The sum-of-squares hierarchy on the sphere, and applications in quantum information theory

Hamza Fawzi; Kun Fang

REVIEW 3 major objections 3 minor 2 cited by

The sum-of-squares hierarchy on the sphere, and applications in quantum information theory

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that the sum-of-squares hierarchy on the sphere converges at rate O(d^2/ℓ^2) and transfers that rate to quantum entanglement bounds.

desk verdict A valuable paper with a genuinely new quadratic rate and a clean DPS/SOS duality, but Section 3 contains two false constant bounds that must be fixed before it is citable as is. read the letter →

arxiv 1908.05155 v1 pith:JE3MNTER submitted 2019-08-14 math.OC cs.CCquant-ph

Kun Fang , Hamza Fawzi This is my paper · ORCID

classification math.OCcs.CCquant-ph MSC 90C2290C2633C4514P10

keywords sum-of-squareshierarchypolynomialoptimizationonthesphereGegenbauerpolynomialsconvergenceratematrix-valuedDPSbestseparablestatequantumentanglement

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 establishes that the sum-of-squares (SOS) hierarchy for maximizing a homogeneous polynomial of degree $2n$ on the unit sphere $\mathbb{S}^{d-1}$ converges at the rate $O(d^2/\ell^2)$ once the level $\ell$ is at least a constant times $d$, where the constants depend only on $n$. This improves the previously known $O(d/\ell)$ rate by a quadratic factor and answers an open question from recent work on SOS upper bounds on the sphere. The same statement holds for matrix-valued polynomials on the sphere, with bounds independent of the matrix size. Via a duality between sums of squares and the DPS hierarchy of separable quantum states, the result yields an $O(d_B^2/\ell^2)$ convergence rate for the Best Separable State problem and extends to polynomials of arbitrary even degree. The proof rests on constructing a polynomial kernel whose Gegenbauer coefficients approach $1$ at the faster rate.

What carries the argument

The engine is the polynomial kernel $K(x,y) = q(\langle x,y\rangle)^2$ on $\mathbb{S}^{d-1} \times \mathbb{S}^{d-1}$. The Funk-Hecke formula diagonalizes this integral operator: its eigenvalues are the Gegenbauer coefficients $\lambda_{2k}$ of the univariate polynomial $\varphi(t)=q(t)^2$, and if those coefficients are close enough to $1$, then $K^{-1}(F+\delta I)$ is nonnegative and $F+\delta I = K K^{-1}(F+\delta I)$ is an explicit sum of squares. Choosing $q$ is recast as a generalized Toeplitz eigenvalue problem: $\rho_{2n}(d,\ell) = \min_{\|e\|=1} \sum_{k=1}^n |(e^T T[C_{2k}/C_{2k}(1)]e)^{-1}-1|$, with $T[h]$ the truncated multiplication matrices for normalized Gegenbauer polynomials. The asymptotic rate comes from lower-bounding the largest eigenvalue of $T[h]$ for $h = \frac{1}{n}\sum_{k=1}^n C_{2k}/C_{2k}(1)$ via the largest root of $C_{\ell+1}$, and the matrix-valued case follows by applying the same harmonic-component bounds entrywise with spectral norms.

What would settle it

Compute the largest zero of $C_{\ell+1}$ for $d=2$, $\ell=10$: it is about $0.9898$, below the lower bound $1 - d^2/(4\ell^2) = 0.99$, so checking the quoted zero bound over the paper's full parameter range would settle whether the proof's constants hold. One can also evaluate $h'(1) = (n+1)(3d+4n-4)/(3(d-1))$ at $n=d$ and compare it with $7n/3$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: if $F(x)$ is a homogeneous symmetric matrix-valued polynomial of degree $2n$ in $d$ variables with $n \le d$ and $0 \le F(x) \le I$ on $\mathbb{S}^{d-1}$, then for every $\ell \ge C_n d$ the polynomial $F + C'_n (d/\ell)^2 I$ is a sum of squares of polynomials of degree at most $\ell$ on the sphere. For the scalar polynomial $F = (p_{\max}-p)/(p_{\max}-p_{\min})$, this gives $(p_\ell-p_{\min})/(p_{\max}-p_{\min}) \le 1 + (C_n d/\ell)^2$, a quadratic improvement over the previously known rate. The more general Theorem 6 bounds every level $\ell$ through a quantity $\rho_{2n}(d,\ell)$ that can be computed for degree $2$ and $4$, and the $O(d^2/\ell^2)$ rate is shown tight for quadratic polynomials. A duality theorem identifies the DPS hierarchy of separable quantum states with a real-sum-of-squares condition on Hermitian polynomials, so the same rate transfers to the Best Separable State problem: $h_{\mathrm{Sep}}(M) \le h_{\mathrm{DPS}_\ell}(M) \le (1 + C d_B^2/\ell^2) h_{\mathrm{Sep}}(M)$ for $\ell \ge C' d_B$.

Load-bearing premise

The proof's explicit constants in the $O(d^2/\ell^2)$ estimate rest on two literature bounds holding throughout the asserted range: the largest zero of the Gegenbauer polynomial $C_{\ell+1}$ is at least $1 - d^2/(4\ell^2)$, and the derivative $h'(1)$ of $h = \frac{1}{n}\sum_{k=1}^n C_{2k}/C_{2k}(1)$ is at most $7n/3$; if either fails, those constants are not established.

Editorial extensions

If this is right

For a homogeneous polynomial of fixed degree $2n$, reaching relative accuracy $\varepsilon$ requires level $\ell = O(d/\sqrt{\varepsilon})$ rather than $O(d/\varepsilon)$.
The certificate level is independent of the size of the matrix-valued polynomial, so operator-valued or multi-constraint problems do not require deeper levels.
The DPS hierarchy approximates the best separable state value with relative error $O(d_B^2/\ell^2)$ at level $\ell \ge C d_B$, with constants independent of the other subsystem dimension.
The duality theorem makes the quantum convergence rate a special case of the SOS rate, so the same bound applies to nonquadratic Hermitian polynomials.

Reading between the lines

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

A direct numerical computation of $\rho_{2n}(d,\ell)$ for small $d$ would test whether the $O(d^2/\ell^2)$ scaling persists outside the proven regime $\ell \ge C_n d$.
The same kernel construction transfers naturally to products of spheres, since a matrix-valued polynomial in $x$ is a bihomogeneous polynomial in $(x,y)$; this would give SOS-based bounds for multilinear optimization.
If the quadratic-case tightness is representative, the $d^2/\ell^2$ rate is likely the true order of the SOS upper-bound hierarchy on the sphere, and further speedups would require a different relaxation family rather than a sharper kernel.

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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

Colleague,

The short version: this is a worthwhile paper that deserves a serious referee, but the proof of the main rate has two numerical inequalities that are simply false as written. The strategy and the theorem are probably right; the constants are not established.

What is new: the kernel optimization idea — tuning a univariate q(t) through a generalized Toeplitz eigenvalue problem to make the Gegenbauer coefficients converge quadratically — is a real improvement over Reznick's d/ℓ kernels. The matrix-valued version is a nice extension, and the formal proof of the DPS/SOS duality (Theorem 12) fills a gap that had been circling the literature. Recovering NOP09 from a SOS argument is a neat byproduct. No fitted parameters, no self-citation games; the NOP09 recovery is an independent sanity check.

Where it wobbles: Proposition 7's closing inequality chain. First, the cited DJ12 bound x_{ℓ+1,ℓ+1} ≥ 1 − d²/(4ℓ²) fails for d=2, where the largest zero is cos(π/(2(ℓ+1))) ≈ 1 − π²/(8ℓ²), smaller than 1 − 1/ℓ². Second, the claim h'(1) ≤ 7n/3 is false: at n=d=2, h'(1)=10 while 7n/3=14/3, and in fact h'(1) > 7n/3 whenever n=d. Both failures are in the last two displayed steps of Proposition 7. These are constant-scale errors, not structural ones: a correct root bound with a slightly larger constant and a uniform bound h'(1) ≤ c n would preserve the O(d²/ℓ²) conclusion. But as written the proof does not establish the constants in Theorems 1 and 2.

The rest of the paper is solid. The duality proof in Appendix B is careful, and the attention to monomial sets in the rsos vs csos distinction is right. The tightness argument for n=2 also looks fine.

Who this is for: anyone working on SOS convergence rates, Lasserre hierarchies, or DPS hierarchies in quantum information. The paper deserves a rigorous referee—send it out. The authors should fix Proposition 7 before publication.

Best,

Referee Report

3 major / 3 minor

Summary. The paper studies the convergence rate of the Lasserre/sum-of-squares hierarchy for maximizing a homogeneous polynomial of degree 2n on the unit sphere S^{d-1}. Its main result, Theorem 2, states that for n ≤ d and a homogeneous matrix-valued polynomial F of degree 2n with 0 ≤ F ≤ I on the sphere, F + C'_n (d/ℓ)^2 I is ℓ-sos whenever ℓ ≥ C_n d, with constants depending only on n; Theorem 1 is the scalar corollary asserting (p_ℓ − p_min)/(p_max − p_min) ≤ 1 + (C_n d/ℓ)^2. The proof uses positive kernels K(x,y) = q(⟨x,y⟩)^2 and analyzes their Gegenbauer coefficients through generalized Toeplitz matrices. The paper also proves a duality theorem (Theorem 12) identifying the dual of the DPS hierarchy in quantum information with real sum-of-squares certificates of the associated Hermitian polynomial, and uses it to transfer the rate result to the Best Separable State problem, recovering and generalizing the quadratic convergence rate of Navascués, Owari and Plenio. A tightness result (Theorem 10) shows the quadratic rate cannot be improved in the quadratic case.

Significance. If the proof is repaired, the paper settles an open question raised by de Klerk and Laurent by improving the known O(d/ℓ) rate for the upper-bound SOS hierarchy on the sphere to O(d^2/ℓ^2). The matrix-valued formulation with constants independent of the matrix size is a real strength, as is the self-contained proof of the DPS/SOS duality, which fills a gap in the literature. The recovery of the known NOP09 rate provides a useful external check, and the tightness analysis for quadratic polynomials is a good addition. However, the proof of the general-degree rate rests on Proposition 7, and two inequalities used there are false as stated. Because Proposition 7 is the only step producing the O(d^2/ℓ^2) estimate on ρ_{2n}, the constants in Theorems 6, 2, and 1 are not established in the present manuscript. The errors concern constants and thresholds rather than the overall proof architecture, so the central claim remains plausible and likely repairable.

major comments (3)

[Section 3, Proposition 7] The proof invokes the bound x_{ℓ+1,ℓ+1} ≥ 1 − d^2/(4ℓ^2) from [DJ12, Section 2.3]. This bound is false for d = 2, which is in the admissible range n ≤ d. For d = 2 the Gegenbauer polynomials are (up to scaling) the Chebyshev polynomials, so the largest zero of C_{ℓ+1} is cos(π/(2(ℓ+1))) = 1 − π^2/(8ℓ^2) + O(ℓ^{-3}), which is strictly smaller than 1 − 1/ℓ^2 for ℓ = 10 and asymptotically for all larger ℓ. The authors need either a valid universal root bound with a larger constant, or an explicit dimension restriction together with a separate argument for the excluded small dimensions.
[Section 3, Proposition 7] The assertion h'(1) ≤ 7n/3, derived from h'(1) = (n+1)(3d+4n−4)/(3(d−1)) and n ≤ d, is false. For n = d = 2 one has h'(1) = 10, while 7n/3 = 14/3. In fact, for d = n the exact value is h'(1) = 7n/3 + (10n−4)/(3(n−1)), which exceeds 7n/3 for every n ≥ 2. Consequently the final lower bound λ_max(T[h]) ≥ 1 − (7n/12)d^2/ℓ^2 is not proven. A correct uniform bound such as h'(1) ≤ 5n for n ≤ d would repair the argument with a different constant.
[Section 3, Theorems 6, 2, and 1] Because Proposition 7 is the only step that bounds ρ̃_{2n}(d,ℓ) by O(d^2/ℓ^2), and because Proposition 9 transfers that bound to ρ_{2n}(d,ℓ), the false inequalities in Proposition 7 leave the constants C_n and C'_n in Theorems 6, 2, and 1 unproven as written. The O(d^2/ℓ^2) rate may well survive with corrected constants, but the current proof is incomplete at this load-bearing point.

minor comments (3)

[Section 4.4, Theorem 15] The phrase “where C, C′ > 0 is some absolute constant” should be “where C, C′ > 0 are absolute constants.”
[Section 3, Eq. (13)] The inverse λ_{2k}^{-1} in the definition of ρ_{2n}(d,ℓ) presupposes λ_{2k} ≠ 0; the authors could state explicitly that the minimization can be restricted to polynomials for which the even Gegenbauer coefficients are positive, since otherwise the objective is not well defined.
[Section 4.1, Eq. (22)] The notation in Eq. (22) uses a bracehtipup/down artefact in the displayed formula; this should be cleaned up to a standard partial-transpose notation such as (I_A ⊗ T_{B_1} ⊗ ⋯ ⊗ T_{B_s} ⊗ I_{B_{s+1}} ⊗ ⋯ ⊗ I_{B_ℓ}).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation is self-contained; Proposition 7's constant gap is a correctness issue, not circularity.

full rationale

I found no circular step in the paper. The central result, Theorem 2, is proved from an explicit kernel construction: for a nonnegative normalized polynomial F, the paper writes F + δ I = K K^{-1}(F + δ I) with K defined by φ(⟨x,y⟩) = q(⟨x,y⟩)^2, and the error ‖K^{-1}(F + δ I) - (F + δ I)‖∞ is controlled by the quantity ρ_{2n}(d,ℓ). The proof of the O(d²/ℓ²) rate then reduces to the analysis of generalized Toeplitz matrices associated with Gegenbauer polynomials (Proposition 7 and surrounding text). This analysis uses external, explicitly cited results on extreme zeros (DJ12, ADGR04) and standard orthogonal-polynomial facts (Par65, DLMF); no parameter is fitted to the target rate, no prior work by the same authors is invoked as load-bearing, and the recovered Navascués-Owari-Plenio DPS rate is presented as an external consistency check rather than an input. Theorem 1 follows from Theorem 2 by the legitimate normalization F = (p_max - p)/(p_max - p_min), which is an application of the theorem, not a disguised restatement of the conclusion. I note a genuine correctness gap in the written constants: the cited bound x_{ℓ+1,ℓ+1} ≥ 1 - d²/(4ℓ²) fails for d=2 at the relevant Chebyshev zeros, and the asserted inequality h'(1) ≤ 7n/3 from the displayed formula h'(1) = (n+1)(3d+4n-4)/(3(d-1)) and n ≤ d is false at n=d=2, where h'(1)=10 > 14/3. That is an arithmetic/root-bound error affecting the stated constants in Proposition 7, but it is not circularity: the conclusion is not equivalent to an input by construction, and a corrected constant would preserve the O(d²/ℓ²) rate. Since circularity is about the reduction of the claimed derivation to its own inputs, and no such reduction occurs, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. The derivation relies on standard orthogonal polynomial theory and cited bounds, of which the Driver-Jordaan root bound is the most fragile.

assumptions (5)

standard math Funk-Hecke formula and Gegenbauer basis diagonalize SO(d)-invariant kernels on the sphere.
Used throughout Sections 2-3 to express the transform Kh in the spherical harmonic basis.
standard math Eigenvalues of T[f] for linear f are the roots of p_{ℓ+1} (Parter).
Proposition 8, used to compute λ_max(T[hbar]) in Proposition 7.
domain assumption Gegenbauer C_i lies above its tangent at t=1 (Proposition 18).
Proved in Appendix A via DLMF properties; needs convexity-type behavior of ultraspherical polynomials on [-1,1].
domain assumption Largest zero bound x_{ℓ+1,ℓ+1} ≥ 1 - d^2/(4ℓ^2) from Driver-Jordaan.
Load-bearing for Proposition 7; appears numerically false for small d, making it the key fragility.
standard math Integral SOS representations can be truncated to finite SOS by convexity.
Used at the end of the proof of Theorem 6 to turn ∫ q^2 H dσ into a finite sum of squares.

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Pith. "Pith review of The sum-of-squares hierarchy on the sphere, and applications in quantum information theory." pith.science (2026). https://pith.science/paper/JE3MNTER

@misc{pith2026190805155,
  author       = {Pith},
  title        = {Pith review of: The sum-of-squares hierarchy on the sphere, and applications in quantum information theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JE3MNTER}},
  note         = {Machine review of arXiv:1908.05155}
}

abstract

We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of Sum-of-Squares (SOS) relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty & Wehner. Specifically, we show that the rate of convergence is no worse than $O(d^2/\ell^2)$ in the regime $\ell \geq \Omega(d)$ where $\ell$ is the level of the hierarchy and $d$ the dimension, solving a problem left open in the recent paper by de Klerk & Laurent (arXiv:1904.08828). Importantly, our analysis also works for matrix-valued polynomials on the sphere which has applications in quantum information for the Best Separable State problem. By exploiting the duality relation between sums of squares and the DPS hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascu\'es, Owari & Plenio.

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Pith tools

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

[1] [1]

I. Area, D. Dimitrov, E. Godoy, and A. Ronveaux. Zeros of G egenbauer and H ermite polynomials and connection coefficients. Mathematics of Computation , 73(248):1937--1951, 2004

work page 1937

[2] [2]

J. V. Baxley. Extreme eigenvalues of T oeplitz matrices associated with certain orthogonal polynomials. SIAM Journal on Mathematical Analysis , 2(3):470--482, 1971

work page 1971

[3] [3]

Barak, F

B. Barak, F. G. Brandao, A. W. Harrow, J. Kelner, D. Steurer, and Y. Zhou. Hypercontractivity, sum-of-squares proofs, and their applications. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing , pages 307--326. ACM, 2012

work page 2012

[4] [4]

Bhattiprolu, M

V. Bhattiprolu, M. Ghosh, V. Guruswami, E. Lee, and M. Tulsiani. Weak decoupling, polynomial folds and approximate optimization over the sphere. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) , pages 1008--1019. IEEE, 2017

work page 2017

[5] [5]

Barak, P

B. Barak, P. K. Kothari, and D. Steurer. Quantum entanglement, sum of squares, and the log rank conjecture. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing , pages 975--988. ACM, 2017

work page 2017

[6] [6]

Blekherman

G. Blekherman. Convexity properties of the cone of nonnegative polynomials. Discrete & Computational Geometry , 32(3):345--371, 2004

work page 2004

[7] [7]

Brickman

L. Brickman. On the field of values of a matrix. Proceedings of the American Mathematical Society , 12(1):61--66, 1961

work page 1961

[8] [8]

Christandl, R

M. Christandl, R. K \"o nig, G. Mitchison, and R. Renner. One-and-a-half quantum de F inetti theorems. Communications in Mathematical Physics , 273(2):473--498, 2007

work page 2007