Measurement-Incompatibility Constraints for Maximal Randomness

Qiongyi He; Tianqi Zheng; Yi Li; Yu Xiang

REVIEW 5 major objections 5 minor 40 references

Measurement-Incompatibility Constraints for Maximal Randomness

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper aims to prove that maximal randomness can be certified by placing the observed distribution on a curved boundary point of the quantum set, and that only one user then needs highly incompatible measurements.

desk verdict A promising direct-optimization construction with a real incompatibility trade-off, but the maximal-randomness claim is not proven because boundary points of the quantum set need not be extreme points. read the letter →

arxiv 2505.17585 v2 pith:XFDKDEYL submitted 2025-05-23 quant-ph

Tianqi Zheng , Yi Li , Yu Xiang , Qiongyi He This is my paper · ORCID

classification quant-ph MSC 81P1581P4081P4594A60 PACS 03.65.Ud03.67.-a03.67.Dd

keywords maximalrandomnessdevice-independentcertificationmeasurementincompatibilityrobustnessquantumprobabilitydistributionsBellnonlocalitymin-entropynetworks

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

This paper is trying to show that maximal randomness can be certified directly from observed probability distributions, without assuming the dimension of the systems or relying on Bell-inequality violations, and that the key resource is the placement of the distribution on the non-flat boundary of the quantum set. In the bipartite case with two inputs and two outputs, the authors exhibit two-qubit pure states and projective measurements whose outputs are perfectly uniform and whose point lies on this boundary, giving guessing probability $1/4$ and $2$ bits of min-entropy. They then derive a quantitative trade-off: maximal randomness is possible as long as one party's measurement incompatibility robustness parameter is sufficiently large, while the other party's parameter can approach the limiting value that corresponds to arbitrarily small incompatibility. The same asymmetric pattern is reported for three parties sharing a GHZ state, with the explicit caveat, stated in the text, that the tripartite maximally-entangled simplification is not rigorously justified and was validated only numerically.

What carries the argument

The engine of the argument is the constrained optimization problem of Eq. (3): optimize one conditional probability, say $P(1,1|2,2)$, subject to the uniform-output constraints $P(a,b|1,1)=1/4$ and extra linear constraints fixing marginals $s$ and $t$, over the quantum set $Q$. Convexity of $Q$ plus the assumption that its boundary is non-flat means the optimizer lands on an extreme, non-decomposable point, which forces Eve's guessing probability to its lower bound. The optimization is made explicit by writing two-qubit pure states in Schmidt form, reducing the problem to the one-dimensional bound $f(A;s,t)$ of Eq. (5), and then to the incompatibility-robustness quantifier of Eq. (6), whose feasible region yields the trade-off curve between the two users' incompatibility parameters. The tripartite version repeats this structure with a GHZ state and a function $g_T(x,z)$ analogous to the bipartite $g(x,z)$.

What would settle it

Take one of the optimized distributions from Eq. (3) with parameters in the allowed region and run a convergent hierarchy of semidefinite relaxations, or an explicit search over decompositions, to test whether the point in $Q$ can be written as $\lambda P_1 + (1-\lambda) P_2$ with $P_1 \neq P_2$ both in $Q$. If such a decomposition exists, the point is not extreme, the guessing probability exceeds $1/4$, and the claimed 2-bit min-entropy is false. For the tripartite claim, the same test should be applied to the reported points, since the paper's only evidence is a single numerical optimization with error below $10^{-5}$.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that globally maximal randomness is achievable whenever the users' joint distribution is uniform on the target outputs and lies on a boundary point of the quantum set $Q$, because the quantum set has a curved (non-polyhedral) boundary and therefore does not admit the decomposition Eve needs. Concretely, the paper identifies a family of two-qubit states and projective measurements solving the constrained optimization problem of Eq. (3); for these, the guessing probability reaches $1/n^2$ in the bipartite case and $1/n^3$ in the tripartite case, i.e. min-entropy $p\log_2 n$ bits for $p$ users. A quantitative relation to measurement incompatibility follows: for maximally entangled bipartite states, the incompatibility parameters of the two users obey a trade-off, and if one user's parameter approaches the limiting value $η \to \sqrt{2}/2$, then the other user's parameter can approach $η \to 1$ while maximal randomness is retained. In the tripartite case the analogous conclusion is that one party, $C'$, needs sufficiently large incompatibility while the others need almost none.

Load-bearing premise

The whole certificate rests on the premise that a point on the boundary of the quantum set is automatically non-decomposable because that boundary is curved; if any constructed distribution sits on a flat face or can be written as a mixture of two other quantum distributions, Eve's guessing probability rises above the uniform value and the claimed maximal randomness fails.

Editorial extensions

If this is right

If the boundary-point criterion holds, maximal randomness can be certified without any Bell-inequality violation, using only the raw probability distribution and simple projective measurements.
Only one user in the network needs highly incompatible measurements; the remaining users can use nearly compatible measurements, which relaxes experimental demands in multi-user protocols.
The method generalizes to arbitrary numbers of users $p$ and outputs $n$, yielding the global min-entropy bound $p \log_2 n$ bits whenever the analogous optimization can be solved.
The trade-off between the users' incompatibility parameters gives a quantitative resource-allocation rule for distributing measurement incompatibility across a quantum network.
The explicit numerical validation of the tripartite case (relative error below $10^{-5}$) suggests, but does not prove, that the same asymmetry extends beyond bipartite systems.

Reading between the lines

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

The paper stops short of proving that its boundary points are extreme points of the quantum set; checking extremality directly, for example with a convergent hierarchy of semidefinite relaxations, would turn the boundary criterion into a fully rigorous certificate.
If the boundary-non-flatness argument transfers, a similar asymmetry between one strong and many weak measurements may hold in steering or prepare-and-measure scenarios, where the relevant sets have different geometries.
The explicit caveat that the GHZ-based tripartite solution lacks rigorous justification invites a search for three-qubit states that are not maximally entangled and could attain the same or smaller objective value, which would strengthen or invalidate the reported trade-off.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things about this paper. The genuinely new part is the direct optimization of probability distributions for randomness certification, without Bell-inequality post-processing, and the explicit projective-measurement constructions for arbitrary bipartite entanglement. The quantitative trade-off between incompatibility robustness values (Fig. 4) is also novel and, if it holds, practically useful: one party can have nearly compatible measurements as long as the other party's incompatibility is sufficiently large. That part is worth engaging with.

What is not established is the central claim of maximal randomness. The certification step in Definitions and Method asserts that a distribution on the boundary of the quantum set cannot be decomposed by Eve because the boundary is "non-flat due to the non-polyhedral nature of the quantum set." That inference is wrong as stated. Non-polyhedral convex sets can have flat faces, and a boundary point can be a convex combination of other boundary points. To get guessing probability 1/n^2 from Eq. (1), you need the constructed distribution to be an extreme point of Q, or at least that every convex decomposition yields a uniform target distribution. The paper proves neither. Also, Eq. (5) optimizes only over two-qubit pure states and projective measurements, so the resulting point may lie on the boundary of that restricted inner approximation, not on the boundary of Q. Without an upper bound over all states and POVMs, condition (2) is unverified.

The tripartite extension is weaker still. The text explicitly says the restriction to GHZ states "lacks rigorous justification" and that correctness is checked numerically with relative error below 10^-5. That is an honest admission, but it means the tripartite certification claim is not a proof.

The bipartite algebra in Eqs. (4)-(7) is explicit and self-consistent, and the numerical dependence on x and z does show nonlocality in the sense of CHSH violation. So this is not a careless paper — the construction is likely correct as a set of quantum behaviors, and the incompatibility trade-off is a real result. The missing piece is the certification step, which is load-bearing for the maximal-randomness conclusion but addressable.

Who gets value from this? People working on DI randomness protocols will want the construction and the trade-off, but they should not treat the certified randomness claim as proven. The paper deserves a serious referee; I would send it to review with a request to fix the extremality argument and provide a genuine upper bound over Q for the bipartite case. The tripartite section should be reframed as numerical evidence, not a theorem.

My recommendation: engage with it, but treat the maximal-randomness certification as open until the boundary-to-extreme-point step is resolved.

Referee Report

5 major / 5 minor

Summary. The paper proposes a framework for certifying maximal randomness directly from observed probability distributions, bypassing the usual step of Bell-inequality violation. It states two conditions for maximal randomness—uniform output distribution and boundary positioning in the quantum set—and applies them to construct bipartite and tripartite distributions from two-qubit and GHZ states with projective measurements. The authors derive analytical bounds for the objective probability, quantify the incompatibility robustness of the users' measurements, and report a trade-off whereby sufficiently large incompatibility of one party permits arbitrarily small incompatibility of others. The central claim is that maximal randomness, with guessing probability 1/n^p and min-entropy p log2 n, is achieved by these distributions.

Significance. If the certification step were sound, the framework would be a useful contribution to device-independent randomness: it works directly with probability distributions, avoids post-selecting on Bell-inequality violations, gives explicit two-qubit analytical formulas, and introduces a quantitative incompatibility trade-off. The manuscript is also transparent about the tripartite restriction to GHZ states and includes numerical checks. However, the foundational gap between boundary and extreme points of the quantum set means the claimed maximal-randomness certification is not established; the paper's own admission that the tripartite restriction 'lacks rigorous justification' further limits the current result. The geometric idea is interesting, but the proof needs substantial additional work before the main conclusion can be accepted.

major comments (5)

[Definitions and Method (conditions after Eq. (2))] The paper treats 'the distribution lies on the boundary of the quantum set' as sufficient for Pg = 1/n^2, with the justification that the boundary is 'non-flat due to the non-polyhedral nature of the quantum set.' This inference is invalid: a boundary point of a convex set need not be an extreme point, and non-polyhedral convex sets can have flat faces (for example, a cylinder is non-polyhedral and has non-extreme boundary points). To bound the guessing probability in Eq. (1), one must prove either that the constructed point is an extreme point of Q or that every convex decomposition of it has uniform target-conditioned distributions. The manuscript does neither, so the central certification step is unproven.
[Results, Eqs. (3)-(5)] The optimization in Eq. (5) is restricted to two-qubit pure states and projective measurements, but the quantum set Q in Eq. (1) includes arbitrary finite dimensions and POVMs. The solution of Eq. (5) therefore gives an extremum of the objective within a restricted two-qubit slice, not necessarily a point on the boundary of the full quantum set. No upper or lower bound over all quantum realizations is supplied, so the 'boundary positioning' condition is not verified even for the bipartite examples shown in Figs. 2-4. Without that verification, the derivations in Eqs. (4)-(7) do not certify maximal randomness.
[Results, discussion of Fig. 3] The sentence 'Their nonlocality guarantees the generation of maximal randomness' is not a valid inference. A CHSH violation rules out local hidden-variable models, but it does not imply that the guessing probability Pg in Eq. (1) equals 1/n^2; nonlocal quantum correlations can generically be decomposed into convex combinations of other nonlocal quantum points, allowing Eve a nontrivial guess. The manuscript needs an explicit proof—for the specific displayed distributions—that Pg = 1/4, rather than relying on nonlocality alone.
[Results, tripartite paragraph after Eq. (8)] The text explicitly states that the restriction to GHZ states and identical A', B' 'lacks rigorous justification' and that correctness is only checked numerically with relative error below 10^-5. A numerical check over a restricted family does not prove that the optimized value lies on the boundary of the full tripartite quantum set, nor that the corresponding distribution certifies 3 log2 2 bits. The tripartite claim in the Conclusion is therefore stronger than what is established in the manuscript.
[Results, Eq. (7)] Equation (7) states sqrt(2 P(1,1|2,2)) = |alpha1 alpha2 + beta1 beta2| <= g(x,z), which is an upper bound on the objective, yet the text identifies this as the 'minimum value' of P(1,1|2,2). For the minimization problem in Eq. (3), a lower bound of the form P >= g^2/2 is needed. Please clarify the intended inequality direction and provide the corresponding derivation; if the displayed direction is correct, the subsequent identification of optimal boundary points in Figs. 3 and 4 does not follow.

minor comments (5)

[Eq. (6)] The compatibility condition on M^eta_{a|x} is not defined in the main text; please specify that for each input x the set {M^eta_{a|x}}_a must be jointly measurable, and state the allowed range of eta (presumably eta in [0,1]).
[Figure 2 caption] The caption says 'The entanglement can be arbitrarily big (A^2 -> 0.5) or arbitrarily small (A^2 -> 0)' without defining A^2; please state explicitly that A is the Schmidt coefficient in the two-qubit state of Eq. (4), since 'entanglement' alone is ambiguous.
[Results, Fig. 3 discussion] The blank region in Fig. 3 is described only as resulting from 'failure to satisfy the constraints'; please list the explicit constraints on x, z, and g(x,z) that define the feasible region, so that the figure can be interpreted quantitatively.
[Tripartite section, Eq. (8)] The tripartite variables x and z are reused without restating their definitions; please either define them again in the tripartite paragraph or refer explicitly to the corresponding equation in the supplementary material.
[Introduction and Conclusion] There are minor language issues: 'Several fundamental questions naturally arises' should be 'naturally arise', and the Conclusion contains 'with as long as' where 'as long as' is intended. Also, the ket in the maximally entangled state is mistyped as '|11)' instead of '|11>'.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the central optimization is self-contained and checked against the CHSH benchmark; the main gaps are unproven boundary/extremality assumptions and an admitted GHZ ansatz, not circular reductions.

full rationale

The derivation chain is self-contained. In the Results section, Eq. (3) defines an optimization problem whose constraints are the uniform-output condition P(a,b|1,1)=1/4 and fixed marginals x,y,z,w; the target P(1,1|2,2) is the objective function, not a fitted input. Equations (4)-(5) then derive the bound f(A;s,t) via Schmidt decomposition and Cauchy-Schwarz, with s, t, x, z, and A^2 acting as constraint parameters or optimization variables. No fitted quantity is later relabeled as a prediction. The boundary-positioning step in Definitions and Method invokes convexity and claims that the quantum set's boundary is 'non-flat due to the non-polyhedral nature of the quantum set'; this is an unsupported geometric inference, since boundary points of non-polyhedral convex sets need not be extreme points. That is a rigor/correctness gap, not circularity. The CHSH-violation check in Fig. 3 provides an external benchmark showing the constructed distributions are nonlocal. The tripartite section explicitly concedes that the GHZ-state restriction 'lacks rigorous justification' and is tested only numerically with relative error below 10^-5, which is an admitted limitation rather than a circular derivation. Self-citations [31] and [40] appear as motivation and notation ('Recently, a necessary and sufficient connection between randomness certification and specific measurement incompatibility structures has been rigorously established [31]'; 'we extend... to a three-user scenario... [39,40]'), while the incompatibility-robustness metric in Eq. (6) is taken from independent standard references [30,33,34]. No central claim reduces, by construction or by self-citation, to its own inputs.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The construction carries four hand-chosen parameters (s, t, A^2, and the constraint probabilities) and four substantive axioms. Two of these axioms are ad hoc: the boundary-extremality conflation and the tripartite GHZ/identical-user assumption. The bipartite result depends mainly on the two-qubit reduction and the boundary claim; the tripartite result depends on the numerically checked but unproved GHZ assumption. No invented entities are introduced.

free parameters (4)

s (Alice's second-measurement marginal) = variable over [0, 0.5]
Introduced in Eq. (3) as s = z + w; enters Eq. (4) and the objective f(A; s, t). Chosen by hand rather than fitted to data.
t (Bob's second-measurement marginal) = variable over [0, 0.5]
Introduced as t = x + y; symmetric role to s in Eq. (4) and Eq. (5).
A^2 (Schmidt coefficient of the two-qubit state) = optimized in [0, 0.5]
The state is |Phi> = A|00> + B|11>; A^2 is varied to maximize/minimize P(1,1|2,2) in Eq. (5), and Fig. 2 maps it across the feasible region.
Constraint probabilities x, z (and y, w) = variables over feasible region in Figs. 3-4
These fix the non-uniform rows of Eq. (3) and define the family of distributions; they are selected, not measured or fitted.

assumptions (4)

standard math The quantum set Q is convex (cited [32]).
Used to argue that Eve's decomposition and boundary reasoning are meaningful; accepted standard result.
ad hoc to paper A uniform-output distribution on the boundary of Q certifies maximal randomness because the boundary is non-flat.
This is the load-bearing step in 'Definitions and Method'; the paper does not prove that the relevant boundary has no flat parts and does not prove the constructed points are extreme points of Q.
domain assumption Extreme or boundary points of the (2,2,2) quantum set can be represented by pure two-qubit states with projective measurements.
The paper states 'we only consider two-qubit states and projective measurements' without proof or citation (e.g., Jordan's lemma); needed for Eqs. (4)-(5) to describe the global optimum.
ad hoc to paper Tripartite maximum is attained within GHZ states with A' and B' identical.
Explicitly admitted: 'this approach lacks rigorous justification'; numerical test with relative error below 1e-5 is the only support.

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Pith. "Pith review of Measurement-Incompatibility Constraints for Maximal Randomness." pith.science (2026). https://pith.science/paper/XFDKDEYL

@misc{pith2026250517585,
  author       = {Pith},
  title        = {Pith review of: Measurement-Incompatibility Constraints for Maximal Randomness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XFDKDEYL}},
  note         = {Machine review of arXiv:2505.17585}
}

read the original abstract

Certifying maximal quantum randomness without assumptions about system dimension remains a pivotal challenge for secure communication and foundational studies. Here, we introduce a generalized framework to directly certify maximal randomness from observed probability distributions across systems with arbitrary user numbers, without relying on the Bell-inequality violations. By analyzing probability distributions directly, we identify a class of quantum states and projective measurements that achieve maximal randomness in bipartite and tripartite scenarios, ensuring practical feasibility. Further analysis reveals a counterintuitive trade-off governing measurement incompatibility among users: sufficient incompatibility for one user permits arbitrarily small incompatibility for others, defying conventional symmetry assumptions in the Bell test. This asymmetry provides a pathway to optimize device-independent protocols by strategically distributing quantum resources. Our results establish a versatile and experimentally accessible route to scalable randomness certification, with implications for quantum cryptography and the physics of nonlocal correlations.

Figures

Figures reproduced from arXiv: 2505.17585 by the authors.

**Figure 2.** Entanglement of the states generating maximal ran [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** The constrained relationship between ηA and ηB. The yellow region bounded below the specified line permits the generation of maximal randomness. for incompatibility robustness is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 3.** Figure 3: The values of CHSH expression I = ⟨A1B1⟩ + ⟨A1B2⟩ + ⟨A2B1⟩ − ⟨A2B2⟩ are illustrated, for different x and z. The blank area results from the failure to satisfy the constraints of x, z and g(x, z). As 2x → 1 and 2z → 0.5 (or vice versa), the violation of inequality (I −…

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

40 extracted references · 33 canonical work pages

[1]

Scarani, H

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Duˇ sek, N. L¨ utkenhaus, and M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys. 81, 1301 (2009)

2009
[2]

Broadbent and C

A. Broadbent and C. Schaffner, Quantum cryptography beyond quantum key distribution, Designs, codes, and cryptography 78, 351 (2016)

work page 2016
[3]

R. L. Harrison, Introduction to monte carlo simulation, 5th International Summer School on Nuclear Physics Methods and Accelerators in Biology and Medicine1204, 17 (2010)

work page 2010
[4]

Pironio, A

S. Pironio, A. Ac ´ ın, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Random numbers certi- fied by bell’s theorem, Nature 464, 1021 (2010)

2010
[5]

Bancal, L

J.-D. Bancal, L. Sheridan, and V. Scarani, More ran- domness from the same data, New Journal of Physics 16, 033011 (2014)

work page 2014
[6]

Y. Z. Law, L. P. Thinh, J.-D. Bancal, and V. Scarani, Quantum randomness extraction for various levels of characterization of the devices, Journal of Physics A: Mathematical and Theoretical 47, 424028 (2014)

2014
[7]

Passaro, D

E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Ac ´ ın, Optimal randomness certification in the quantum steer- ing and prepare-and-measure scenarios, New Journal of Physics 17, 113010 (2015)

work page 2015
[8]

Ioannou, B

M. Ioannou, B. Longstaff, M. V. Larsen, J. S. Neergaard- Nielsen, U. L. Andersen, D. Cavalcanti, N. Brunner, and J. B. Brask, Steering-based randomness certification with squeezed states and homodyne measurements, Phys. Rev. A 106, 042414 (2022)

work page 2022

Show all 40 references

[9]

Ac ´ ın and L

A. Ac ´ ın and L. Masanes, Certified randomness in quan- tum physics, Nature 540, 213 (2016)

2016
[11]

Nieto-Silleras, S

O. Nieto-Silleras, S. Pironio, and J. Silman, Using complete measurement statistics for optimal device- independent randomness evaluation, New Journal of Physics 16, 013035 (2014)

2014
[12]

Y. Liu, Q. Zhao, M.-H. Li, J.-Y. Guan, Y. Zhang, B. Bai, W. Zhang, W.-Z. Liu, C. Wu, X. Yuan, H. Li, W. J. Munro, Z. Wang, L. You, J. Zhang, X. Ma, J. Fan, Q. Zhang, and J.-W. Pan, Device-independent quantum random-number generation, Nature 562, 548 (2018)

2018
[13]

Y. Liu, X. Yuan, M.-H. Li, W. Zhang, Q. Zhao, J. Zhong, Y. Cao, Y.-H. Li, L.-K. Chen, H. Li, T. Peng, Y.-A. Chen, C.-Z. Peng, S.-C. Shi, Z. Wang, L. You, X. Ma, J. Fan, Q. Zhang, and J.-W. Pan, High-speed device- independent quantum random number generation with- out a detecti...

2018
[14]

Wooltorton, P

L. Wooltorton, P. Brown, and R. Colbeck, Tight analytic bound on the trade-off between device-independent ran- domness and nonlocality, Phys. Rev. Lett. 129, 150403 (2022)

2022
[15]

Ac ´ ın, S

A. Ac ´ ın, S. Massar, and S. Pironio, Randomness ver- sus nonlocality and entanglement, Phys. Rev. Lett. 108, 100402 (2012)

2012
[16]

Ramanathan, Y

R. Ramanathan, Y. Liu, and S. Pironio, Maxi- mum quantum non-locality is not always sufficient for device-independent randomness generation (2024), arXiv:2408.03665 [quant-ph]

2024 arXiv
[17]

Dhara, G

C. Dhara, G. Prettico, and A. Ac ´ ın, Maximal quantum 6 randomness in bell tests, Phys. Rev. A88, 052116 (2013)

2013
[18]

Pironio, Lifting bell inequalities, Journal of mathemat- ical physics 46, 062112 (2005)

S. Pironio, Lifting bell inequalities, Journal of mathemat- ical physics 46, 062112 (2005)

2005
[19]

M. L. Almeida, J.-D. Bancal, N. Brunner, A. Ac ´ ın, N. Gisin, and S. Pironio, Guess your neighbor’s input: A multipartite nonlocal game with no quantum advan- tage, Phys. Rev. Lett. 104, 230404 (2010)

2010
[20]

Mao, Z.-D

Y.-L. Mao, Z.-D. Li, S. Yu, and J. Fan, Test of genuine multipartite nonlocality, Phys. Rev. Lett. 129, 150401 (2022)

2022
[21]

ˇSupi´ c, J.-D

I. ˇSupi´ c, J.-D. Bancal, and N. Brunner, Quantum non- locality in networks can be demonstrated with an arbi- trarily small level of independence between the sources, Phys. Rev. Lett. 125, 240403 (2020)

2020
[22]

F. J. Curchod, M. L. Almeida, and A. Ac ´ ın, A versa- tile construction of bell inequalities for the multipartite scenario, New Journal of Physics 21, 023016 (2019)

2019
[23]

Svetlichny, Distinguishing three-body from two-body nonseparability by a bell-type inequality, Phys

G. Svetlichny, Distinguishing three-body from two-body nonseparability by a bell-type inequality, Phys. Rev. D 35, 3066 (1987)

1987
[24]

N. D. Mermin, Extreme quantum entanglement in a su- perposition of macroscopically distinct states, Phys. Rev. Lett. 65, 1838 (1990)

1990
[25]

Wooltorton, P

L. Wooltorton, P. Brown, and R. Colbeck, Expanding bi- partite bell inequalities for maximum multi-partite ran- domness (2024), arXiv:2308.07030 [quant-ph]

2024
[26]

Woodhead, J

E. Woodhead, J. m. k. Kaniewski, B. Bourdoncle, A. Salavrakos, J. Bowles, A. Ac ´ ın, and R. Augusiak, Maximal randomness from partially entangled states, Phys. Rev. Res. 2, 042028 (2020)

2020
[27]

Tavakoli, M

A. Tavakoli, M. Farkas, D. Rosset, J.-D. Bancal, and J. Kaniewski, Mutually unbiased bases and sym- metric informationally complete measurements in bell experiments, Science Advances 7, eabc3847 (2021), https://www.science.org/doi/pdf/10.1126/sciadv.abc3847

2021 doi
[28]

G. M. D’Ariano, P. Lo Presti, and P. Perinotti, Clas- sical randomness in quantum measurements, Journal of Physics A: Mathematical and General 38, 5979 (2005)

2005
[29]

Senno, T

G. Senno, T. Strohm, and A. Ac ´ ın, Quantifying the intrinsic randomness of quantum measurements, Phys. Rev. Lett. 131, 130202 (2023)

2023
[30]

Designolle, M

S. Designolle, M. Farkas, and J. Kaniewski, Incompat- ibility robustness of quantum measurements: a unified framework, New Journal of Physics 21, 113053 (2019)

2019
[31]

Y. Li, Y. Xiang, J. Tura, and Q. He, Necessary and suffi- cient condition for randomness certification from incom- patibility (2024), arXiv:2409.14991 [quant-ph]

2024
[32]

Brunner, D

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

2014
[33]

Heinosaari, J

T. Heinosaari, J. Kiukas, and D. Reitzner, Noise robust- ness of the incompatibility of quantum measurements, Phys. Rev. A 92, 022115 (2015)

2015
[34]

G¨ uhne, E

O. G¨ uhne, E. Haapasalo, T. Kraft, J.-P. Pellonp¨ a¨ a, and R. Uola, Colloquium: Incompatible measurements in quantum information science, Rev. Mod. Phys. 95, 011003 (2023)

2023
[35]

Cavalcanti, M

D. Cavalcanti, M. L. Almeida, V. Scarani, and A. Ac ´ ın, Quantum networks reveal quantum nonlocality, Nature communications 2, 184 (2011)

2011
[36]

Andreoli, G

F. Andreoli, G. Carvacho, L. Santodonato, R. Chaves, and F. Sciarrino, Maximal qubit violation of n-locality in- equalities in a star-shaped quantum network, New Jour- nal of Physics 19, 113020 (2017)

2017
[37]

Contreras-Tejada, C

P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Genuine multipartite nonlocality is intrinsic to quantum networks, Phys. Rev. Lett. 126, 040501 (2021)

2021
[38]

Sarkar, Network quantum steering enables randomness certification without seed randomness, Quantum 8, 1419 (2024)

S. Sarkar, Network quantum steering enables randomness certification without seed randomness, Quantum 8, 1419 (2024)

2024
[39]

Woodhead, B

E. Woodhead, B. Bourdoncle, and A. Ac ´ ın, Randomness versus nonlocality in the Mermin-Bell experiment with three parties, Quantum 2, 82 (2018)

2018
[40]

Y. Li, Y. Xiang, X.-D. Yu, H. C. Nguyen, O. G¨ uhne, and Q. He, Randomness certification from multipartite quan- tum steering for arbitrary dimensional systems, Phys. Rev. Lett. 132, 080201 (2024)

2024
[41]

Ac ´ ın, A

A. Ac ´ ın, A. Andrianov, L. Costa, E. Jan´ e, J. I. La- torre, and R. Tarrach, Generalized schmidt decomposi- tion and classification of three-quantum-bit states, Phys. Rev. Lett. 85, 1560 (2000)

2000

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Scarani, H

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Duˇ sek, N. L¨ utkenhaus, and M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys. 81, 1301 (2009)

2009

[2] [2]

Broadbent and C

A. Broadbent and C. Schaffner, Quantum cryptography beyond quantum key distribution, Designs, codes, and cryptography 78, 351 (2016)

work page 2016

[3] [3]

R. L. Harrison, Introduction to monte carlo simulation, 5th International Summer School on Nuclear Physics Methods and Accelerators in Biology and Medicine1204, 17 (2010)

work page 2010

[4] [4]

Pironio, A

S. Pironio, A. Ac ´ ın, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Random numbers certi- fied by bell’s theorem, Nature 464, 1021 (2010)

2010

[5] [5]

Bancal, L

J.-D. Bancal, L. Sheridan, and V. Scarani, More ran- domness from the same data, New Journal of Physics 16, 033011 (2014)

work page 2014

[6] [6]

Y. Z. Law, L. P. Thinh, J.-D. Bancal, and V. Scarani, Quantum randomness extraction for various levels of characterization of the devices, Journal of Physics A: Mathematical and Theoretical 47, 424028 (2014)

2014

[7] [7]

Passaro, D

E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Ac ´ ın, Optimal randomness certification in the quantum steer- ing and prepare-and-measure scenarios, New Journal of Physics 17, 113010 (2015)

work page 2015

[8] [8]

Ioannou, B

M. Ioannou, B. Longstaff, M. V. Larsen, J. S. Neergaard- Nielsen, U. L. Andersen, D. Cavalcanti, N. Brunner, and J. B. Brask, Steering-based randomness certification with squeezed states and homodyne measurements, Phys. Rev. A 106, 042414 (2022)

work page 2022