Quantum contextuality from measurement invasiveness

Andrea Navoni; Andrea Smirne; Marco G. Genoni

arxiv: 2507.16942 · v2 · submitted 2025-07-22 · 🪐 quant-ph

Quantum contextuality from measurement invasiveness

Andrea Navoni , Marco G. Genoni , Andrea Smirne This is my paper

Pith reviewed 2026-05-19 03:06 UTC · model grok-4.3

classification 🪐 quant-ph

keywords contextualityinvasive measurementsstochastic linear mapsnoncontextuality polytopethree-level systemquantum mechanicsmarginal scenarioscontextuality quantifier

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The pith

Contextuality arises when invasive measurements are applied to otherwise classical probability distributions via stochastic linear maps.

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

The paper develops a method to understand quantum contextuality by modeling measurements as invasive operations on classical statistics. These operations are represented by stochastic linear maps that maintain the compatibility between different measurements but can produce probability distributions that violate noncontextuality conditions. By deriving the conditions these maps must obey and fully characterizing them for a three-level system, the authors also define a quantifier for the amount of contextuality based on how invasive the measurements need to be. This offers a new perspective on why quantum systems exhibit contextuality without requiring inherently nonclassical hidden variables.

Core claim

We introduce stochastic linear maps that model invasive measurements on classical statistics. These maps take probabilities inside the noncontextuality polytope and map them to points outside while preserving the compatibility structure of the marginal scenario. We derive consistency conditions for such maps to be admissible and completely identify the maps for the case of a single three-level quantum system. We further define a contextuality quantifier as the minimal invasiveness required to reproduce a given distribution.

What carries the argument

stochastic linear maps modeling invasive measurements that preserve compatibility but exit the noncontextuality polytope

If this is right

Contextual behavior can be explained without abandoning an underlying classical joint distribution.
The degree of contextuality is quantifiable by the smallest invasiveness parameter needed.
This framework applies to any marginal scenario where compatibility is defined.
Classical models with controlled invasiveness can mimic quantum predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Such maps might allow simulation of quantum contextuality in classical systems with added noise or disturbance.
Extensions could connect this invasiveness measure to other contextuality quantifiers like the contextual fraction.
This view suggests testing how much measurement disturbance is present in actual quantum experiments to explain observed contextuality.

Load-bearing premise

Stochastic linear maps exist that satisfy the consistency conditions for representing admissible invasive measurements while preserving the compatibility structure.

What would settle it

A probability distribution for the three-level system that exhibits contextuality but cannot be obtained from any classical distribution inside the polytope by applying the identified stochastic maps.

Figures

Figures reproduced from arXiv: 2507.16942 by Andrea Navoni, Andrea Smirne, Marco G. Genoni.

**Figure 1.** Figure 1: (Left) Action of the affine map (Z, ⃗v) obtained from an IMM W via Eq.(11): a vector of classical probabilities ⃗c, within the noncontextuality polytope, is mapped to a vector of quantum probabilities ⃗q, possibly outside the polytope; the facets of the noncontextuality polytope are depicted as straight lines – solid and dashed – while the border of the set of quantum probabilities as solid lines – straig… view at source ↗

**Figure 2.** Figure 2: Contextual fraction CF(⃗q) as a function of λ and a, for the same family of quantum states as those in Fig.1 of the main text. (Inset) Invasiveness cost T = IC(⃗q) (solid line) and contextual fraction T = CF(⃗q) (dashed line) as a function of a for the probabilities given by the quantum state |ψa⟩; IC(⃗q) is divided by 15 to have a comparable scale between the two quantities [PITH_FULL_IMAGE:figures/full… view at source ↗

read the original abstract

Contextuality is a defining feature that separates the quantum from the classical descriptions of physical systems. Within the marginal-scenario framework, noncontextual models are characterized by the existence of a single joint probability distribution consistent with all measurable contexts, while contextual models violate this condition. Building on this approach, we introduce a general method to analyze contextuality in terms of stochastic linear maps that effectively model invasive measurements on an otherwise classical statistics. These maps transform probabilities within the noncontextuality polytope, which includes all classical probabilities, into probabilities that may lie outside the polytope, while preserving the compatibility structure of the scenario at hand. We derive general consistency conditions that such maps must satisfy to represent admissible invasive measurements, and we fully identify them for a paradigmatic example of contextuality for a single three-level quantum system. Furthermore, we introduce a quantifier of contextuality based on the minimal invasiveness required to reproduce a given probability distribution, which offers a distinct approach on how to evaluate the degree of contextuality in a general scenario.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper recasts contextuality as the minimal invasiveness of measurements modeled by stochastic linear maps on classical probabilities, with explicit maps worked out for a three-level system.

read the letter

The main thing to know is that this work treats contextuality as arising from invasive measurements that can be represented by stochastic linear maps acting on classical probability distributions. The maps preserve compatibility relations but push the probabilities outside the noncontextuality polytope, and the authors derive consistency conditions the maps must obey plus a quantifier based on the smallest such invasiveness needed to match a target distribution. They then identify the maps completely for the standard three-level example. That explicit construction is the concrete advance here and gives something a reader can check directly against the polytope picture. The approach stays grounded in the marginal-scenario framework and does not introduce hidden state assumptions or break compatibility, so the internal logic holds up on the stated terms. The quantifier itself is operational and distinct from some earlier measures, though its optimization-based definition does tie the value closely to the observed distribution, which limits how predictive it can be without further calibration. This is a minor rather than fatal issue and does not undermine the modeling tool. The paper is aimed at researchers already working with noncontextual polytopes and operational characterizations of contextuality in quantum foundations. Someone in that subfield would get direct value from the consistency conditions and the worked example. It is grounded enough, with a clear new element and a reproducible case, to deserve a serious referee rather than a desk reject. I would send it for peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a general method for analyzing quantum contextuality by employing stochastic linear maps to model the effects of invasive measurements on an otherwise classical probability distribution. These maps are designed to transform probabilities within the noncontextuality polytope to those outside it while maintaining the compatibility structure of the measurement scenario. The authors derive general consistency conditions for admissible invasive measurements and claim to fully identify such maps for a paradigmatic three-level quantum system. Additionally, they propose a quantifier of contextuality based on the minimal invasiveness needed to reproduce a given probability distribution.

Significance. If the derivations and identifications hold, this work contributes a distinct perspective on contextuality by framing it in terms of measurement invasiveness rather than solely through joint probability inconsistencies. The preservation of compatibility and the explicit treatment of a paradigmatic case are notable strengths that could aid in developing quantitative measures of contextuality. This approach may offer new tools for distinguishing classical and quantum behaviors in foundational studies.

major comments (2)

[§3 (General method)] §3 (General method): The derivation of the general consistency conditions for the stochastic linear maps should be presented with explicit equations to allow verification that the identified maps for the three-level system satisfy them without violating the noncontextual assumptions.
[§5 (Quantifier)] §5 (Quantifier): The definition of the contextuality quantifier via minimization of invasiveness over admissible maps risks a degree of circularity, as the minimal value is determined by optimizing to match the target distribution; this could be clarified by showing how it provides independent information beyond the distribution itself.

minor comments (1)

[Abstract] Abstract: The abstract mentions 'full identification' for the three-level system but could briefly indicate what this identification entails, such as the form of the maps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us identify areas for improvement in clarity and presentation. We address each major comment below and outline the revisions we will make to the manuscript.

read point-by-point responses

Referee: §3 (General method): The derivation of the general consistency conditions for the stochastic linear maps should be presented with explicit equations to allow verification that the identified maps for the three-level system satisfy them without violating the noncontextual assumptions.

Authors: We agree that the current presentation of the consistency conditions in Section 3 would benefit from greater explicitness. In the revised manuscript we will insert the full set of linear equations that define admissible stochastic maps, including the conditions that preserve the compatibility structure and map the noncontextuality polytope into itself or its complement as appropriate. We will then explicitly verify that the maps identified for the three-level system satisfy these equations and do not inadvertently introduce noncontextual assumptions that contradict the target contextual behavior. revision: yes
Referee: §5 (Quantifier): The definition of the contextuality quantifier via minimization of invasiveness over admissible maps risks a degree of circularity, as the minimal value is determined by optimizing to match the target distribution; this could be clarified by showing how it provides independent information beyond the distribution itself.

Authors: We acknowledge the concern about possible circularity. The quantifier is not intended to be a tautological restatement of the distribution; rather, it measures the smallest invasiveness (in the sense of the chosen norm on the stochastic maps) that is required to push a given distribution outside the noncontextuality polytope while respecting the scenario’s compatibility relations. In the revision we will add a paragraph in Section 5 that (i) states the optimization problem formally, (ii) shows that the resulting minimal value is a functional of the distribution that is independent of any particular choice of joint distribution, and (iii) illustrates with a concrete numerical example how the quantifier distinguishes distributions that are equidistant from the polytope boundary under different metrics. This establishes that the measure supplies new quantitative information beyond the raw probabilities. revision: partial

Circularity Check

0 steps flagged

Derivation remains self-contained; no load-bearing reduction to inputs

full rationale

The paper derives general consistency conditions for stochastic linear maps modeling invasive measurements, then explicitly identifies the admissible maps for the three-level system while preserving compatibility structure. The minimal-invasiveness quantifier is introduced as a distinct evaluation tool based on the smallest such map reproducing a target distribution. No quoted step equates a claimed prediction or first-principles result to its own fitted parameters or prior self-citation by construction; the central construction operates on the noncontextuality polytope as an independent input and does not collapse into a renaming or self-referential definition. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard marginal-scenario definition of noncontextuality and introduces stochastic linear maps as a modeling device; no free parameters or new physical entities are explicitly introduced in the abstract.

axioms (1)

domain assumption Noncontextual models are characterized by the existence of a single joint probability distribution consistent with all measurable contexts.
This is the foundational definition of the marginal-scenario framework invoked at the start of the abstract.

pith-pipeline@v0.9.0 · 5702 in / 1249 out tokens · 48082 ms · 2026-05-19T03:06:05.167215+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 3 internal anchors

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Quantum contextuality from measurement invasiveness

as one of the key properties distinguishing quantum me- chanics from any classical model, contextuality is now recog- nized as a cornerstone of the foundations of quantum mechan- ics [2], and has become a subject of growing interest across different fields, especially due to its emerging key role in under- standing the origin of quantum computational adva...

work page internal anchor Pith review Pith/arXiv arXiv 2025
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M. Ara´ujo, M. T. Quintino, C. Budroni, M. T. Cunha, and A. Ca- bello, All noncontextuality inequalities for the n-cycle scenario, Phys. Rev. A 88, 022118 (2013)

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S. Abramsky, R. S. Barbosa, and S. Mansfield, Contextual frac- tion as a measure of contextuality, Phys. Rev. Lett.119, 050504 (2017)

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B. Amaral, A. Cabello, M. T. Cunha, and L. Aolita, Noncontex- tual wirings, Phys. Rev. Lett. 120, 130403 (2018)

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K. Horodecki, J. Zhou, M. Stankiewicz, R. Salazar, P. Horodecki, R. Raussendorf, R. Horodecki, R. Ramanathan, and E. Tyhurst, The rank of contextuality, New Journal of Physics 25, 073003 (2023)

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G. Vitagliano and C. Budroni, Leggett-Garg macrorealism and temporal correlations, Phys. Rev. A 107, 040101 (2023). More details about the KCBS polytope The KCBS scenario includes 5 observables, G = {A1, . . . , A5} where each Ai has outcomes Oi = {−1, 1}, and it is defined by the contexts FKCBS = {(A1, A2), (A2, A3), (A3, A4), (A4, A5), (A5, A1)}. (S1) T...

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[1] [1]

Quantum contextuality from measurement invasiveness

as one of the key properties distinguishing quantum me- chanics from any classical model, contextuality is now recog- nized as a cornerstone of the foundations of quantum mechan- ics [2], and has become a subject of growing interest across different fields, especially due to its emerging key role in under- standing the origin of quantum computational adva...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[2] [2]

Quantum Reservoir Com- puting (QuReCo)

– see Eq.(6) – [ 24]. As anticipated, a violation of the KCBS inequality is realized in quantum mechanics for a qutrit, H = C 3, with the observables described by five pairwise compatible projective measurements. The latter are associated with self-adjoint operators bAi = 2 |vi⟩⟨vi| − 1 C 3 , fixed by pairwise orthogonal vectors – see [24] – ⟨vi|vi+1⟩ = 0...

work page 2022

[3] [3]

Kochen and E

S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, J. Math. Mech. 17, 59 (1967)

work page 1967

[4] [4]

Budroni, A

C. Budroni, A. Cabello, O. G ¨uhne, M. Kleinmann, and J.-A. Larsson, Kochen-Specker contextuality, Rev. Mod. Phys. 94, 045007 (2022)

work page 2022

[5] [5]

Raussendorf, Contextuality in measurement-based quantum computation, Phys

R. Raussendorf, Contextuality in measurement-based quantum computation, Phys. Rev. A 88, 022322 (2013)

work page 2013

[6] [6]

Howard, J

M. Howard, J. Wallman, V . Veitch, and J. Emerson, Contextual- ity supplies the ‘magic’for quantum computation, Nature510, 351 (2014)

work page 2014

[7] [7]

Bermejo-Vega, N

J. Bermejo-Vega, N. Delfosse, D. E. Browne, C. Okay, and R. Raussendorf, Contextuality as a resource for models of quan- tum computation with qubits, Phys. Rev. Lett. 119, 120505 (2017)

work page 2017

[8] [8]

Bravyi, D

S. Bravyi, D. Gosset, R. K¨onig, and M. Tomamichel, Quantum advantage with noisy shallow circuits, Nature Physics 16, 1040 (2020)

work page 2020

[9] [9]

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

work page 2005

[10] [10]

Khrennikov, Contextual Approach to Quantum Formalism (Springer, Netherland, 2009)

A. Khrennikov, Contextual Approach to Quantum Formalism (Springer, Netherland, 2009)

work page 2009

[11] [11]

Abramsky and A

S. Abramsky and A. Brandenburger, The sheaf-theoretic struc- ture of non-locality and contextuality, New Journal of Physics 13, 113036 (2011)

work page 2011

[12] [12]

Kurzy´nski, R

P. Kurzy´nski, R. Ramanathan, and D. Kaszlikowski, Entropic test of quantum contextuality, Phys. Rev. Lett. 109, 020404 (2012)

work page 2012

[13] [13]

Chaves and T

R. Chaves and T. Fritz, Entropic approach to local realism and noncontextuality, Phys. Rev. A85, 032113 (2012)

work page 2012

[14] [14]

Cabello, S

A. Cabello, S. Severini, and A. Winter, Graph-theoretic approach to quantum correlations, Phys. Rev. Lett. 112, 040401 (2014)

work page 2014

[15] [15]

E. N. Dzhafarov, V . H. Cervantes, and J. V . Kujala, Contextuality in canonical systems of random variables, Phil. Trans. Royal Soc. A 375, 20160389 (2017)

work page 2017

[16] [16]

M. P. M¨uller and A. J. P. Garner, Testing quantum theory by generalizing noncontextuality, Phys. Rev. X13, 041001 (2023)

work page 2023

[17] [17]

Fritz and R

T. Fritz and R. Chaves, Entropic inequalities and marginal prob- lems, IEEE Transactions on Information Theory 59, 803 (2013)

work page 2013

[18] [18]

Brunner, D

N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys.86, 419 (2014)

work page 2014

[19] [19]

Fine, Hidden variables, joint probability, and the Bell inequal- ities, Phys

A. Fine, Hidden variables, joint probability, and the Bell inequal- ities, Phys. Rev. Lett. 48, 291 (1982)

work page 1982

[20] [20]

A. A. Klyachko, M. A. Can, S. Binicio˘glu, and A. S. Shumovsky, Simple test for hidden variables in spin-1 systems, Phys. Rev. Lett. 101, 020403 (2008)

work page 2008

[21] [21]

Holevo, Probabilistic and Statistical Aspects of Quantum Theory, 1st ed., Publications of the Scuola Normale Superiore (Edizioni della Normale Pisa, 2011)

A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, 1st ed., Publications of the Scuola Normale Superiore (Edizioni della Normale Pisa, 2011)

work page 2011

[22] [22]

Heinosaari and M

T. Heinosaari and M. Ziman, The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement (Cam- bridge University Press, 2014)

work page 2014

[23] [23]

D. Avis, H. Imai, T. Ito, and Y . Sasaki, Deriving tight Bell inequalities for 2 parties with many 2-valued observables from facets of cut polytopes (2004), arXiv:quant-ph/0404014 [quant- ph]

work page internal anchor Pith review Pith/arXiv arXiv 2004

[24] [24]

Richter, A

M. Richter, A. Smirne, W. Strunz, and D. Egloff, Classical inva- sive description of informationally-complete quantum processes, Annalen der Physik 536, 2300304 (2024)

work page 2024

[25] [25]

We are not taking into account explicitly the trivial contexts made of one single observable, since the corresponding prob- abilities can always be recovered via marginalization from the probabilities of the five contexts considered

work page

[26] [26]

Supplemental Information

work page

[27] [27]

Ara´ujo, M

M. Ara´ujo, M. T. Quintino, C. Budroni, M. T. Cunha, and A. Ca- bello, All noncontextuality inequalities for the n-cycle scenario, Phys. Rev. A 88, 022118 (2013)

work page 2013

[28] [28]

Lapkiewicz, P

R. Lapkiewicz, P. Li, C. Schaeff, N. K. Langford, S. Ramelow, M. Wie´sniak, and A. Zeilinger, Experimental non-classicality of an indivisible quantum system, Nature 474, 490 (2011)

work page 2011

[29] [29]

This is the case since we are not setting any restriction on the (finite) dimension of the Hilbert space where states and measure- ments defining ⃗Q are settled

work page

[30] [30]

Kleinmann, O

M. Kleinmann, O. G ¨uhne, J. R. Portillo, J.- ˚A. Larsson, and A. Cabello, Memory cost of quantum contextuality, New Journal of Physics 13, 113011 (2011)

work page 2011

[31] [31]

Grudka, K

A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, P. Joshi, W. Kłobus, and A. W´ojcik, Quantifying contextuality, Phys. Rev. Lett.112, 120401 (2014)

work page 2014

[32] [32]

Abramsky, R

S. Abramsky, R. S. Barbosa, and S. Mansfield, Contextual frac- tion as a measure of contextuality, Phys. Rev. Lett.119, 050504 (2017)

work page 2017

[33] [33]

On geometrical aspects of the graph approach to contextuality

B. Amaral and M. T. Cunha, On geometrical aspects of the graph approach to contextuality (2017), arXiv:1709.04812 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[34] [34]

Amaral, A

B. Amaral, A. Cabello, M. T. Cunha, and L. Aolita, Noncontex- tual wirings, Phys. Rev. Lett. 120, 130403 (2018)

work page 2018

[35] [35]

J. V . Kujala and E. N. Dzhafarov, Measures of contextuality and non-contextuality, Phil. Trans. Royal Soc. A 377, 20190149 (2019)

work page 2019

[36] [36]

Horodecki, J

K. Horodecki, J. Zhou, M. Stankiewicz, R. Salazar, P. Horodecki, R. Raussendorf, R. Horodecki, R. Ramanathan, and E. Tyhurst, The rank of contextuality, New Journal of Physics 25, 073003 (2023)

work page 2023

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G. Vitagliano and C. Budroni, Leggett-Garg macrorealism and temporal correlations, Phys. Rev. A 107, 040101 (2023). More details about the KCBS polytope The KCBS scenario includes 5 observables, G = {A1, . . . , A5} where each Ai has outcomes Oi = {−1, 1}, and it is defined by the contexts FKCBS = {(A1, A2), (A2, A3), (A3, A4), (A4, A5), (A5, A1)}. (S1) T...

work page 2023