Pith. sign in

REVIEW 4 major objections 4 minor 41 references

ND-Photonic QRNGs in a Noisy Environment

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 This paper proves that a photonic 3D QRNG with imperfect, unsharp measurements still produces maximally unpredictable outputs, because the sharp pointer outcome—not the unsharp system observable—carries the value-indefiniteness guarantee.

desk verdict Theorem 5's pointer-state strategy is a genuinely new angle for unsharp KS, but its proof rests on an unproven purity assumption; the paper deserves review, yet its central guarantee is conditional. read the letter →

arxiv 2608.10053 v1 pith:IPQVYSQ3 submitted 2026-08-10 quant-ph

J. M. Agüero Trejo , Cristian S. Calude , O. C. Stoica This is my paper · ORCID
classification quant-ph PACS 03.65.Ta42.50.Ex
keywords quantumrandomnumbergeneratorvalueindefinitenessKochen-SpeckertheoremphotonicimplementationunsharpmeasurementStinespringdilationdepolarizingchannelrandomnesscertification
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

This paper asks whether a photonic three-dimensional quantum random number generator (QRNG) still produces maximally unpredictable outputs when its components are imperfect—lossy beam splitters, imprecise phase shifters, detector noise, and unsharp measurements generally. The authors try to establish that the guarantee survives: under a pure-total-state assumption and the observable fact that at least two macroscopically distinct pointer outcomes occur with nonzero probability, the pointer observable is value indefinite even though the measured system observable is unsharp. If true, this means the strongest known certification of quantum randomness, based on the Located Kochen-Specker Theorem, applies to a practical integrated-photonics device that does not need cryogenic cooling. The paper also models the effect of a depolarizing channel on the output distribution, giving explicit probability shifts that in the ternary protocol become $1/4+3p/32$, $1/2-3p/16$, and $1/4+3p/32$.

What carries the argument

The machinery is the Located Kochen-Specker Theorem (Theorem 1) applied to pointer states, together with the measurement-dilation theorem (a Stinespring dilation) showing that the actual measurement taking place on system-plus-pointer is sharp even when the observable inferred on the system alone is unsharp. The paper also uses the decomposition of an imperfect Mach–Zehnder interferometer into error-offset angles $\alpha,\beta$ and the depolarising channel with Kraus operators on a qutrit to estimate output distributions. In equations (10)–(11) the purity assumption on the total state $|\Psi\rangle$ yields the crucial inequality $0<|\langle\Psi|\Phi\rangle|<1$ from $|\Psi\rangle=a|\Phi\rangle+a'|\Phi'\rangle+\dots$ with mutually orthogonal macroscopic pointer branches.

What would settle it

Run the device many times with identical preparations and record the pointer-outcome statistics: if a single detector or pointer branch fires with probability exactly 1 (or never fires) across all runs, the inequality $0<|\langle\Psi|\Phi\rangle|<1$ fails and the certified unpredictability is lost. Alternatively, a controlled test in which the pointer is coupled to a thermal environment that demonstrably produces a mixed total state should, if the paper's claim is wrong, allow a classical predictor to beat the value-indefiniteness bound.

Watch

Extended reading notes

Core claim

The central claim is Theorem 5, an unsharp version of the Located Kochen-Specker Theorem: for possibly unsharp measurements, if identically prepared runs produce nontrivial probabilities in at least two macroscopically distinct total pointer states $|\Phi\rangle$ and $|\Phi'\rangle$, then the projection observable $P_\Phi$ is value indefinite. The argument applies the original Located Kochen-Specker Theorem not to the state of the observed photonic system but to the pointer states in the total Hilbert space of system plus measuring device. The key step is that the pointer observable is always sharp—macroscopically distinct states are orthogonal—so the only needed condition is $0<|\langle\Psi|\Phi\rangle|<1$, which follows from the purity of the total state and the coexistence of two nonzero-outcome branches. Consequently the random digit is generated by the sharp pointer outcome, not by the unsharp observable of the photons, and the restrictive commutativity assumption of earlier unsharp Kochen-Specker results is not needed.

Load-bearing premise

The whole proof hinges on there physically existing a larger system—the observed photons plus the pointer plus any environment—whose total state is pure before and after the measurement; if the total state is a mixture, Theorem 5 does not follow.

Editorial extensions

If this is right

  • For a noisy photonic 3D QRNG, the random digit is certified to be maximally unpredictable even with unsharp measurements, because the pointer outcome is the sharp observable that satisfies the Located Kochen-Specker Theorem.
  • The certification reduces to an empirically checkable condition: identical preparations must yield nonzero probability for at least two macroscopically distinct pointer outcomes.
  • Under a depolarising-channel noise model, the ternary output probabilities become $1/4+3p/32$, $1/2-3p/16$, and $1/4+3p/32$, and the binary probabilities become $1/2\pm 3p/16$, so the bias is computable and can be compensated.
  • The result removes the need for the commutativity assumption in earlier unsharp Kochen-Specker arguments, widening the class of noisy devices that can be certified.
  • The same argument applies to any unitarily equivalent operator with the same eigenstates, so the implementation choice can be optimised to avoid forbidden beam-splitter splitting ratios without losing the guarantee.

Reading between the lines

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

  • If the pure-total-state assumption is the true load-bearing premise, then a practical certification protocol should also verify purity of the joint state; without it, a mixed-state adversary model could restore partial predictability even when two pointer branches fire.
  • The same pointer-state reasoning could in principle certify value indefiniteness for other non-cryogenic platforms such as trapped ions or solid-state defects, since only macroscopic pointer distinctness and total-state purity are needed, not the specific photonic hardware.
  • Treating unsharpness as an observer's restriction rather than a physical blur suggests a possible connection between the generalised contextuality of single POVMs and pointer value indefiniteness; the paper hints at this but does not develop a proof.
  • A direct experimental extension would be to measure the output-distribution shifts as a function of the estimated error rate $p$ and compare with the depolarising-channel prediction; agreement would support the model, while systematic deviation would reveal non-depolarising noise such as correlated losses that the current analysis does not cover.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 paper studies a photonic implementation of a 3D quantum random number generator whose security is certified by the Located Kochen-Specker Theorem, and investigates whether measurement unsharpness and depolarising noise destroy that certification. The authors model the measurement as an open quantum system, invoke the Ozawa/Stinespring measurement dilation to argue that the physically sharp observable is the pointer observable rather than the unsharp observable of the observed system, and state Theorem 5, an "unsharp Located Kochen-Specker Theorem" intended to guarantee value indefiniteness of the pointer outcome whenever identically prepared runs produce nontrivial probabilities in two macroscopically distinct pointer states. The paper also models errors as a depolarising channel on a qutrit and computes the resulting outcome probabilities for the ternary and binary protocols.

Significance. If fully established, the paper would provide a valuable extension of the security argument for photonic 3D QRNGs, showing that at least one class of practical imperfections neither destroys value indefiniteness nor changes the desired outcome distribution beyond a calculable bias. The strategy of applying the Located Kochen-Specker Theorem to pointer states rather than to the unsharp observable is well motivated, and the discussion of why Breuer's unsharp Kochen-Specker theorems are insufficient is informative. The depolarising-channel calculation in Section 9 is correct and useful for estimating the outcome distribution. However, the central Theorem 5 is proven only under a pure-total-state assumption that is not derived from the device physics, and the transfer of the Kochen-Specker conditions to pointer states is not justified in detail. The significance is therefore conditional on closing that gap.

major comments (4)
  1. [Section 8, Assumption 1 and Theorem 5] The proof of Theorem 5 depends crucially on the total state of system plus pointer being a pure vector |Ψ⟩, from which Eq. (10) yields 0<|⟨Ψ|Φ⟩|<1. The paper does not derive this purity from the physical model; the justification that "in most formulations of quantum mechanics, at least the total state of the universe is assumed to be a pure state" is an interpretive premise, not a consequence of the device dynamics or of the proposed verification tests. For a mixed total state such as ρ = 1/2|Φ⟩⟨Φ| + 1/2|Φ′⟩⟨Φ′| on the pointer subspace, both pointer probabilities are nonzero, yet the pointer context admits the admissible assignment v(P_Φ)=1, v(P_Φ′)=0, v(P_Φ″)=0; the eigenstate principle has no mixed-state counterpart, so value indefiniteness does not follow. Since Theorem 5 is the load-bearing result for the noise-robustness claim, this gap must be addressed, either by deriving purity from the experimental preparation or by reformulating the theorem for mixed states.
  2. [Section 8, proof of Theorem 5] The proof applies Theorem 1 to the pointer states but does not verify that the three hypotheses of Theorem 1—admissibility, non-contextuality, and the eigenstate principle—hold for the pointer observables in the noisy setting. In particular, the eigenstate principle is a condition on prepared states, whereas the pointer state after the measurement is a correlated state of system and apparatus; the paper needs a dedicated argument that a value assignment function restricted to the pointer projectors is admissible and non-contextual on the total Hilbert space H = H_S ⊗ H_A. Merely having 0<|⟨Ψ|Φ⟩|<1 is not sufficient to invoke Theorem 1 without explicitly checking its hypotheses.
  3. [Section 4.2, Theorem 2] Theorem 2 is stated without proof, and its conclusion that "unsharp measurements do not affect the value indefiniteness of the outcomes" is ambiguous because it does not distinguish between the observed-system observable and the pointer observable. The proof would need to establish exactly the transfer of value indefiniteness to pointer states that Theorem 5 later attempts, but no proof is provided and the logical relation between Theorem 2 and Theorem 5 is not spelled out. As written, Theorem 2 is an unsupported assertion.
  4. [Section 9, Eqs. (27)-(28)] The depolarising-channel calculation is algebraically correct, but it does not by itself show that the conditions of Theorem 5 are satisfied. Equations (27)-(28) give outcome probabilities for the reduced state of the qutrit, whereas the inequality 0<|⟨Ψ|Φ⟩|<1 in Theorem 5 requires a global pure state of system plus pointer; the paper does not verify that, for some Stinespring dilation compatible with the depolarising channel, the pointer states have nonzero and non-unit overlap with the global state. The calculation is thus a useful estimate of outcome bias but not a proof that the value-indefiniteness guarantee survives the noise.
minor comments (4)
  1. [Section 3, Theorem 1] The statement of Theorem 1 contains an incomplete sentence: "for every value assignment function v: O → {0,1} If the above three conditions are satisfied..." should read "... {0,1}, if the above three conditions are satisfied, then...".
  2. [Section 8, Eq. (10)] The second inner product in Eq. (10) is typeset incorrectly; the term |Ψ|Φ′> should be |⟨Ψ|Φ′⟩|².
  3. [Section 10] The concluding paragraph says "the depolarising channel acting on a single qubit", but the analysis in Section 9 treats a qutrit; this should be corrected.
  4. [Section 9] The Kraus operators A–H in Eq. (24) are said to represent the depolarising channel, but it is not stated that they form a unitary error basis for qutrits; adding this clarification would help readers verify the normalization of the channel.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity: Theorem 5 applies the independently published Located Kochen-Specker Theorem under an explicit purity assumption; minor self-citation only for deferred experimental verification.

full rationale

The paper's central new result, Theorem 5, is not derived from its own assumptions by construction. It assumes an explicitly stated pure total state (Sec. 8, Assumption 1) and nontrivial pointer probabilities, uses Eqs. (10)-(11) to obtain 0<|⟨Ψ|Φ⟩|<1, and then invokes Theorem 1, the Located Kochen-Specker Theorem, which is an externally published mathematical result by Abbott, Calude and Svozil. The conclusion that P_Φ is value indefinite does not appear in the assumptions; the nontrivial-probabilities premise alone is compatible with non-contextual definite values on a mixed state, so the purity assumption is genuinely additional and load-bearing. That means Theorem 5 is conditional and its physical scope is limited, but the inference is a legitimate logical derivation rather than a renaming or a fitted prediction. The paper's reliance on [10] (an unpublished same-author CDMTCS report) for the experimental verification of unitarity is a minor self-citation, but it does not enter the proof chain of Theorem 5; it supports the separate claim that the device can be tested. No parameter is fitted to force value indefiniteness, and no equation reduces to the target claim. Hence no significant circularity is present; score 2 reflects only the minor self-citation for deferred experimental validation.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on one fitted parameter (p) and on several substantive domain assumptions. No new particles, forces, or dimensions are introduced; the ancilla and environment in the Stinespring dilation are presented as physical parts of the measuring device, not invented entities.

free parameters (1)
  • p (error probability)
    The depolarising channel outcome probabilities (Eqs. 27-28) depend linearly on p, which is a free parameter 'estimated by assessing the efficiency of the physical implementation'. No calibration data are provided.
assumptions (6)
  • domain assumption The total state of the universe (including observed system, pointer, and environment) is pure before and after the measurement.
    Assumption 1 in Section 8; used to derive 0<|⟨Ψ|Φ⟩|<1 via Eq. (10). If the total state is mixed, Theorem 5 does not follow.
  • domain assumption Macroscopically distinguishable pointer states are orthogonal.
    Remark 1 in Section 8; used to expand |Ψ> as a sum over mutually orthogonal pointer eigenstates and to guarantee that the pointer observable is sharp.
  • domain assumption The Located Kochen-Specker Theorem applies to pointer states in the total Hilbert space, including admissibility, non-contextuality, and the eigenstate principle.
    Invoked in the proof of Theorem 5. These conditions are assumed to hold for the extended pointer observable without a dedicated proof.
  • domain assumption Photon losses and multi-photon events can be ignored without biasing the output when a heralded single-photon source is used.
    Section 3, paragraph on loss. The claim that undetected or double-click events can simply be discarded is load-bearing for the noise model but not experimentally demonstrated here.
  • ad hoc to paper The effect of experimental imperfections can be modeled as a depolarising channel on a qutrit.
    Eqs. (23)-(26). This is a modeling choice presented to 'broadly model' errors; it is not derived from first principles or from measured component fidelities in this paper.
  • domain assumption Stinespring dilation is physically realized by the measurement process (measurement dilation).
    Section 7, Remark 7. The paper argues that the larger Hilbert space is not fictitious, which is necessary for Theorem 5 to have physical meaning. This is a substantive interpretative assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of ND-Photonic QRNGs in a Noisy Environment." pith.science (2026). https://pith.science/paper/IPQVYSQ3

@misc{pith2026260810053,
  author       = {Pith},
  title        = {Pith review of: ND-Photonic QRNGs in a Noisy Environment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPQVYSQ3}},
  note         = {Machine review of arXiv:2608.10053}
}
read the original abstract

Standard pseudo-random generators have weaknesses that have led to the development of quantum random number generators (QRNGs). However, common QRNG validation methods, whether based on quantum indeterminism or statistical tests, are insufficient to guarantee high-quality randomness. In contrast, a mathematical theory based on the Located Kochen-Specker Theorem proves that 3D QRNGs produce maximally unpredictable outputs without using entanglement, and both theory and experiments have supported their security. The paper focuses on a practical photonic implementation of a 3D QRNG that is easier to deploy than cryogenic superconducting implementations. As any physical implementation is subject to various measurement errors, it is important to study theoretically and experimentally the type and role of errors in 3D QRNGs. In this paper, we will model the photonic 3D QRNG as an open quantum system, constructed as an arrangement of imperfect beam-splitters with a range of losses based on the fidelity of its components, and we will show that under certain conditions, the process remains within the scope of the Kochen-Specker Theorem, which guarantees maximum unpredictability.

Figures

Figures reproduced from arXiv: 2608.10053 by the authors.

Figure 1
Figure 1. A and B arrangements. where ϱ is a positive semi-definite operator acting on the Hilbert space H such that Φ(ϱ) = X i KiϱK † i and X i K † iKi ≤ I. (21) Note that for ϱ 2 = ϱ, we have the projection operator; that is, there exists a |ψ⟩ ∈ H such that ϱ = |ψ⟩ ⟨ψ|. We call {Ki} a Kraus operator which need not be unitary (note that if it is unitary we have ϱ 7→ UϱU † ). Theorem 6 (Stinespring dilation [39]) Let A be a … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 36 canonical work pages

  1. [1]

    A. A. Abbott, C. S. Calude, J. Conder, and K. Svozil. Strong Kochen-Specker theorem and incom- putability of quantum randomness.Physical Review A, 86(062109), Dec 2012

    work page 2012

  2. [2]

    A. A. Abbott, C. S. Calude, and K. Svozil. Value indefiniteness is almost everywhere.Physical Review A, 89(3):032109–032116, 2014

    2014

  3. [3]

    A. A. Abbott, C. S. Calude, and K. Svozil. A non-probabilistic model of relativised predictability in physics.Information, 6(4):773–789, 2015. 18

    work page 2015

  4. [4]

    A. A. Abbott, C. S. Calude, and K. Svozil. A variant of the Kochen-Specker theorem localising value indefiniteness.Journal of Mathematical Physics, 56:102201, Oct 2015

    work page 2015

  5. [5]

    J. M. Agüero Trejo and C. S. Calude. A new quantum random number generator certified by value indefiniteness.Theoretical Computer Science, 862:3–13, Mar. 2021

    work page 2021

  6. [6]

    J. M. Agüero Trejo and C. S. Calude. Photonic ternary quantum random number generators.Proc. R. Soc. A, 479:1–16, 2023

    work page 2023

  7. [7]

    J. M. Agüero Trejo and C. S. Calude. AnN–dimensional quantum random number generator. Report CDMTCS-583, Centre for Discrete Mathematics and Theoretical Computer Science, Uni- versity of Auckland, Auckland, New Zealand, May 2025

    work page 2025

  8. [8]

    J. M. Agüero Trejo and C. S. Calude. Photonic ternary quantum random number genera- tors.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 479(2273):20220543, May 2023

    work page 2023

Show all 41 references
  1. [9]

    J. M. Agüero Trejo, C. S. Calude, M. J. Dinneen, A. Fedorov, A. Kulikov, R. Navarathna, and K. Svozil. How real is incomputability in physics?Theoretical Computer Science, 1003:114632, 2024

    2024

  2. [10]

    J. M. Agüero Trejo, C. S. Calude, and O. C. Stoica. Testing the 3D QRNG by undoing. Report CDMTCS-587, Centre for Discrete Mathematics and Theoretical Computer Science, University of Auckland, Auckland, New Zealand, Dec. 2025

    2025

  3. [11]

    Araki and M

    H. Araki and M. Yanase. Measurement of quantum mechanical operators.Phys. Rev., 120(2):622, 1960

    1960

  4. [12]

    Beneduci

    R. Beneduci. Positive operator valued measures and feller markov kernels.J. Math. Anal. Appl., 442(1):50–71, 2016

    2016

  5. [13]

    T. Breuer. Kochen-Specker theorem for finite precision spin-one measurements.Physical review letters, 88(24):240402, 2002

    2002

  6. [14]

    T. Breuer. A Kochen-Specker theorem for unsharp spin 1 observables. InNon-locality and Modal- ity, pages 195–203. Springer, Heidelberg, 2002

    2002

  7. [15]

    T. Breuer. Another no-go theorem for hidden variable models of inaccurate spin 1 measurements. Philosophy of Science, 70(5):1368–1379, 2003

    2003

  8. [16]

    No information without disturbance

    P. Busch. “No information without disturbance”: Quantum limitations of measurement. In W. Myr- vold, J. Christian, and P. Pearle, editors,Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, volume 73 ofWestern Ontario Series in Philosophy of Science, pag...

    2009

  9. [17]

    Die Messung quantenmechanischer Operatoren

    P. Busch. Translation of “Die Messung quantenmechanischer Operatoren” by EP Wigner.Arxiv preprint quant-ph/1012.4372, pages 1–10, December 2010. arXiv:quant-ph/1012.4372

    2010 arXiv

  10. [18]

    Busch, P

    P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen.Quantum measurement, volume 23. Springer, Switzerland, 2016

    2016

  11. [19]

    A. Cabello. Finite-precision measurement does not nullify the Kochen-Specker theorem.Phys. Rev. A, 65(5):052101, 2002

    2002

  12. [20]

    C. S. Calude and K. Svozil. Binary quantum random number generator based on value indefinite observables.Scientific Reports, 14(1):12845, June 2024

    2024

  13. [21]

    Cilluffo

    D. Cilluffo. Commentary on the decomposition of universal multiport interferometers: how it works in practice.Arxiv preprint quant-ph/2412.11955, pages 1–13, 2024. arXiv:quant- ph/2412.11955. 19

    2024 arXiv

  14. [22]

    W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley. Optimal design for universal multiport interferometers.Optica, 3(12):1460–1465, 2016

    2016

  15. [23]

    d’Espagnat.Veiled reality: An analysis of present-day quantum mechanical concepts

    B. d’Espagnat.Veiled reality: An analysis of present-day quantum mechanical concepts. CRC Press, Boca Raton, FL, USA, 2018

    2018

  16. [24]

    d’Espagnat.Conceptual Foundations of Quantum Mechanics

    B. d’Espagnat.Conceptual Foundations of Quantum Mechanics. W.A. Benjamin, New York, 1976

    1976

  17. [25]

    d’Espagnat.On physics and philosophy

    B. d’Espagnat.On physics and philosophy. Princeton University Press, Princeton, NJ, 2006

    2006

  18. [26]

    Grudka and P

    A. Grudka and P. Kurzy ´nski. Is there contextuality for a single qubit?Phys. Rev. Lett., 100(16):160401, 2008

    2008

  19. [27]

    Heisenberg.Physics and Philosophy: The Revolution in Modern Science

    W. Heisenberg.Physics and Philosophy: The Revolution in Modern Science. Harper, New York, 1958

    1958

  20. [28]

    Holevo.Probabilistic and statistical aspects of quantum theory, volume 1 ofPSNS

    A. Holevo.Probabilistic and statistical aspects of quantum theory, volume 1 ofPSNS. Springer Basel, Basel, Switzerland, 2011

    2011

  21. [29]

    What is the Q in QRNG?idQuan- tique, Geneva, Switzerland, May 2020

    ID Quantique SA.Random Number Generation – White Paper. What is the Q in QRNG?idQuan- tique, Geneva, Switzerland, May 2020

    2020

  22. [30]

    Kulikov, M

    A. Kulikov, M. Jerger, A. Potoˇcnik, A. Wallraff, and A. Fedorov. Realization of a quantum random generator certified with the Kochen-Specker theorem.Phys. Rev. Lett., 119:240501, Dec 2017

    2017

  23. [31]

    R. Kunjwal. Beyond the Cabello-Severini-Winter framework: Making sense of contextuality with- out sharpness of measurements.Quantum, 3:184, 2019

    2019

  24. [32]

    Landsman.Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, volume 188 ofFundamental Theories of Physics

    K. Landsman.Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, volume 188 ofFundamental Theories of Physics. Springer, Cham, Switzerland, 2017

    2017

  25. [33]

    Loveridge

    L. Loveridge. A relational perspective on the Wigner-Araki-Yanase theorem.Journal of Physics: Conference Series, 1638(1):012009, 2020

    2020

  26. [34]

    Measurement of quantum mechanical operators

    L. Loveridge and P. Busch. “Measurement of quantum mechanical operators” revisited.Eur. Phys. J. D, 62(2):297–307, 2011

    2011

  27. [35]

    Mannalath, S

    V . Mannalath, S. Mishra, and A. Pathak. A comprehensive review of quantum random number gen- erators: concepts, classification and the origin of randomness.Quantum Information Processing, 22:439, 2023

    2023

  28. [36]

    M. Naimark. About second-kind self-adjoint extensions of symmetrical operator.Izv. Akad. Nauk USSR, Ser. Mat, 4:53–104, 1940

    1940

  29. [37]

    M. Ozawa. Quantum measuring processes of continuous observables.J. Math. Phys., 25(1):79–87, 1984

    1984

  30. [38]

    Spekkens

    R. Spekkens. Contextuality for preparations, transformations, and unsharp measurements.Phys. Rev. A, 71(5):052108, 2005

    2005

  31. [39]

    Stinespring

    W. Stinespring. Positive functions on C*-algebras.Proc. Amer. Math. Soc., 6(2):211–216, 1955

    1955

  32. [40]

    O. Stoica. Can we accurately read or write quantum data?Physica Scripta, 100(8):085108, 2025

    2025

  33. [41]

    E. Wigner. Die Messung quantenmechanischer Operatoren.Zeitschrift für Physik A Hadrons and nuclei, 133(1):101–108, 1952. 20

    1952

Pith tools

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