Pith. sign in

REVIEW 4 major objections 4 minor 50 references

Anomalous values, Fisher information, and contextuality, in generalized quantum measurements

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For qubits, the paper shows that anomalous values, enhanced Fisher information, and contextuality are not the same phenomenon in generic postselected measurements; only in the strict weak-value limit do they coincide.

desk verdict The theoretical point about decoupling is sound and worth taking seriously, but the experimental test of the Pusey inequality uses a suboptimal disturbance parameter and the error analysis is too thin to fully support the experimental claims. read the letter →

arxiv 1909.00755 v1 pith:OARFP7QH submitted 2019-09-02 quant-ph

Valeria Cimini , Ilaria Gianani , Fabrizio Piacentini , Ivo P. Degiovanni , Marco Barbieri This is my paper · ORCID
classification quant-ph
keywords weakmeasurementsanomalousvaluesFisherinformationcontextualityPuseyinequalitypostselectionqubitmetrologygeneralized
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

Postselected weak measurements on qubits are known to produce three effects: values outside the observable's spectrum, Fisher information that exceeds the standard quantum limit, and probabilities that can resist non-contextual models. This paper asks whether the three are the same underlying phenomenon. It shows they are not: for the generalized qubit measurements defined by Eq. (1), there is no one-to-one mapping among anomalous values, enhanced Fisher information, and contextuality. The link exists only in the strict weak-value limit, where the disturbance vanishes. A photonic experiment with intermediate measurement strength confirms the decoupling: anomalous values appear while the Pusey inequality remains satisfied, and the Fisher-information enhancement tracks the functional shape of the postselected value, not its anomaly.

What carries the argument

The machinery is the one-parameter family of measurement operators $M_0$, $M_1$ of Eq. (1), realized by a controlled-sign gate coupling the qubit to a meter qubit, with postselection after the interaction. The argument runs through two identities: the anomalous-value definition $\sigma_w = (p^c_0-p^c_1)/\kappa$, and the Fisher information $F_{\rm ps} = \kappa^2(\partial_\theta\sigma_w)^2/(1-\kappa^2\sigma_w^2)$, whose denominator shows that only $\kappa\sigma_w$ matters. The third ingredient is the Pusey inequality $I_0<0$, extended in the appendix from continuous meters to this positive-operator-valued measurement, which supplies the quantitative test for non-contextual models, where measurement outcomes are predetermined independently of the context. These pieces let the paper scan states and show that the conditions fail to align.

What would settle it

The claim would be falsified by an exhaustive numerical scan of the family of Eq. (1) at some fixed $\kappa>0$ showing that the three conditions $|\sigma_w|>1$, $F_{\rm ps}>Q$, and a Pusey violation hold together for every preparation and postselection, since the paper asserts that no such one-to-one correspondence exists; the present experiment already demonstrates the decoupling for one strength.

Watch

Extended reading notes

Core claim

Within the family of qubit measurements (1) followed by postselection on $\langle\varphi|$, the paper establishes that the three signatures are independent. The anomalous value $\sigma_w=(p^c_0-p^c_1)/\kappa$ can exceed the $[-1,1]$ spectrum of the measured observable while the Pusey-type inequality (5) is satisfied, so no non-contextual model is ruled out; the experiment observes exactly this. The conditional Fisher information $F_{\rm ps}=\kappa^2(\partial_\theta\sigma_w)^2/(1-\kappa^2\sigma_w^2)$ depends only on the combination $\kappa\sigma_w$, so its enhancement is a response to the changed functional dependence of $\sigma_w$ on $\theta$, not to the presence of anomalies. The paper's conclusion is a negative universal statement: apart from the strict weak-value regime, anomalous values, improved Fisher information, and contextuality must be assessed separately, and claims that couple them are ordering-dependent, as seen in the difference from the postselection-before-measurement scheme.

Load-bearing premise

The load-bearing premise is that the one-parameter family of qubit measurements followed by postselection, studied here, fairly represents generic postselected measurements; if another family restored a one-to-one link, the general claim would fail.

Editorial extensions

If this is right

  • For generic postselected qubit measurements, an anomalous value is not a witness of contextuality: the experiment records anomalous $\sigma_w$ while the Pusey inequality stays satisfied.
  • A conditional Fisher information above the standard quantum limit does not certify an advantage per prepared event; the postselection probability cancels the gain in total resource counting.
  • The connection between anomalies, Fisher information, and contextuality is restored only in the strict weak-value limit $\kappa\ll1$, so claims valid there cannot be extrapolated to stronger measurements.
  • The generalized measurement can be viewed as an effective non-Hermitian (PT-symmetric) measurement, with postselection corresponding to ignoring outcomes, matching behaviour reported in other settings.

Reading between the lines

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

  • Beyond the paper, the qubit-specific decoupling leaves open whether higher-dimensional meters or multi-outcome postselection could re-establish a one-to-one link at finite strength; the paper's proof technique does not obviously extend to those cases.
  • The functional form $F_{\rm ps}=\kappa^2(\partial_\theta\sigma_w)^2/(1-\kappa^2\sigma_w^2)$ suggests that metrological optimization should target the slope of $\sigma_w(\theta)$ rather than the magnitude of the anomaly, a strategy testable by varying $\kappa$ continuously.
  • The order dependence the paper notes between its scheme and the postselection-before-measurement scheme implies that contextuality claims for postselected measurements are operationally relational: any comparison must specify the ordering of measurement and postselection.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript considers postselected generalized measurements on qubits, implemented with a qubit meter coupled through a C-Sign gate, and studies the relations among three quantum signatures: anomalous values of the postselected observable, enhanced conditional Fisher information, and contextuality as probed by the Pusey inequality. The main theoretical results are Eq. (6), which expresses the postselected Fisher information in terms of kappa*sigma_w, and an extension of Pusey's non-contextuality inequality to the discrete measurement operators of Eq. (1). The paper reports an experiment measuring weak values, Fisher-information-related variances, and the Pusey quantity for two postselections, and concludes that outside the strict weak-value regime there is no one-to-one mapping among the three phenomena. The central claim is that for qubits these effects must be assessed independently.

Significance. If established, the result is a useful clarification of the relations among anomalous values, enhanced Fisher information, and contextuality in postselected quantum measurements. The derivation of Eq. (6) is clean and the extension of the Pusey inequality to a discrete qubit-meter setting is a valuable technical contribution. The experimental implementation is relevant and the paper explicitly acknowledges limitations about the scope of the no-one-to-one claim. However, the experimental support for the contextuality decoupling is weakened by the use of a non-optimal disturbance probability, and the Fisher-information comparison lacks uncertainty estimates. With these points addressed, the manuscript would be suitable for publication.

major comments (4)
  1. [Eq. (5), Appendix, Fig. 3] The Pusey-inequality test uses pd = 1 - sqrt(1 - kappa^2), but direct algebra on the operators of Eq. (1) shows that S - c|phi><phi| is positive semidefinite for every c <= (1 + sqrt(1 - kappa^2))/2, so the smallest valid disturbance probability is pd,min = (1 - sqrt(1 - kappa^2))/2. The inequality with pd,min is strictly stronger, and the caption of Fig. 3 claims the data are evaluated under 'the best circumstances for disproving the non-contextual model,' which is not the case with the reported pd. Concretely, I0_opt = I0_reported + (1 - sqrt(1 - kappa^2))/(2 p_phi), so any data point with -(1 - sqrt(1 - kappa^2))/(2 p_phi) < I0_reported < 0 would violate the optimal Pusey inequality. The conclusion that anomalous values can appear without contextuality is therefore not established unless the data are reanalyzed with the optimal pd or a clear justification for the weaker choice is provided.
  2. [Abstract and Conclusions] The abstract and introduction state that 'when performing generic postselected measurements there exist no one-to-one mapping' among anomalous values, enhanced Fisher information, and contextuality. The proof and the experimental demonstration, however, are confined to the specific two-outcome qubit-meter family of Eq. (1) with postselection after the measurement. The authors acknowledge some caveats later, but the abstract's use of 'generic' overstates the scope. Please qualify the claim in the abstract and conclusions so that it is clear the result is established for this family and this ordering, and that other measurement strategies may behave differently.
  3. [Fig. 3 and Conclusions] Satisfying the Pusey inequality is a necessary condition for the existence of a non-contextual model, not a sufficient one. The statement in the Conclusions that 'a non-contextual model becomes appropriate' is therefore too strong: the experimental data can at most show that this particular inequality does not rule out non-contextual models. Rephrase to avoid implying that a non-contextual model has been constructed or established, and make clear that the conclusion is about the failure of one specific test.
  4. [Table I and Fig. 2] The comparison between measured variances and the Cramer-Rao bound is central to the Fisher-information discussion, but Table I lists no uncertainties on the measured Delta^2 theta values, making it difficult to judge whether the differences from sigma_CR are significant. In addition, the theoretical curves in Fig. 2 rely on parameters v = 0.78, T_H = 0.98, and T_V = 0.34, whose origin is not stated in the text; the text also mentions a visibility limited to 97%, which is inconsistent with v = 0.78. Please provide error bars for Table I and specify how the imperfection parameters were determined.
minor comments (4)
  1. [Conclusions] In the Conclusions, 'more pondering' should read 'more pondered' or 'considered more carefully'.
  2. [Eq. (5) and Appendix] The notation for pd is inconsistent between Eq. (5), where it is given as 1 - sqrt(1 - kappa^2), and the Appendix, where the printed text can be read as 1 - 2*sqrt(1 - kappa^2); please harmonize the notation and ensure the intended expression is unambiguous.
  3. [Fig. 3] The figure caption should state whether the error bars on I0 include propagation of both the coincidence-count statistics and the uncertainty on kappa.
  4. [Experimental setup] The acronyms PPBS and C-Sign should be defined at first use, and the relation between the meter angle mu and kappa = sin(4 mu) should be stated clearly in the main text rather than only in the figure caption.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found: central derivation extends Pusey's inequality in a self-contained way; self-citations are background only.

full rationale

The central claim is supported by a self-contained extension of Pusey's inequality in the Appendix, which starts from the POVM elements (8), constructs S = M0|phi><phi|M0^dagger + M1|phi><phi|M1^dagger, and proves the inequality without invoking the paper's own conclusions. The contextuality test in Eq. (5) uses directly measured joint probabilities and an analytically computed pd; no fitted parameter is renamed as a prediction. The Fisher-information relation Eq. (6) is an algebraic identity following from the definition of sigma_w in Eq. (3), so the reported tenuous link is a consequence of definitions, not a circular import. Self-citations (Refs. [27], [29], [32], [46]) are used as background or for previously established experimental context; none is load-bearing for the no-one-to-one mapping claim. The skeptic's point that the pd used in Eq. (5) may not be the minimal disturbance probability concerns the tightness of the contextuality test, not circularity: a weaker valid inequality does not reduce the derivation to its inputs. The paper itself acknowledges the limitation that inequality (5) 'does not exhaust all possible scenarios,' further indicating that the claim is empirical and conditional rather than circular.

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

The central derivation rests on standard quantum measurement and Fisher information formalism, plus Pusey's non-contextuality assumptions. The only adjustable inputs are three experimental imperfection parameters used to model the data curves; their provenance (calibration vs fit) is not stated.

free parameters (3)
  • Visibility v = 0.78
    Used to model the reduced interference visibility in the theoretical predictions of Fig. 2; the text does not state whether this is calibrated independently or fitted.
  • PPBS transmission TH = 0.98
    Used to model imperfect beam-splitter transmission in the theoretical predictions; not stated whether measured or fitted.
  • PPBS transmission TV = 0.34
    Used to model imperfect beam-splitter transmission in the theoretical predictions; not stated whether measured or fitted.
assumptions (3)
  • domain assumption Sharp measurements have deterministic outcomes in ontic models
    Used in the Appendix to derive the Pusey inequality (5) for the generalized measurement; this is the standard assumption in Pusey's non-contextuality framework.
  • standard math The measurement operations are described by POVM elements in Eq. (8)
    Basis for the derivation of the inequality; standard quantum measurement formalism.
  • standard math Fisher information for postselected conditional probabilities follows Eq. (4)
    Definition of Fisher information used to derive Eq. (6); standard statistics.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Anomalous values, Fisher information, and contextuality, in generalized quantum measurements." pith.science (2026). https://pith.science/paper/OARFP7QH

@misc{pith2026190900755,
  author       = {Pith},
  title        = {Pith review of: Anomalous values, Fisher information, and contextuality, in generalized quantum measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OARFP7QH}},
  note         = {Machine review of arXiv:1909.00755}
}
read the original abstract

Postselection following weak measurements has long been investigated for its peculiar manifestation of quantum signatures. In particular, the postselected events can give rise to anomalous values lying outside the spectrum of the measured quantity, and may provide enhanced Fisher information. Furthermore, the Pusey inequality highlights that, for extremely weak measurements, non-contextual models can account for the outcome probabilities. It is then interesting to investigate whether these are linked in a unified framework. Here we discuss on the existence of a possible connection in the case of qubits. We show that when performing generic postselected measurements there exist no one-to-one mapping between them, an instance that leads to drawing more involved considerations.

Figures

Figures reproduced from arXiv: 1909.00755 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Measured weak values for both postselections. Each [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Verification of Pusey inequality for both postselec [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 35 canonical work pages

  1. [1]

    consolidated measurement

    (9) This can be decomposed as S = (1−pd)|φ⟩⟨φ| +pdEd, (10) for some POVM with elementsEd, andI−Ed. Explicit calculations show thatpd = 1− 2 √ 1−κ2. Finally, we consider the cases for whichpφ > 0. Under these circumstances, one can demonstrate that non-contextualmodelsimplytheinequality (5); theproof follows closely the original one due to Pusey [28], with...

    work page

  2. [2]

    Aharonov, D

    Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)

    1988

  3. [3]

    G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, Phys. Rev. Lett.94, 220405 (2005)

    work page 2005

  4. [4]

    A. J. Leggett, Phys. Rev. Lett.62, 2325 (1989)

    work page 1989

  5. [5]

    Peres, Phys

    A. Peres, Phys. Rev. Lett.62, 2326 (1989)

    work page 1989

  6. [6]

    I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D40, 2112 (1989)

    1989

  7. [7]

    Vaidman, Philosophical Transactions of the Royal So- ciety A: Mathematical, Physical and Engineering Sci- ences 375, 20160395 (2017)

    L. Vaidman, Philosophical Transactions of the Royal So- ciety A: Mathematical, Physical and Engineering Sci- ences 375, 20160395 (2017)

    work page 2017

  8. [8]

    Vaidman, A

    L. Vaidman, A. Ben-Israel, J. Dziewior, L. Knips, M. Weißl, J. Meinecke, C. Schwemmer, R. Ber, and H. Weinfurter, Phys. Rev. A96, 032114 (2017)

    2017

Show all 50 references
  1. [9]

    Dressel, M

    J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Rev. Mod. Phys.86, 307 (2014)

    2014

  2. [10]

    N. K. Langford, T. J. Weinhold, R. Prevedel, K. J. Resch, A. Gilchrist, J. L. O’Brien, G. J. Pryde, and A. G. White, Phys. Rev. Lett.95, 210504 (2005)

    2005

  3. [11]

    Kiesel, C

    N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. We- infurter, Phys. Rev. Lett.95, 210505 (2005)

    2005

  4. [12]

    R.Okamoto, H.F.Hofmann, S.Takeuchi, andK.Sasaki, Phys. Rev. Lett.95, 210506 (2005)

    2005

  5. [13]

    Mancino, M

    L. Mancino, M. Sbroscia, E. Roccia, I. Gianani, F. Somma, P. Mataloni, M. Paternostro, and M. Barbi- eri, NPJ Quantum Information4, 20 (2018)

    2018

  6. [14]

    Aharonov, S

    Y. Aharonov, S. Popescu, D. Rohrlich, and P. Skrzypczyk, New Journal of Physics 15, 113015 (2013)

    2013

  7. [15]

    Denkmayr, H

    T. Denkmayr, H. Geppert, S. Sponar, H. Lemmel, A. Matzkin, J. Tollaksen, and Y. Hasegawa, Nature Communications 5, 4492 EP (2014)

    2014

  8. [16]

    Danan, D

    A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, Phys. Rev. Lett.111, 240402 (2013). 6

    2013

  9. [17]

    Corrêa, M

    R. Corrêa, M. F. Santos, C. H. Monken, and P. L. Sal- danha, New Journal of Physics17, 053042 (2015)

    2015

  10. [18]

    Ferrie and J

    C. Ferrie and J. Combes, Phys. Rev. Lett.113, 120404 (2014)

    2014

  11. [19]

    Vaidman, arXiv:1409.5386 (2014)

    L. Vaidman, arXiv:1409.5386 (2014)

    2014 arXiv

  12. [20]

    Aharonov and D

    Y. Aharonov and D. Rohrlich, arXiv:1409.8555 (2014)

    2014 arXiv

  13. [21]

    Hofmann, M

    H. Hofmann, M. Iinuma, and Y. Shikano, arXiv:1410.7126 (2014)

    2014 arXiv

  14. [22]

    Brodutch, Phys

    A. Brodutch, Phys. Rev. Lett.114, 118901 (2015)

    2015

  15. [23]

    Ferrie and J

    C. Ferrie and J. Combes, Phys. Rev. Lett.114, 118902 (2015)

    2015

  16. [24]

    A. N. Jordan, A. N. Korotkov, and M. Büttiker, Phys. Rev. Lett. 97, 026805 (2006)

    2006

  17. [25]

    N. S. Williams and A. N. Jordan, Phys. Rev. Lett.100, 026804 (2008)

    2008

  18. [26]

    Palacios-Laloy, F

    A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet, D. Vion, D. Esteve, and A. N. Korotkov, Nature Physics 6, 442 EP (2010)

    2010

  19. [27]

    M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J.L.O’Brien, A.G.White, andG.J.Pryde,Proceedings of the National Academy of Sciences108, 1256 (2011)

    2011

  20. [28]

    Avella, F

    A. Avella, F. Piacentini, M. Borsarelli, M. Barbieri, M. Gramegna, R. Lussana, F. Villa, A. Tosi, I. P. De- giovanni, and M. Genovese, Phys. Rev. A 96, 052123 (2017)

    2017

  21. [29]

    M. F. Pusey, Phys. Rev. Lett.113, 200401 (2014)

    2014

  22. [30]

    Piacentini, A

    F. Piacentini, A. Avella, M. P. Levi, R. Lussana, F. Villa, A. Tosi, F. Zappa, M. Gramegna, G. Brida, I. P. Degio- vanni, and M. Genovese, Phys. Rev. Lett.116, 180401 (2016)

    2016

  23. [31]

    J. P. Torres and L. J. Salazar-Serrano, Scientific Reports 6, 19702 EP (2016)

    2016

  24. [32]

    Zhang, A

    L. Zhang, A. Datta, and I. A. Walmsley, Phys. Rev. Lett. 114, 210801 (2015)

    2015

  25. [33]

    H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, Phys. Rev. A86, 040102 (2012)

    2012

  26. [34]

    Ferrie and J

    C. Ferrie and J. Combes, Phys. Rev. Lett.112, 040406 (2014)

    2014

  27. [35]

    Hosten and P

    O. Hosten and P. Kwiat, Science319, 787 (2008)

    2008

  28. [36]

    P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett.102, 173601 (2009)

    2009

  29. [37]

    Pang and T

    S. Pang and T. A. Brun, Phys. Rev. Lett.115, 120401 (2015)

    2015

  30. [38]

    A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, Phys. Rev. X4, 011031 (2014)

    2014

  31. [39]

    G. C. Knee and E. M. Gauger, Phys. Rev. X4, 011032 (2014)

    2014

  32. [40]

    Harris, R

    J. Harris, R. W. Boyd, and J. S. Lundeen, Phys. Rev. Lett. 118, 070802 (2017)

    2017

  33. [41]

    Feizpour, M

    A. Feizpour, M. Hallaji, G. Dmochowski, and A. M. Steinberg, Nature Physics11, 905 EP (2015)

    2015

  34. [42]

    Dziewior, L

    J. Dziewior, L. Knips, D. Farfurnik, K. Senkalla, N. Ben- shalom, J. Efroni, J. Meinecke, S. Bar-Ad, H. Weinfurter, and L. Vaidman, Proceedings of the National Academy of Sciences 116, 2881 (2019)

    2019

  35. [43]

    D. R. Arvidsson-Shukur, N. Y. Halpern, H. V. Lep- age, A. A. Lasek, C. H. Barnes, and S. Lloyd, arXiv:1903.02563 (2019)

    2019 arXiv

  36. [44]

    T. C. Ralph, S. D. Bartlett, J. L. O’Brien, G. J. Pryde, and H. M. Wiseman, Phys. Rev. A73, 012113 (2006)

    2006

  37. [45]

    G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, Phys. Rev. Lett.92, 190402 (2004)

    2004

  38. [46]

    M. G. A. Paris, International Journal of Quantum Infor- mation 07, 125 (2009)

    2009

  39. [47]

    Altorio, M

    M. Altorio, M. G. Genoni, M. D. Vidrighin, F. Somma, and M. Barbieri, Phys. Rev. A92, 032114 (2015)

    2015

  40. [48]

    C. M. Bender and S. Boettcher, Phys. Rev. Lett.80, 5243 (1998)

    1998

  41. [49]

    Y. Wu, W. Liu, J. Geng, X. Song, X. Ye, C.-K. Duan, X. Rong, and J. Du, arXiv:1812.05226 (2018)

    2018 arXiv

  42. [50]

    Naghiloo, M

    M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. W. Murch, arXiv:1901.07968 (2019)

    2019 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.