Nearly tight weighted 2-designs in complex projective spaces of every dimension

Dustin G. Mixon; John Jasper

REVIEW 1 major objections 3 minor 18 references

Nearly tight weighted 2-designs in complex projective spaces of every dimension

T0 review · 1 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A single Sidon-set construction gives nearly minimal weighted 2-designs in every dimension.

desk verdict First general nearly-tight weighted 2-design bound via Sidon sets; clean proof, only real dependency is a deep prime-gap theorem. read the letter →

arxiv 2501.14938 v1 pith:EPDANINJ submitted 2025-01-24 math.FA math.CO

John Jasper , Dustin G. Mixon This is my paper · ORCID

classification math.FAmath.CO MSC 05B3011B7542C15

keywords weighted2-designscomplexprojectivespacesSidonsetsZauner'sconjectureSIC-POVMsentanglementbreakingrankprimegapsfiniteabeliangroups

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

Zauner's conjecture asserts that every complex projective space $\mathbb{CP}^{d-1}$ carries $d^2$ equally spaced points, a SIC, which is equivalent to the smallest weighted $2$-design having size $n(d)=d^2$. The paper proves $n(d)\le d^2+O(d^{1.525})$ for every dimension $d$, the first general upper bound that is $o(d^4)$ and therefore within a lower-order term of the $d^2$ lower bound. The proof converts any Sidon set of size $d$ in a finite abelian group $G$ into an explicit weighted $2$-design of size $|G|+d$, then uses dense Sidon sets in groups of order near $d^2$ together with the cited prime-gap estimate. A dimension-by-dimension table through $d=150$ shows the construction improving on all previous bounds except where a SIC is already known.

What carries the argument

The machine that carries the argument is a map from any subset $S$ of a finite abelian group $G$ to a sequence of $|G|+|S|$ unit vectors in $\mathbb{C}^{|S|}$: for each character $\alpha$ of $G$, take $x_\alpha(s)=\alpha(s)/\sqrt{|S|}$; for each $r\in S$, take the standard basis vector $e_r$. The proof shows that when $S$ is a Sidon set (all pairwise sums $a+b$ with $a,b\in S$ distinct), the sum of rank-one projections from the characters equals $(|G|/|S|^2)(A+B)$ and the basis vectors supply $B$, so the symmetric projection $P$ on $(\mathbb{C}^{|S|})^{\otimes 2}$ is the nonnegative combination $P=\frac{|S|^2}{2|G|}X+\frac{1}{2}E$; that is exactly a weighted $2$-design. Dense Sidon sets with $|G|=d^2+O(d^{1.525})$ come from the families collected in Proposition 9, and the least-prime bound converts the group order into the theorem's exponent.

What would settle it

Numerically test the construction on a dense Sidon set of size $d$ not covered by Table 1, for example the Erdős–Turán set in $(\mathbb{F}_p)^2$ with $p=10^4+7$: form the $|G|+|S|$ vectors, compute the weighted sum of rank-one projections, and check equality with the symmetric projection; one mismatch for any Sidon set would disprove the paper's main reduction $n(d)\le m(d)+d$, while agreement supports it.

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Extended reading notes

Core claim

The central claim is Theorem 4: $n(d)\le d^2+O(d^{1.525})$, where $n(d)$ is the smallest size of a weighted $2$-design for $\mathbb{CP}^{d-1}$ (equivalently, the entanglement-breaking rank of a certain quantum channel). The route is Corollary 8, $n(d)\le m(d)+d$, with $m(d)$ the smallest order of a finite abelian group containing a Sidon set of size $d$. Applying the group-character construction of Section 2 to any Sidon set gives a weighted $2$-design of size $|G|+|S|$; choosing $S$ of size $d$ from a dense Sidon set in a group of order essentially $d^2$, and bounding the least prime $p(d)\ge d$ by $d+O(d^{0.525})$ via the cited prime-gap theorem, yields the theorem. This is the first known universal upper bound on $n(d)$ that is $o(d^4)$.

Load-bearing premise

The bound's stated exponent rests on the cited prime-gap theorem that the smallest prime $p(d)\ge d$ satisfies $p(d)\le d+O(d^{0.525})$ for every $d$; the paper does not prove this analytic number theory input.

Editorial extensions

If this is right

For every dimension $d$, a weighted $2$-design of size within $O(d^{1.525})$ of the lower bound $d^2$ can now be written down explicitly, not merely shown to exist.
In all dimensions $d\le150$ displayed in Table 1, the new bound is the best known unless an exact SIC is already available, and the strict improvements come from taking subsets of the Hughes-type Sidon sets.
The construction cannot prove Zauner's conjecture: any group carrying a Sidon set of size $d$ has order at least $d^2-d+1$, so these designs always have size at least $d^2+1$.
Better prime-gap results would immediately sharpen the exponent, while reaching $d^2+o(d\log^2 d)$ would require a fundamentally different approach under the usual heuristic for prime gaps.

Reading between the lines

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

Extending the paper's construction, the same Sidon-set pipeline plausibly yields weighted $k$-designs for larger $k$ by replacing the symmetric projection with the projection onto $k$-th order symmetric tensors, giving small tomographically complete measurement sets in every dimension.
Because the table shows Hughes-type Sidon subsets providing the strict improvements, it is likely that multiplicative Sidon sets in groups of the form $(\mathbb{F}_q^\times)^2$ are the most efficient raw material known; examining larger $q$ would test whether this advantage persists asymptotically.
The prime-gap dependence couples a problem in quantum information (entanglement breaking rank) to the short-interval distribution of primes; a lower bound on $n(d)$ of the form $d^2+\omega(d)$ would therefore constrain the density of Sidon sets in finite abelian groups, a connection the paper leaves implicit.

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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

The paper does something genuinely new: it shows that any Sidon set in any finite abelian group gives a weighted projective 2-design via the Bodmann–Haas construction, and then it leverages the densest known Sidon-set families to get n(d) ≤ d^2 + O(d^{1.525}) for every dimension d. That is the first general upper bound that is o(d^4), let alone within lower-order terms of the d^2 lower bound. The connection to Hughes Sidon sets in particular is new and, as the table shows, it drives most of the concrete improvements below dimension 150.

What the paper does well: Theorem 7 is a direct computation and it is correct. The averaging over characters gives X = (|G|/|S|^2)(A+B), the standard-basis part gives E = B, and the weights are nonnegative. Corollary 8 then makes the whole strategy transparent: n(d) ≤ m(d)+d, where m(d) is the smallest group with a Sidon set of size d. The use of Erdős–Turán sets and Baker–Harman–Pintz is clean and the O(d^{1.525}) follows immediately. I also appreciate that the authors are upfront in Section 4 that this approach cannot reach d^2, since the construction always gives at least d^2+1 points. That is a real limitation and they state it plainly.

The soft spots are mostly minor. The main external input, Baker–Harman–Pintz, is a deep analytic number theory theorem, but it is a published theorem and not something the paper needs to reprove. If a weaker prime-gap bound were used, the exponent would degrade but the construction would still be o(d^4), so the core contribution does not stand or fall on that theorem. The table is large and the choices of which Hughes subset to use are computational, but they are reproducible from the stated parameters. One small thing: the text says the weighted design has size |G|+|S|, and the lower bound from the pigeonhole principle gives |G| ≥ d^2−d+1, so indeed |G|+|S| ≥ d^2+1; the paper's claim that the method cannot prove Zauner's conjecture is accurate.

Who is this for? Anyone working on weighted 2-designs, entanglement-breaking rank, or quantitative approaches to Zauner's conjecture. It is a solid, honest paper with a correct central proof and a meaningful new bound. It deserves a serious referee and should be published, probably with only minor revisions. I would cite it if I were writing in this area.

My recommendation: send it to peer review; it will pass.

Referee Report

1 major / 3 minor

Summary. The paper constructs weighted projective 2-designs in CP^{d-1} of size d^2 + O(d^{1.525}) for every d, giving the first general upper bound on the minimal design size n(d) that is o(d^4). The construction applies the Bodmann-Haas idea to Sidon sets: Theorem 7 proves directly that for any finite abelian group G and Sidon set S in G, the |G|+|S| unit vectors x_alpha (alpha in hat G) and e_r (r in S) form a weighted 2-design for CP^{|S|-1}, with nonnegative weights |S|^2/(2|G|) and 1/2; the proof is an explicit entrywise computation of the projection onto the symmetric tensors. Corollary 8 then gives n(d) <= m(d) + d, where m(d) is the minimal size of a group containing a Sidon set of size d. Combining the Erdos-Turan Sidon set (x,x^2) in F_p^2 with the Baker-Harman-Pintz prime gap bound p(d) <= d + O(d^{0.525}) yields Theorem 4. The paper also tabulates the resulting bounds for d <= 150, compares them with all previous bounds, and candidly explains why the method cannot reach Zauner's conjecture.

Significance. If the results stand, Theorem 4 is a genuine quantitative advance: it is the first dimension-uniform bound of the form d^2 + o(d^2), improving on the previous universal bound (binom(d+1,2))^2 = Theta(d^4) and approaching the SIC lower bound n(d) >= d^2. The proof is a clean, fully checkable direct computation with no fitted parameters, and the external inputs (classical Sidon families and the published Baker-Harman-Pintz theorem) are independent of the paper's claims. The explicit nonnegative weights and the honest statement of limitations in Section 4 are additional strengths. Spot-checks of Table 1 are consistent with the formulas in Proposition 3. One caveat: Proposition 9(c) misstates the Bose-Chowla construction, and since Table 1 and parts of Section 3 use that family, this component needs correction; Theorem 4 itself does not depend on it.

major comments (1)

[Section 3, Proposition 9(c)] Proposition 9(c) is false as stated. For the relative trace Tr_{F_{q^2}/F_q}(x) = x + x^q, the set S = {x in (F_{q^2})^x : Tr(x) = 0} has exactly q-1 elements, not q, because the trace-zero set is a 1-dimensional F_q-subspace that includes 0. It is also not Sidon in the multiplicative group (F_{q^2})^x: for odd q >= 3, any nonzero trace-zero omega satisfies (-omega)^2 = omega^2 with -omega != omega, so the distinct unordered pairs {-omega,-omega} and {omega,omega} have equal product; for even q >= 4 the set is F_q^x, where 1*g^2 = g*g for g != 1. The intended Bose-Chowla Sidon set of size q is the affine hyperplane {x in (F_{q^2})^x : Tr(x) = c} for a fixed nonzero c in F_q, for instance c = 1; the standard verification is that if a,b,c,d in S satisfy ab = cd and t := a/c = d/b, then applying the trace to a = tc and d = tb gives c(t - t^q) = 1 - t^q and b(t - t^q) = 1 - t^q, forcing either t = 1 or b = c, and hence {a,b} = {c,d} as multisets. This correction is load-bearing for Table 1, whose rows using Bose(q) (for instance d = 2, 7, 11, 13, 16, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149) presume a Sidon set of size q in a group of order q^2 - 1, and for the discussion's assertion that m(d) is never achieved by the Erdos-Turan family. Theorem 4 is unaffected because its proof uses only Proposition 9(a) and the prime-gap bound, but Proposition 9(c) must be corrected and the dependent sentences and table entries re-verified accordingly.

minor comments (3)

[Section 3, after Corollary 8] The sentence 'Corollary 8 is the sharpest known upper bound on n(d) for all but finitely many d' is imprecise, because for the infinite families of dimensions with d-1 a prime power or d a prime power the bound only ties Proposition 3(c) and (d), respectively; please rephrase as 'at least as sharp as, and for all but finitely many dimensions strictly sharper than, the previously known bounds'.
[Table 1] The claim 'For every d <= 150, exactly one of three things happens' depends on the completeness of the list of known SIC dimensions taken from the website [8] plus four added references; since the paper does not reproduce the dimension set, the reader cannot check the rows marked (a). Please include the full list of dimensions with known SICs or state explicitly how the list was assembled.
[Proof of Theorem 4] For readability, the proof should add one sentence explaining that the Baker-Harman-Pintz bound on consecutive prime gaps implies p(d) <= d + O(d^{0.525}) for p(d) the least prime >= d, by applying the gap bound to the largest prime strictly below d.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 7 verifies the design condition by direct computation, and the main bound uses only the external Baker–Harman–Pintz prime gap theorem.

full rationale

The paper's central derivation is self-contained. Theorem 7 computes the matrices X and E for the Bodmann–Haas construction and shows, using only the Sidon property, that P = (|S|^2/(2|G|))X + (1/2)E, which is exactly the weighted 2-design condition with nonnegative weights. Corollary 8 then bounds n(d) by m(d) + d, and Theorem 4 follows from the Erdős–Turán Sidon set in F_p^2 together with the externally proved Baker–Harman–Pintz bound on prime gaps. No parameter is fitted to the target quantity, no conclusion is assumed from the desired result, and the only self-citation ([12], used for Proposition 3(e) and background equivalence) is not load-bearing for the main theorem. The construction is verified against the definition of a weighted 2-design by explicit algebra, so there is no circular reduction.

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

The paper introduces no free parameters and no new postulated entities. Its argument rests on standard theorems from additive combinatorics (Sidon set families) and analytic number theory (prime gaps), plus the prior equivalence between weighted 2-designs and entanglement breaking rank.

assumptions (3)

standard math Baker-Harman-Pintz prime gap estimate: p(d) <= d + O(d^{0.525}) for the smallest prime p(d) >= d
Used in the proof of Theorem 4 to bound m(d) <= p(d)^2; cited from [4].
standard math Existence of the Sidon set constructions in Proposition 9 (Erdos-Turan, Singer, Bose, Spence, Hughes)
These families are cited from the literature; the paper relies on them to bound m(d) for all d.
domain assumption The equivalence between the size of the smallest weighted 2-design and the entanglement breaking rank n(d)
This equivalence, from [12] (Iverson-King-Mixon), identifies n(d) with the design size; the paper's progress is stated in terms of n(d).

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Cite this review

Pith. "Pith review of Nearly tight weighted 2-designs in complex projective spaces of every dimension." pith.science (2026). https://pith.science/paper/EPDANINJ

@misc{pith2026250114938,
  author       = {Pith},
  title        = {Pith review of: Nearly tight weighted 2-designs in complex projective spaces of every dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EPDANINJ}},
  note         = {Machine review of arXiv:2501.14938}
}

read the original abstract

We use dense Sidon sets to construct small weighted projective 2-designs. This represents quantitative progress on Zauner's conjecture.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 16 canonical work pages

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Appleby, S

M. Appleby, S. T. Flammia, G. S. Kopp, A Constructive Approach t o Zauner’s Conjecture via the Stark Conjectures, arXiv:2501.03970 (2025)

arXiv 2025
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Appleby, S

M. Appleby, S. Flammia, G. McConnell, J. Yard, SICs and algebraic n umber theory, Found. Phys. (2017) 1–18

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R. C. Baker, G. Harman, J. Pintz, The diﬀerence between conse cutive primes, II, Proc. London Math. Soc. 83 (2001) 532–562

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arXiv 2024
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B. G. Bodmann, J. Haas, Achieving the orthoplex bound and cons tructing weighted complex projective 2-designs with Singer sets, Linear Alge bra Appl. 511 (2016) 54–71

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G. Zauner, Quantum designs, PhD thesis, U. Vienna, 1999. 7 Table 1: Best known bounds on n(d) d d2 Prior bound on n(d) Best known bound on m(d) + d Prop. 3 part Sidon set Group 1 1 1 2 (a) {0} { 0} 2 4 4 5 (a) Bose(2) Z3 3 9 9 10 (a) Singer(2) Z7 4 16 16 17 (a) Singer(3) Z13 5...

1999

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Appleby, I

M. Appleby, I. Bengtsson, M. Grassl, M. Harrison, G. McConnell, SIC-POVMs from Stark units: Prime dimensions n2 + 3, J. Math. Phys. 63 (2022) 112205. 6

work page 2022

[2] [2]

Appleby, S

M. Appleby, S. T. Flammia, G. S. Kopp, A Constructive Approach t o Zauner’s Conjecture via the Stark Conjectures, arXiv:2501.03970 (2025)

arXiv 2025

[3] [3]

Appleby, S

M. Appleby, S. Flammia, G. McConnell, J. Yard, SICs and algebraic n umber theory, Found. Phys. (2017) 1–18

work page 2017

[4] [4]

R. C. Baker, G. Harman, J. Pintz, The diﬀerence between conse cutive primes, II, Proc. London Math. Soc. 83 (2001) 532–562

work page 2001

[5] [5]

Bengtsson, M

I. Bengtsson, M. Grassl, G. McConnell, SIC-POVMs from Stark u nits: Dimen- sions n2 + 3 = 4 p, p prime, arXiv:2403.02872 (2024)

arXiv 2024

[6] [6]

B. G. Bodmann, J. Haas, Achieving the orthoplex bound and cons tructing weighted complex projective 2-designs with Singer sets, Linear Alge bra Appl. 511 (2016) 54–71

work page 2016

[7] [7]

Eberhard, F

S. Eberhard, F. Manners, The Apparent Structure of Dense S idon Sets, Elec- tronic J. Combin. 30 (2023)

work page 2023

[8] [8]

Flammia, Exact SIC ﬁducial vectors, physics.usyd.edu.au/~sflammia/SIC/

S. Flammia, Exact SIC ﬁducial vectors, physics.usyd.edu.au/~sflammia/SIC/

work page