). It further connects a nonvanishing quantum part to strange weak values and quantum contextuality (Theorem 1) and interprets the KD-nonreality quantum uncertainty for rank-1 PVMs as state disturbance (Proposition 3).

Significance. The conceptual goal is valuable: a decomposition of measurement uncertainty into operationally accessible quantum and classical parts is a natural and useful target, and the weak-value formulation in Eqs. (23a)-(23b) is a genuine strength, since it gives a direct experimental route to the proposed quantum parts. The inequality half of Proposition 1 is proven with standard trace-norm and Cauchy-Schwarz arguments, and Theorem 1 relies on established strange-weak-value contextuality results. If the full set of claims were correct, the paper would provide a clean operational link among KD quasiprobabilities, measurement uncertainty, impurity, and contextuality. However, the advertised infimum result is false as stated: near-trivial POVMs make the total uncertainty arbitrarily small for every state, so the claimed equality with the quantum impurity cannot hold. This is a central advertised result of the abstract and conclusion, and it also underlies the proposed QCD6 requirement.

major comments (2)
  1. [Appendix B, Eqs. (B14)-(B15)] The claimed infimum in Proposition 2 is false. For any state ρ and any ε∈(0,1), the operators M^1=(1−ε)I and M^2=εI form a POVM basis according to Definition 1. The outcome probabilities are Pr(1)=1−ε and Pr(2)=ε, so Eqs. (10a)-(10b) give total uncertainties S=2√(ε(1−ε)) and T=√(1−ε)+√ε−1, both of which tend to 0 as ε→0. Hence the infimum over all POVM bases is 0 for every state, whereas Eqs. (21a) and (21b) claim positive values for every mixed state (for example √(d−1) for ρ=I/d). The proof fails at the first inequality in Eq. (E2): it uses (∑_k λ_k q_k)^2 ≤ ∑_k λ_k^2 q_k, which requires ∑_k q_k ≤ 1, but here q_k=Tr{Π_{λ_k} M^a} sums to Tr{M^a}, which is not bounded by 1. For M^1=(1−ε)I, the left side is (1−ε)^2 while the right side is (1−ε)∑_k λ_k^2; for a mixed state ∑_k λ_k^2<1, so the inequality fails for small ε. This invalidates Proposition 2, the QCD6 requirement, and the abstract claim that the minimum of the total measurement uncertainty over all POVM measurements is the quantum impurity.
  2. [Section IV, Theorem 1] The equality half of Proposition 1 for the KD-nonclassicality is not established. The proof assumes that for every pure state |ψ⟩, every PVM basis {|a⟩}, and every basis {|c⟩} containing |ψ⟩, there exists a rank-1 PVM basis {|b*⟩} that is mutually unbiased with both {|a⟩} and {|c⟩}. The text cites the existence of a triple of mutually unbiased bases, but that existence does not imply a common mutually unbiased basis for two arbitrary given bases. Without a proof of this stronger statement, Eq. (B15) does not follow, and the property QCD1 (vanishing classical uncertainty for a rank-1 PVM on any pure state) is unsupported for the KD-nonclassicality decomposition. A different argument, or a proof of the common-MUB claim, is needed for this load-bearing equality.
minor comments (4)
  1. [Appendix B, Eq. (B14)] There are numerous typographical errors in the abstract and introduction, including 'unpredict able', 'meausuure', and 'quantum' for 'quantum'; the manuscript needs a careful proofreading pass.
  2. [Appendix A] In Eq. (E6), the final summation uses the index j but the summand is written with λ_a(ρ); the notation should be made consistent, for example by summing over the eigenprojector index a.
  3. [Section III, notation] The proof of NComm1 contains a malformed expression in Eq. (A1), namely 'Tr {M b√ M a2 ̺)' with unmatched parentheses; please correct the mathematical display.
  4. [Section III, Prop. 2] The same symbol S is used both for the entropy functional S({Pr(a|ρ,M^a)}) in Eqs. (9a) and (10a) and for the quantum impurity S(ρ) in Eq. (21a); this can lead to confusion and should be disambiguated.

Circularity Check

1 steps flagged · score 3.0 of 10

Localized circular dependency in Eq. (8)/NComm1 proof; otherwise the decomposition is definitional and self-contained.

  1. other [Section II, Eq. (8); Appendix A, Proof of NComm1]
    "“as an implication of NComm1, the KD nonreality in a state ρ relative to a POVM {M^a} is vanishing if and only if the corresponding KD nonclassicality in ρ relative to {M^a} is vanishing, i.e., ... (8)” ... “Now let us prove that the converse is also true. Due to Eq. (8) it is sufficient to assume that U Quant KD−NRe(ρ; {M a}) = 0.”"

    Eq. (8) is introduced as a consequence of NComm1, but the proof of NComm1's converse in Appendix A invokes Eq. (8) to reduce the nonclassicality case to the nonreality case. Thus the equivalence of vanishing KD-nonreality and vanishing KD-nonclassicality is being used to prove the very faithfulness property from which it was said to follow. This is a genuine but localized circular dependency: it supports NComm1/QCD2 and the NRe↔NCl equivalence used in Theorem 1, although the equivalence is plausibly provable by a direct argument. It is not a fitted-input or self-citation reduction, and the main additive decomposition remains independent.

full rationale

The central additive separation is not circular by construction: the quantum parts are defined directly from KD quasiprobabilities (Eqs. (4a)-(4b)), the total uncertainty is defined from the outcome probabilities (Eqs. (10a)-(10b)), and the classical part is literally the difference (Eqs. (16a)-(16b)). Proposition 1's inequality is derived via Cauchy-Schwarz/Jensen-type estimates rather than assumed, and the pure-PVM equality is an explicit calculation, though Appendix B rests on an unproven MUB-triple existence assumption. No parameter is fitted and then renamed a prediction. The same-author citations (Refs. [41,43] for QU4 and Ref. [53] for lower bounds) are auxiliary; the contextuality claim rests on independent theorems [36-38]. The one true circular dependency is Eq. (8): it is stated as an implication of NComm1, but Appendix A's proof of NComm1 uses Eq. (8) to prove the converse, so the NRe↔NCl equivalence is presupposed in the proof of the faithfulness property that supposedly implies it. This is localized and reparable, so it raises the score only mildly. Separate correctness risks, including the invalid inequality in Eq. (E2) for Tr{M^a}>1 that undermines Proposition 2's infimum claim, are serious mathematical issues but are not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

No free parameters appear; the paper is a definitional framework built on standard quantum measurement theory. It introduces two new quantum uncertainty quantifiers with operational weak-value handles. The main implicit assumptions are finite-dimensionality, existence of suprema over rank-1 PVM bases, and an unproven common-MUB existence assertion in the equality part of Proposition 1.

assumptions (5)
  • domain assumption Quantum states are density operators on a finite-dimensional Hilbert space and measurements are POVMs.
    Used throughout, starting in Section II.
  • standard math The Kirkwood-Dirac quasiprobability Tr{M_b M_a rho} is normalized, yields correct marginals, and its nonreality or negativity indicates noncommutativity.
    Equation (1) and the surrounding discussion in Section II.
  • standard math The supremum over rank-1 PVM bases in Definitions 1 is attained.
    Several proofs, including Appendix B, select a basis achieving the supremum; compactness of the set of orthonormal bases makes this plausible, but it is not stated as an assumption.
  • ad hoc to paper For any pure state |psi> and PVM basis {|a>}, there exists a rank-1 PVM basis achieving the equality in Proposition 1 via a common mutually unbiased basis.
    Used in Appendix B to prove the equality half of Proposition 1 for KD-nonclassicality; no proof or citation for this stronger statement is given.
  • domain assumption Strange weak values imply generalized contextuality via weak measurement with postselection.
    External result cited as Refs. [36-38], used in Theorem 1.
invented entities (2)
  • KD-nonreality quantum uncertainty U_Quant_KD-NRe independent evidence
    purpose: Quantify the genuine quantum part of measurement uncertainty in the first decomposition.
    A new theoretical quantifier, not a physical entity; it is operationally accessible because it is expressed as an average over weak values in Eq. (23a), which can be measured in the laboratory.
  • KD-nonclassicality quantum uncertainty U_Quant_KD-NCl independent evidence
    purpose: Quantify the genuine quantum part of measurement uncertainty in the second decomposition.
    A new theoretical quantifier with the same weak-value measurement handle via Eq. (23b); no new particle or force is introduced.

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Pith. "Pith review of Separation of measurement uncertainty into quantum and classical parts based on Kirkwood-Dirac quasiprobability and generalized entropy." pith.science (2026). https://pith.science/paper/M6RYZQKK

@misc{pith2026241210619,
  author       = {Pith},
  title        = {Pith review of: Separation of measurement uncertainty into quantum and classical parts based on Kirkwood-Dirac quasiprobability and generalized entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M6RYZQKK}},
  note         = {Machine review of arXiv:2412.10619}
}
read the original abstract

Measurement in quantum mechanics is notoriously unpredictable. The uncertainty in quantum measurement can arise from the noncommutativity between the state and the measurement basis which is intrinsically quantum, but it may also be of classical origin due to the agent's ignorance. It is of fundamental as well as practical importance to cleanly separate the two contributions which can be directly accessed using laboratory operations. Here, we propose two ways of decomposition of the total measurement uncertainty additively into quantum and classical parts. In the two decompositions, the total uncertainty of a measurement described by a POVM (positive-operator-valued measure) over a quantum state is quantified respectively by two generalized nonadditive entropies of the measurement outcomes; the quantum parts are identified, respectively, by the nonreality and the nonclassicality | which captures simultaneously both the nonreality and negativity | of the associated generalized Kirkwood-Dirac quasiprobability relative to the POVM of interest and a PVM (projection-valued measure) and maximized over all possible choices of the latter; and, the remaining uncertainties are identified as the classical parts. Both decompositions are shown to satisfy a few plausible requirements. The minimum of the total measurement uncertainties in the two decompositions over all POVM measurements are given by the impurity of the quantum state quantified by certain generalized quantum entropies, and are entirely classical. We argue that nonvanishing genuine quantum uncertainty in the two decompositions are sufficient and necessary to prove quantum contextuality via weak measurement with postselection. Finally, we suggest that the genuine quantum uncertainty is a manifestation of a specific measurement disturbance.

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Works this paper leans on

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    However, unlike conventional probability, KD quasiprobability may assume nonreal value, and its real part, called Terletsky-Barut-Margenou-Hill (TBMH) q uasiprobability [45–47], may be negative or larger than one. Such anomalous values manifest quantum noncom- mutativity. Namely, assuming any two of the ingredients for the defi nition of the KD quasiprobab...

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    Proof of Proposition 1 for the KD nonclassicality in a state relative to a POVM Given a state ̺ and a POVM measurement basis {M a}, we first have from the definition in Eq. (4b), U Quant KD−NCl(̺; {M a}) + 1 = ∑ a sup {Π b}∈Mr1PVM(H) ∑ b ⏐ ⏐ ⏐Tr{Π bM a̺} Tr{Π b̺} ⏐ ⏐ ⏐Tr{Π b̺} ≤ ∑ a ( ∑ b∗ ⏐ ⏐ ⏐Tr{Π b∗M a̺} Tr{Π b∗̺} ⏐ ⏐ ⏐ 2 Tr{Π b∗̺} ) 1/2 (B11) = ∑ a ( ∑ ...

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