bibkey: sharma2010variance
authors: Rajesh Sharma, M. Gupta, G. Kapoor
year: 2010
title: Some better bounds on the variance with applications
doi: null
url: https://jmi.ele-math.com/04-32/Some-better-bounds-on-the-variance-with-applications
claim: Popoviciu's range-squared-over-four bound gives the spectral half-width variance bound, which combines with Cauchy-Schwarz and finite support counting to bound the sparse covariance sum.
strata_touched:
  - D5/S3/Quantum/Information/CovarianceSumBound
license: citation-only
triage: anchor

Popoviciu’s bound and the spectral covariance sum

Verified locator

The exact upstream locator is: https://jmi.ele-math.com/04-32/Some-better-bounds-on-the-variance-with-applications

The publisher’s article page was retrieved successfully. It gives the title, the three authors above, DOI 10.7153/jmi-04-32 and the volume-4, issue-3 placement in Journal of Mathematical Inequalities. This note binds the publisher URL.

The round-17 seat reports the citation as Journal of Mathematical Inequalities 4(3), 355–363 (2010), and specifically printed page 355: equation (1.3) defines r = M−m and equation (1.4) states Popoviciu’s bound S² ≤ r²/4. These page and equation readings are seat-reported, not checked in the PDF here.

Scope of the dependency

For a Hermitian A with spectrum in [c−Δ,c+Δ], Δ ≥ 0, a positive trace-one density D induces a probability distribution on the spectral values. Its variance is the repository’s Var_D(A). The weighted form of the range bound can be checked directly: for a random variable X in [m,M], with mean μ,

E[(X−m)(M−X)] ≥ 0,
Var(X) ≤ (M−μ)(μ−m) ≤ (M−m)²/4.

This argument works for arbitrary nonnegative spectral weights, including zero weights, so it does not assume an invertible density or a uniform sample. Taking m = c−Δ and M = c+Δ gives Var_D(A) ≤ Δ². The repository realizes this same bound through continuous functional calculus and the positive expectation of a centered square. Its existing variance_le_half_width_sq Mathlib citation remains intact.

For covariance_sum_le, apply that variance bound to every R_x and then use the Cauchy-Schwarz and support-counting chain written out in D5/L/Quantum/axler2024innerproduct:

|Cov_D(R_x,R_y)| ≤ sqrt(Var_D(R_x) Var_D(R_y)) ≤ Δ²,
Σ_{x,y∈Q} |Cov_D(R_x,R_y)| ≤ |Q| b Δ².

The support size b is supplied as a hypothesis. The paper is cited for the explicit variance bound, not for a named sparse quantum covariance theorem or a locality law. The weighted spectral translation and finite summation are standard direct consequences spelled out here.

What this note does and does not attest

Attested by this repository’s own retrieval: the publisher’s article-page title, authors, DOI and volume/issue placement above.

Not attested here: the article’s PDF, printed page 355, equations (1.3) and (1.4), and the page range 355–363. Those are the round-17 seat’s report, relayed in the implementation brief and issue #6298, and remain ASSUMED-UNVERIFIED as interior readings. That seat explicitly reported checking the 2010 paper’s statement of Popoviciu’s bound. Neither that report nor this note claims a page-by-page verification of Popoviciu’s 1935 original paper. No priority or earliest-appearance claim is made.