No Finite NPA Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario

Anubhav Chaturvedi

REVIEW 1 major objections 4 minor 22 references

No finite level of the NPA semidefinite hierarchy reproduces the complete set of quantum correlations in the (2,2,2) Bell scenario.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-02 01:43 UTC pith:KPLCIWKK

load-bearing objection The paper very likely proves that no finite NPA level is exact in the simplest Bell scenario, but the load-bearing determinant identity (Supplemental Eq. S45) needs machine verification before the proof is fully trustworthy. the 1 major comments →

arxiv 2607.14569 v1 pith:KPLCIWKK submitted 2026-07-16 quant-ph

No Finite NPA Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario

Anubhav Chaturvedi This is my paper

classification quant-ph PACS 03.65.Ud

keywords NPA hierarchyquantum correlationsBell scenariosum of squaresMotzkin polynomialtilted CHSHsemidefinite relaxationquantum set

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Most quantum-versus-classical questions in the simplest Bell scenario have finite answers: the first level of the standard semidefinite hierarchy gives the exact quantum maximum of the CHSH inequality, and the intermediate level 1+AB gives exact maxima for all one-sided tilted CHSH inequalities. This paper proves those are coincidences. It shows that no finite level of the hierarchy—and no relaxation built from any fixed finite list of measurement-projector words—reproduces the full eight-dimensional set of quantum behaviors in the (2,2,2) scenario. The proof homes in on the symmetric doubly tilted CHSH functional near its critical endpoint: the quantum advantage over the local value has the exact cubic form (4/3)(1-α)^3, while every fixed level's error grows faster, so the gap between the NPA bound and the quantum value diverges relative to the violation. A scaled expectation of a positive operator converges to the Motzkin polynomial, a nonnegative polynomial that cannot be a sum of squares; a bounded NPA error would make it one, forcing the contradiction. The result matters because it separates 'finite-level exactness for a particular Bell inequality' from 'exact finite description of the quantum set,' settling the open question negatively in the minimal scenario.

Core claim

The paper claims that for every fixed NPA level L, the level-L upper bound on the symmetric doubly tilted CHSH functional exceeds the exact quantum maximum near the critical point by an amount that grows faster than (1-α)^3, while the quantum advantage itself is exactly (4/3)(1-α)^3 to leading order. Concretely, ω_Q(1-T)=4-2T+(4/3)T^3-(4/9)T^4+... and (ω_L(1-T)-ω_Q(1-T))/T^3 → +∞ for every L. Since every standard finite-word-list NPA relaxation contains some level d, its feasible set strictly contains the quantum set Q, and the containing behaviors can be placed arbitrarily close to the local deterministic point. Thus the exactness of low levels for CHSH and one-sided tilts does not extend t

What carries the argument

The proof works by scaling analysis at α=1. A standard reduction of two binary observables to two-dimensional invariant blocks reduces the quantum optimization to 4×4 blocks parameterized by two local cosines; an exact determinant identity shows the optimal angles are of order (1-α)^{1/2} and the quantum advantage is of order (1-α)^3, with coefficient 4/3 fixed by the exact quartic formula for ω_Q. Writing T=t^2 and scaling the measurement angles by t, the positive operator ω_Q(1-t^2)I-H_t evaluated on an explicit three-parameter vector ξ_t gives, after division by t^6, the Motzkin polynomial 4/3 M(x/2,y/2). On the NPA side, strict primal feasibility and finite-dimensional duality give an at

Load-bearing premise

The load-bearing step is the exact determinant identity q_T(σ;u,v)=σP_T(σ;u,v)+8uvR_T(u,v) in the Supplemental Material, which the paper says follows by direct evaluation but does not verify by an explicit expansion; if that identity carries an algebraic error, the localization u+v<18T, the cubic coefficient 4/3, and the Motzkin limit would not follow.

What would settle it

Compute both sides of the Supplemental determinant identity q_T(σ;u,v)=σP_T+8uvR_T at, say, T=0.1, u=1, v=1, σ=0.5; any nonzero difference would invalidate the proof's endpoint scaling. Alternatively, solve the level-1+AB NPA semidefinite program for the doubly tilted CHSH functional at α=0.999 and check whether (ω_L-ω_Q)/(1-α)^3 keeps increasing past any bound; a finite limit would contradict Theorem 1.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

No finite NPA level, and no relaxation built from a fixed finite list of measurement-projector words with standard NPA constraints, has a feasible set equal to the quantum set in the (2,2,2) scenario.
The almost quantum set (level 1+AB) strictly contains the complete quantum set, with postquantum feasible behaviors accumulating at a single local deterministic behavior.
For the symmetric doubly tilted CHSH family, every fixed finite level fails near α=1; the finite-level exactness of CHSH at level 1 and of one-sided tilted CHSH at level 1+AB is a special property of those functionals, not of the quantum set.
Any exact finite description of the quantum set in this scenario would have to leave the standard NPA framework—for instance by letting the word list grow with the desired accuracy.

Where Pith is reading between the lines

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

The same staircase of divergences should appear for any Bell functional whose maximizers tend to a point where both measurement pairs become compatible; for such families, fixed-level NPA errors will generically overwhelm the quantum advantage near the endpoint.
The result does not show the quantum set in (2,2,2) is non-semialgebraic; it shows only that it is not a single fixed spectrahedral shadow, leaving room for a convergent level-dependent description.
Practically, the divergence means near-critical NPA upper bounds can exceed the true quantum value by an arbitrarily large multiple of the quantum violation, so small violations reported from finite-level relaxations should be checked against level growth.
A direct algebraic test: expand both sides of the determinant identity q_T(σ;u,v)=σP_T+8uvR_T to confirm the asserted 'direct evaluation', since this single identity supports the u+v<18T localization and the cubic coefficient.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The paper very likely proves that no finite NPA level is exact in the simplest Bell scenario, but the load-bearing determinant identity (Supplemental Eq. S45) needs machine verification before the proof is fully trustworthy.

read the letter

I read the full manuscript and supplement. The main result is real and the proof is mostly convincing. It shows that for the symmetric doubly tilted CHSH family, the quantum maximum exceeds the local value by (4/3)T^3, while every fixed NPA level's error blows up relative to that scale. It then upgrades from Bell values to behavior sets, showing that any fixed finite word-list relaxation strictly contains the quantum set, and even constructs postquantum behaviors that accumulate at a local deterministic point. That is a clean answer to an open question, and the proof strategy — block determinant, exact quartic from Ref. [7], Motzkin polynomial with Newton-polytope obstruction, fixed-level compactness — is coherent.

The stress-test note is on point. Supplemental Eq. (S45) is load-bearing. The identity q_T = sigma P_T + 8uv R_T is stated as following by "direct evaluation" with no derivation and no computer-algebra file. It drives the localization u+v<18T, the O(T^3) bound on the quantum excess, the limit S = UV(1-(U+V)/6), and the cubic coefficient 4/3. If that identity has a sign or factor error, the fixed-level contradiction never gets off the ground. I do not see an actual error, but for a proof whose whole point is to rule out exactness, an unverified algebraic linchpin is a real gap. A referee should ask for a machine-checked expansion or a symbolic verification file.

The dependence on Ref. [7] is less concerning. The quartic equation is published, and the expansion follows by implicit differentiation; the determinant identity independently fixes the cubic scale, so the import mainly quantifies higher-order remainders. The disclosure of the two concurrent preprints is honest and appropriate. It does reduce novelty, but the work appears independent and the proof is more self-contained in some respects.

The rest of the proof holds up on my read. The compactness argument is careful, the trace identity is clean, and the extraction of valid behaviors from level-d moment matrices is sound.

This is a within-field result, but an important one. Anyone working on the NPA hierarchy, semidefinite descriptions of quantum correlations, or almost-quantum correlations will want to know it. I would cite it once the determinant identity is verified.

Recommendation: send to a serious referee. The paper deserves refereeing, not desk rejection, but the referee should be asked to verify Eq. (S45), ideally with a symbolic computation. If that checks out, this is an accept.

Referee Report

1 major / 4 minor

Summary. The paper claims a negative resolution of the finite-level exactness question for the NPA hierarchy in the minimal (2,2,2) Bell scenario. For the symmetric doubly tilted CHSH functional h_α with α=1−T, it proves (i) the exact quantum maximum has the expansion ω_Q(1−T)=4−2T+(4/3)T^3−(4/9)T^4+(4/81)T^5+O(T^6); (ii) for every fixed level L, the level-L NPA upper bound exceeds ω_Q by an amount whose ratio to T^3 diverges as T↓0; and (iii) consequently every standard finite-level NPA relaxation, and more generally every relaxation based on a fixed finite list of projectors, strictly contains the full quantum set, with post-quantum feasible behaviors accumulating at a local deterministic point. The proof proceeds by reducing the quantum optimization to two-qubit blocks via Jordan's lemma, fixing the cubic endpoint scale through an exact determinant identity, constructing a scaled positive-operator expectation that converges to the Motzkin polynomial, and then proving that a bounded fixed-level error would force the Motzkin polynomial plus a nonnegative constant to be a sum of squares, which is ruled out by a Newton-polytope argument. Full proofs are in the Supplemental Material.

Significance. If correct, the paper settles an open question: no finite level of the standard NPA hierarchy, and no relaxation built from a fixed finite list of measurement-projector words, reproduces the complete quantum correlation set even in the simplest Bell scenario. The proof is fully analytical, avoids numerical fitting, treats all finite levels uniformly, and includes a constructive argument placing post-quantum behaviors arbitrarily close to a local deterministic point. A notable strength is the explicit endpoint analysis: the exact cubic coefficient 4/3 and the Motzkin-polynomial limit are derived rather than inferred numerically. The extension from Bell-value gaps to strict inclusion of behavior sets is also carefully addressed. The main weakness is that one load-bearing algebraic identity is asserted by direct evaluation without a machine-checkable derivation; however, my spot checks of special cases are consistent, and the issue appears to be repairable by supplying verification.

major comments (1)

[Supplemental Eq. (S45)] The determinant identity q_T(σ;u,v)=σP_T(σ;u,v)+8uvR_T(u,v) is load-bearing for the entire endpoint analysis. It is used to prove the localization u_T+v_T<18T, the O(T^3) bound on the quantum excess, the limit S=UV(1−(U+V)/6), and hence the matching cubic coefficient 4/3. The manuscript says only that this follows by 'direct evaluation' and gives no derivation, factorization, or computer-algebra file. An algebraic error here would break the proof before the Motzkin polynomial enters. I checked the T=0 cases u=v=0 and u=v=2, which are consistent, but the general identity is complicated enough that a sign or coefficient error is not implausible. Please provide a derivation (for example, by factoring at u=0 and v=0 and comparing finitely many coefficients) or attach a symbolic-verification script.

minor comments (4)

[Supplemental Definition S10 / Corollary S14] The proof of Corollary S14 uses convexity of the feasible behavior set N_S, which is true because N_S is a projection of a spectrahedral cone, but this is not stated explicitly. A one-sentence justification would help the reader.
[Supplemental Lemma S8] The notation π_{t,x,y}(w_i)ξ_t is terse. Since π_{t,x,y} is the 4×4 representation defined by the observables in Eq. (S104), a brief clarifying sentence would improve reproducibility, especially for readers who do not immediately connect the abstract group-algebra evaluation to the concrete qubit matrices.
[Main text, Eq. (10)] The text says the 'unique choice cancelling both squares' is ξ_t; strictly, C is fixed earlier by the vanishing of the t^4 term, and Eq. (10) uniquely fixes A and B. Consider rewording to 'the unique choice of A and B that cancels the squares, together with the previously fixed C'.
[Supplemental Eq. (S71)] The exact quartic F(λ,α) is imported from Ref. [7]. Since the present proof depends on this result, it would be helpful to state explicitly that the formula covers all 0≤α≤1 and that the relevant root is the one tending to 4 as α↑1. This is presumably clear from the reference, but a sentence in the Supplemental Material would make the dependence precise.

Circularity Check

0 steps flagged

No circularity: the quantum-value input and the NPA-certificate limit are independent; only a same-author citation and an unverified determinant identity are noted as risks.

full rationale

The main claim is not circular. Theorem 1 asserts a divergence of fixed NPA upper bounds from the quantum maximum; the paper uses the quantum maximum as an input and derives the NPA-side contradiction independently through the trace identity, fixed-level compactness, and the Motzkin-polynomial Newton-polytope obstruction. No parameter is fitted to the target conclusion, and the quantum maximum is not defined in terms of the NPA sets. The cubic coefficient 4/3 is derived in-paper from the determinant identity S45 and an explicit lower bound, before Ref. [7] is invoked for the finer analytic expansion. Ref. [7] shares an author with the present paper, but it is a published, parameter-free exact quartic equation for the quantum maximum; the target result of this paper is not assumed there, and the same cubic coefficient is independently established in this manuscript. The note about overlapping preprints does not introduce circularity. One genuine but non-circular risk is Supplemental Eq. (S45): the paper says 'Direct evaluation of the determinant gives the exact polynomial identity' and gives no machine-checkable expansion. That identity is load-bearing for the localization u_T+v_T<18T and the O(T^3) scale, so an algebraic error there would affect the proof; this is a verification/correctness concern, not a reduction of the conclusion to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No fitted parameters: all expansion coefficients come from exact polynomial identities (Eq. S79) and determinant bounds. The paper relies on background results: the exact quantum maximum from Ref. [7], Jordan's lemma, the Motzkin/Newton-polytope facts, and finite-dimensional SDP duality. No new physical entities are postulated.

axioms (5)

domain assumption The exact quantum maximum of h_α is the largest real root of quartic F(λ,α) (Eq. S71) and the optimal cosine is given by Eq. (S72), as derived in Ref. [7].
Imported from prior work by the same author and co-authors; used to obtain the analytic expansion (S80) and the remainder bound. Not re-derived in this paper.
standard math The determinant identity q_T(σ;u,v)=σP_T+8uvR_T (Eq. S45) is exactly correct.
Direct algebra asserted in Supplemental Material; if wrong, the endpoint scale, the O(T^3) bound, and the cubic coefficient 4/3 do not follow.
standard math Motzkin polynomial M(X,Y) is nonnegative but not a sum of squares, and M+c for c≥0 is not SOS (Newton polytope half property).
Known result from Hilbert/Motzkin/Reznick [15-17]; used for the contradiction that rules out bounded fixed-level error.
standard math Finite-dimensional SDP duality holds with Slater condition and dual attainment (Prop S3).
Used to get the Gram certificate and the trace identity that bounds certificate matrices.
standard math Jordan's lemma decomposition into 2D blocks (Lemma S2).
Used to reduce the quantum optimization to 4×4 block matrices and derive the endpoint determinant.

pith-pipeline@v1.3.0-alltime-deepseek · 246 in / 16479 out tokens · 223034 ms · 2026-08-02T01:43:29.800420+00:00 · methodology

0 comments

read the original abstract

The Navascu\'es--Pironio--Ac\'in (NPA) hierarchy gives the standard semidefinite outer approximations to quantum behaviors. Whether \emph{any} finite level can already equal the quantum set has remained open even in the bipartite scenario with two binary measurements per party. We demonstrate that \emph{no finite level} is exact. For the symmetric doubly tilted CHSH functional $h_\alpha=A_0B_0+A_0B_1+A_1B_0-A_1B_1+\alpha(A_0+B_0)$, set $T=1-\alpha$. Its quantum maximum satisfies $[\omega_{\rm Q}(1-T)-(4-2T)]/T^3\to4/3$, whereas every fixed NPA level satisfies $[\omega_L(1-T)-\omega_{\rm Q}(1-T)]/T^3\to+\infty$. Under the corresponding boundary rescaling, an explicit expectation of the positive operator $\omega_{\rm Q}(1-t^2)I-H_t$ converges to the Motzkin polynomial. A bounded fixed-level error would therefore make the Motzkin polynomial plus a nonnegative constant a sum of squares, which is impossible. Consequently, every standard NPA relaxation based on a fixed finite list of words in the measurement projectors strictly contains the complete quantum set, and its nonquantum behaviors accumulate at a local deterministic behavior. Thus, the finite-level exactness of CHSH and all one-sided tilted CHSH maxima does not extend to an exact finite-level description of the complete quantum set in the minimal scenario.

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 2 canonical work pages

[1]

(7) gives the exact cubic coefficient of the quantum advantage

Since the local value is 4 − 2T , Eq. (7) gives the exact cubic coefficient of the quantum advantage. The Motzkin polynomial.The cubic advantage fixes the boundary scale. Write T = t2 and vary the two local angles independently, A0 =Z, A 1 = cos(tx)Z+ sin(tx)X, B0 =Z, B 1 = cos(ty)Z+ sin(ty)X.(9) Let Ht(x, y) be the matrix of h1−t2 in this representation....

Pith/arXiv arXiv 2026
[2]

Navascu´ es, S

M. Navascu´ es, S. Pironio, and A. Ac ´ ın, Bounding the set of quantum correlations, Phys. Rev. Lett.98, 010401 (2007)

2007
[3]

Navascu´ es, S

M. Navascu´ es, S. Pironio, and A. Ac ´ ın, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations, New J. Phys.10, 073013 (2008)

2008
[4]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett.23, 880 (1969)

1969
[5]

B. S. Tsirelson, Quantum generalizations of Bell’s inequal- ity, Lett. Math. Phys.4, 93 (1980)

1980
[6]

Ac ´ ın, S

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

2012
[7]

Bamps and S

C. Bamps and S. Pironio, Sum-of-squares decompositions for a family of Clauser–Horne–Shimony–Holt-like inequal- ities and their application to self-testing, Phys. Rev. A 91, 052111 (2015)

2015
[8]

Gigena, E

N. Gigena, E. Panwar, G. Scala, M. Ara´ ujo, M. Farkas, and A. Chaturvedi, Self-testing tilted strategies for maxi- mal loophole-free nonlocality, npj Quantum Inf.11, 82 (2025)

2025
[9]

L. J. Landau, Empirical two-point correlation functions, Found. Phys.18, 449 (1988)

1988
[10]

T. P. Le, C. Meroni, B. Sturmfels, R. F. Werner, and T. Ziegler, Quantum correlations in the minimal scenario, Quantum7, 947 (2023)

2023
[11]

Mikos-Nuszkiewicz and J

A. Mikos-Nuszkiewicz and J. Kaniewski, Extremal points of the quantum set in the Clauser–Horne–Shimony–Holt scenario: Conjectured analytical solution, Phys. Rev. A 108, 012212 (2023)

2023
[12]

Barizien and J.-D

V. Barizien and J.-D. Bancal, Quantum statistics in the minimal Bell scenario, Nat. Phys.21, 577–582 (2025)

2025
[13]

Ishizaka, NPA hierarchy and extremal criterion in the simplest Bell scenario, Entropy27, 182 (2025), doi:10.3390/e27020182

S. Ishizaka, NPA hierarchy and extremal criterion in the simplest Bell scenario, Entropy27, 182 (2025), doi:10.3390/e27020182

work page doi:10.3390/e27020182 2025
[14]

Ara´ ujo, I

M. Ara´ ujo, I. Klep, A. J. P. Garner, T. V´ ertesi, and M. Navascu´ es, First-order optimality conditions for non- commutative optimization problems, Found. Comput. Math. (2026), doi:10.1007/s10208-026-09761-x

work page doi:10.1007/s10208-026-09761-x 2026
[15]

P. R. Halmos, Two subspaces, Trans. Am. Math. Soc. 144, 381 (1969)

1969
[16]

Hilbert, ¨Uber die Darstellung definiter Formen als Summe von Formenquadraten, Math

D. Hilbert, ¨Uber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann.32, 342 (1888)
[17]

T. S. Motzkin, The arithmetic-geometric inequality, in Inequalities, edited by O. Shisha (Academic Press, New York, 1967), pp. 205–224

1967
[18]

Reznick, Extremal PSD forms with few terms, Duke Math

B. Reznick, Extremal PSD forms with few terms, Duke Math. J.45, 363 (1978)

1978
[19]

Navascu´ es, Y

M. Navascu´ es, Y. Guryanova, M. J. Hoban, and A. Ac ´ ın, Almost quantum correlations, Nat. Commun.6, 6288 (2015)

2015
[20]

Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen, MIP∗ = RE, Commun. ACM64, 131–138 (2021), doi:10.1145/3485628

doi:10.1145/3485628 2021
[21]

Fanizza, L

M. Fanizza, L. Kroell, A. Mehta, C. Paddock, D. Ro- chette, W. Slofstra, and Y. Zhao, The NPA hierarchy does not always attain the commuting operator value, arXiv:2510.04943 [quant-ph] (2025)

arXiv 2025
[22]

No Finite NP A Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev.38, 49–95 (1996), doi:10.1137/1038003. 5 Supplemental Material for “No Finite NP A Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario” Anubhav Chaturvedi Division of Quantum Optics and Information, Institute of Theoretical Physics and Astrophysics, Faculty of Mathemati...

doi:10.1137/1038003 1996

[1] [1]

(7) gives the exact cubic coefficient of the quantum advantage

Since the local value is 4 − 2T , Eq. (7) gives the exact cubic coefficient of the quantum advantage. The Motzkin polynomial.The cubic advantage fixes the boundary scale. Write T = t2 and vary the two local angles independently, A0 =Z, A 1 = cos(tx)Z+ sin(tx)X, B0 =Z, B 1 = cos(ty)Z+ sin(ty)X.(9) Let Ht(x, y) be the matrix of h1−t2 in this representation....

Pith/arXiv arXiv 2026

[2] [2]

Navascu´ es, S

M. Navascu´ es, S. Pironio, and A. Ac ´ ın, Bounding the set of quantum correlations, Phys. Rev. Lett.98, 010401 (2007)

2007

[3] [3]

Navascu´ es, S

M. Navascu´ es, S. Pironio, and A. Ac ´ ın, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations, New J. Phys.10, 073013 (2008)

2008

[4] [4]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett.23, 880 (1969)

1969

[5] [5]

B. S. Tsirelson, Quantum generalizations of Bell’s inequal- ity, Lett. Math. Phys.4, 93 (1980)

1980

[6] [6]

Ac ´ ın, S

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

2012

[7] [7]

Bamps and S

C. Bamps and S. Pironio, Sum-of-squares decompositions for a family of Clauser–Horne–Shimony–Holt-like inequal- ities and their application to self-testing, Phys. Rev. A 91, 052111 (2015)

2015

[8] [8]

Gigena, E

N. Gigena, E. Panwar, G. Scala, M. Ara´ ujo, M. Farkas, and A. Chaturvedi, Self-testing tilted strategies for maxi- mal loophole-free nonlocality, npj Quantum Inf.11, 82 (2025)

2025

[9] [9]

L. J. Landau, Empirical two-point correlation functions, Found. Phys.18, 449 (1988)

1988

[10] [10]

T. P. Le, C. Meroni, B. Sturmfels, R. F. Werner, and T. Ziegler, Quantum correlations in the minimal scenario, Quantum7, 947 (2023)

2023

[11] [11]

Mikos-Nuszkiewicz and J

A. Mikos-Nuszkiewicz and J. Kaniewski, Extremal points of the quantum set in the Clauser–Horne–Shimony–Holt scenario: Conjectured analytical solution, Phys. Rev. A 108, 012212 (2023)

2023

[12] [12]

Barizien and J.-D

V. Barizien and J.-D. Bancal, Quantum statistics in the minimal Bell scenario, Nat. Phys.21, 577–582 (2025)

2025

[13] [13]

Ishizaka, NPA hierarchy and extremal criterion in the simplest Bell scenario, Entropy27, 182 (2025), doi:10.3390/e27020182

S. Ishizaka, NPA hierarchy and extremal criterion in the simplest Bell scenario, Entropy27, 182 (2025), doi:10.3390/e27020182

work page doi:10.3390/e27020182 2025

[14] [14]

Ara´ ujo, I

M. Ara´ ujo, I. Klep, A. J. P. Garner, T. V´ ertesi, and M. Navascu´ es, First-order optimality conditions for non- commutative optimization problems, Found. Comput. Math. (2026), doi:10.1007/s10208-026-09761-x

work page doi:10.1007/s10208-026-09761-x 2026

[15] [15]

P. R. Halmos, Two subspaces, Trans. Am. Math. Soc. 144, 381 (1969)

1969

[16] [16]

Hilbert, ¨Uber die Darstellung definiter Formen als Summe von Formenquadraten, Math

D. Hilbert, ¨Uber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann.32, 342 (1888)

[17] [17]

T. S. Motzkin, The arithmetic-geometric inequality, in Inequalities, edited by O. Shisha (Academic Press, New York, 1967), pp. 205–224

1967

[18] [18]

Reznick, Extremal PSD forms with few terms, Duke Math

B. Reznick, Extremal PSD forms with few terms, Duke Math. J.45, 363 (1978)

1978

[19] [19]

Navascu´ es, Y

M. Navascu´ es, Y. Guryanova, M. J. Hoban, and A. Ac ´ ın, Almost quantum correlations, Nat. Commun.6, 6288 (2015)

2015

[20] [20]

Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen, MIP∗ = RE, Commun. ACM64, 131–138 (2021), doi:10.1145/3485628

doi:10.1145/3485628 2021

[21] [21]

Fanizza, L

M. Fanizza, L. Kroell, A. Mehta, C. Paddock, D. Ro- chette, W. Slofstra, and Y. Zhao, The NPA hierarchy does not always attain the commuting operator value, arXiv:2510.04943 [quant-ph] (2025)

arXiv 2025

[22] [22]

No Finite NP A Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev.38, 49–95 (1996), doi:10.1137/1038003. 5 Supplemental Material for “No Finite NP A Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario” Anubhav Chaturvedi Division of Quantum Optics and Information, Institute of Theoretical Physics and Astrophysics, Faculty of Mathemati...

doi:10.1137/1038003 1996