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Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products

T0 review · 1 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that the strengthened Cauchy–Schwarz inequality is preserved by integer Hadamard powers, and generalizes it to matrices, eigenvalue comparisons, and three or more vectors.

desk verdict Main results are new and likely correct, but the proof of Lemma 3.1 has a repairable gap that holds up the x>=1 case of the central theorem. read the letter →

arxiv 2507.10327 v1 pith:PV7F3FCR submitted 2025-07-14 math.FA quant-ph

Nathaniel Johnston , Sarah Plosker , Charles Torrance , Luis M. B. Varona This is my paper · ORCID
classification math.FAquant-ph MSC 97H6015A1815A45
keywords Cauchy–SchwarzinequalityHadamardpowerstensorproductsorthogonalprojectionsmatrixinequalitiessumofsquaresrealignmentcriterionmultipartiteentanglement
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

The paper's target is the 2019 inequality $\|\mathbf{v}^2\|\|\mathbf{w}^2\| - \langle\mathbf{v}^2,\mathbf{w}^2\rangle \leq \|\mathbf{v}\|^2\|\mathbf{w}\|^2 - \langle\mathbf{v},\mathbf{w}\rangle^2$ and the question of why it is true. It offers three new proofs and several generalizations. The central new result is Theorem 3.2: for every real $x \geq 1$, the function $f_x(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|^x\|\mathbf{w}\|^x - |\langle\mathbf{v},\mathbf{w}\rangle|^x$ is non-increasing under orthogonal projections. This one fact yields the Hadamard-power generalization for every integer $p \geq 1$ and, through tensor products, multi-vector versions; it also shows the original exponent $2$ is not special. The paper shows the inequality fails in the intervals $(0,1)$ and $(1,2)$, supplies a sum-of-squares certificate for even exponents, and conjectures that it holds for all real $p \geq 2$.

What carries the argument

The load-bearing object is the family $f_x(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|^x\|\mathbf{w}\|^x - |\langle\mathbf{v},\mathbf{w}\rangle|^x$ for real $x \geq 1$, together with Lemma 3.1: if $a,b,c,d \geq 0$ satisfy $\max\{a,b\} \geq \max\{c,d\}$ and $a+b \geq c+d$, then $a^x + b^x - c^x - d^x \geq 0$ for all $x \geq 1$. Lemma 3.1 lifts the projection inequality from the $x=1$ case to all $x>1$. The tensor-product projection that converts this into Hadamard-power statements is the orthogonal projection onto $\operatorname{span}\{\mathbf{e}_j^{\otimes p} : 1 \leq j \leq n\} \subset (\mathbb{R}^n)^{\otimes p}$, which extracts the entrywise product $\mathbf{x}_1 \odot \cdots \odot \mathbf{x}_p$ from the tensor product $\mathbf{x}_1 \otimes \cdots \otimes \mathbf{x}_p$. A second independent mechanism is Theorem 3.5, a sum-of-squares decomposition of $\|\mathbf{v}\|^{2k}\|\mathbf{w}\|^{2k} - \langle\mathbf{v},\mathbf{w}\rangle^{2k}$ that generalizes Lagrange's identity and certifies positivity for even exponents.

What would settle it

Pick random nonnegative $a,b,c,d$ satisfying $\max\{a,b\} \geq \max\{c,d\}$ and $a+b \geq c+d$, and evaluate $a^x+b^x-c^x-d^x$ for $x \geq 1$; any negative value refutes Lemma 3.1 and with it Theorem 3.2. Equivalently, search for vectors $\mathbf{v},\mathbf{w}$ and an orthogonal projection $P$ with $f_x(P\mathbf{v},P\mathbf{w}) > f_x(\mathbf{v},\mathbf{w})$. The paper reports that in the conjectured range $p \geq 2$ with nonnegative vectors, more than $10^{10}$ random tests found no counterexample, so a concrete pair with non-integer $p \geq 2$ violating Inequality (5.2) would decisively refute Conjecture 5.1.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 3.2: for every real $x \geq 1$, every pair of vectors $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$, and every orthogonal projection $P$, one has $f_x(P\mathbf{v},P\mathbf{w}) \leq f_x(\mathbf{v},\mathbf{w})$, where $f_x(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|^x\|\mathbf{w}\|^x - |\langle\mathbf{v},\mathbf{w}\rangle|^x$. Since $f_2$ is the squared area of the Gram parallelogram and $f_1$ is the angle-dependent quantity from Section 2, this says one mechanism—orthogonal projections cannot increase these generalized deficits—governs the original inequality. Applying the theorem to tensor-product vectors and the projection onto the subspace spanned by $\mathbf{e}_j^{\otimes p}$ gives Corollary 3.3, and specializing all vectors to the same pair recovers the Hadamard-power inequality $\|\mathbf{x}^p\|\|\mathbf{y}^p\| - |\langle\mathbf{x}^p,\mathbf{y}^p\rangle| \leq \|\mathbf{x}\|^p\|\mathbf{y}\|^p - |\langle\mathbf{x},\mathbf{y}\rangle|^p$ for every integer $p \geq 1$.

Load-bearing premise

The whole extension from exponent one to all $x>1$ rests on Lemma 3.1: for nonnegative $a,b,c,d$ with $\max\{a,b\} \geq \max\{c,d\}$ and $a+b \geq c+d$, the difference $a^x+b^x-c^x-d^x$ stays nonnegative for every $x \geq 1$; if that comparison failed, the projection theorem and the Hadamard-power inequalities would fail with it.

Editorial extensions

If this is right

  • For every integer $p \geq 1$, the inequality $\|\mathbf{x}^p\|\|\mathbf{y}^p\| - |\langle\mathbf{x}^p,\mathbf{y}^p\rangle| \leq \|\mathbf{x}\|^p\|\mathbf{y}\|^p - |\langle\mathbf{x},\mathbf{y}\rangle|^p$ holds, so the original $p=2$ inequality is one member of an infinite family.
  • For matrices, diagonal entries obey $\|\operatorname{diag}(X)\||\|\operatorname{diag}(Y)\| - \langle\operatorname{diag}(X),\operatorname{diag}(Y)\rangle \leq \|X\|_F\|Y\|_F - \langle X,Y\rangle_F$, and for symmetric matrices this bounds diagonals against eigenvalues sorted in opposite order.
  • Even Hadamard powers have an explicit positivity certificate: $\|\mathbf{v}\|^{2k}\|\mathbf{w}\|^{2k} - \langle\mathbf{v},\mathbf{w}\rangle^{2k}$ is a sum of squares indexed by multinomial partitions, so the polynomial is certified nonnegative.
  • The multipartite tensor-product argument produces new inequalities for three or more vectors, such as Corollary 4.2, obtained by choosing permutations of the realignment map.
  • If Conjecture 5.1 is correct, the inequality $\|\mathbf{v}^p\|\|\mathbf{w}^p\| - \langle\mathbf{v}^p,\mathbf{w}^p\rangle \leq \|\mathbf{v}\|^p\|\mathbf{w}\|^p - \langle\mathbf{v},\mathbf{w}\rangle^p$ holds for every real $p \geq 2$, so the integer-exponent result is not the true boundary.

Reading between the lines

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

  • The projection-monotonicity theorem suggests a general principle: any unitarily invariant quantity built monotonically from $\|\mathbf{v}\|$, $\|\mathbf{w}\|$, and $|\langle\mathbf{v},\mathbf{w}\rangle|$ may satisfy the same projection inequality, potentially generating further Cauchy–Schwarz-type inequalities beyond the $f_x$ family.
  • Because the three-vector corollary arises from one permutation of the realignment map, the same construction likely yields a family of multipartite entanglement criteria, one per permutation class of $S_{2p}$; the paper only computes the classes for $p=2$ and $p=3$.
  • The conjecture for real $p \geq 2$ could be attacked by refining the binomial-series expansion used in Example 1: if the $O(\varepsilon^2)$ coefficient can be shown nonnegative for all $p \geq 2$, the two-entry case would be settled and the full conjecture would reduce to a reduction argument.
  • The sum-of-squares identity may extend to odd integers or to matrix-valued variables by replacing scalar squares with inner products of matrices, which would connect the inequality to semidefinite-programming certificates for polynomial positivity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper investigates a Cauchy-Schwarz-type inequality of Johnston and MacLean, namely ||v^2||||w^2|| - <v^2,w^2> <= ||v||^2||w||^2 - <v,w>^2 for real vectors, and presents several new proofs and generalizations. It gives a geometric/area interpretation (Section 2), an algebraic proof via orthogonal projections and a scalar lemma (Section 3), a sum-of-squares identity that yields an even-integer exponent generalization (Section 3.1), a tensor-product/realignment proof inspired by quantum information (Section 4), and a multipartite generalization to more than two vectors (Section 4.2). It also identifies ranges of exponents for which the natural generalization fails and formulates a conjecture for non-integer p >= 2.

Significance. If the central results hold, the paper provides a satisfying conceptual explanation of the original inequality and extends it substantially: Theorem 3.2 gives projection monotonicity for all x >= 1, Corollary 3.4 establishes the Hadamard-power generalization for all integer p >= 1, Theorem 3.5 gives an exact sum-of-squares identity of independent interest, and Theorem 4.1 yields a multipartite realignment inequality. The paper is honest about its limitations, including explicit counterexamples for p in (0,1) and (1,2) and a clearly labeled conjecture for non-integer p >= 2. The proofs are mostly detailed and self-contained, and the numerical testing is extensive. The main obstacle to acceptance is a genuine proof gap in Lemma 3.1, which Theorem 3.2 explicitly uses for the x > 1 case; this gap is localized and appears repairable, but it must be fixed before the central generalization is rigorous.

major comments (1)
  1. [Section 3, Lemma 3.1] The proof of Lemma 3.1 is incomplete. After scaling a+b=1, the hypotheses give c+d <= 1, not c+d = 1 as the proof assumes. The sentence 'decreasing c and/or d increases the value of f(x), so it suffices to prove the lemma when c+d=1 as well' justifies only reducing c and d to reach a smaller total; reaching c+d=1 requires increasing c and/or d, which decreases f(x) and can destroy the condition max{a,b} >= max{c,d}. For example, with a=0.6, b=0.4, c=0.4, d=0.1, the proportional rescaling to c+d=1 gives c'=0.8, d'=0.2, and c' > a. The subsequent monotonicity argument around Eq. (3.2) proves g(a) >= g(c) only in the normalized case with a >= c >= 1/2; the cases c < 1/2 and c+d < 1 are not handled. Since Theorem 3.2 invokes Lemma 3.1 to lift the projection inequality from x=1 to all x > 1, this is a load-bearing gap. The lemma itself appears true: after sorting, (a,b) weakly majorizes (c,d) and t^x is increasing and convex for x >= 1, so a repair by weak majorization should be possible, but the proof as written is not rigorous.
minor comments (4)
  1. [Section 2, Fact 1] The geometric proof of Fact 1 is only partial. Equation (2.3) relates f(v,w) to the area g(x,y) of a parallelogram built from square-root length vectors, but projecting v and w does not induce the corresponding projection of those auxiliary vectors, so the statement 'orthogonal projections cannot increase areas' does not by itself give f(Pv,Pw) <= f(v,w). The footnote on page 3 acknowledges that an algebraic proof is deferred to Theorem 3.2; I recommend explicitly labeling the Section 2 argument as heuristic and pointing to Theorem 3.2 for the rigorous proof.
  2. [Section 3, Theorem 3.2] In the proof of the x=1 case, the sign function sign(z)=z/|z| is used in the vectors (v1, sign(v1)||Pv||) and (w1, sign(w1)||Pw||), but it is undefined when v1=0 or w1=0. This case can be handled by a separate argument or by defining sign(0)=0, but it should be addressed explicitly.
  3. [Section 4.2, after Corollary 4.2] The claims that for p=2 all 24 permutations lead to only two possible values and that for p=3 all 720 permutations lead only to trivial inequalities or permutations of Corollary 4.2 are stated as 'direct computation' without proof or accompanying code. These classification claims are not needed for the validity of the inequalities themselves, but they should be proved or else relegated to a clearly described computational appendix.
  4. [Section 5] The numerical statement 'after more than 10 10 randomly-generated vectors' is ambiguous; it should read '10^10' or 'ten billion'. The same rendering issue affects the figure captions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper re-derives the target inequality from independent projection, tensor/realignment, and sum-of-squares arguments.

full rationale

The central derivation chain is self-contained. The target inequality (1.1) is not assumed: it is re-derived in Section 2 from projection monotonicity of parallelogram areas, in Section 4.1 from the external realignment criterion, and in Section 3.1 from Lagrange's identity and a new sum-of-squares decomposition. Theorem 3.2 establishes the x=1 case directly by Cauchy-Schwarz and triangle-inequality manipulations, then lifts to x>1 via Lemma 3.1, which is proved in the paper as an elementary scalar inequality; the bootstrapping uses only the already-proved x=1 case plus this lemma. Self-citations [Joh18, JM19] appear only as the historical source of the target inequality, not as evidence for the new theorems. There are no fitted parameters, no predictions of fitted quantities, and no imported uniqueness or ansatz claims. The possible gap noted in the proof of Lemma 3.1 about normalizing c+d=1 is a proof-repair issue, not a circularity: it does not make any conclusion identical to an input by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No parameters are fitted; the paper proves inequalities. The external results (realignment criterion, von Neumann trace inequality, the eigenvalue inner-product inequality) are standard theorems cited from the literature. The only domain restriction is the non-negativity assumption in Section 5 for non-integer exponents. No new entities are postulated.

assumptions (8)
  • standard math Cauchy-Schwarz inequality for real and complex inner product spaces
    Used throughout, e.g., in the proof of the x=1 case of Theorem 3.2 and in bounding inner products.
  • standard math Lagrange's identity: ||v||^2||w||^2 - <v,w>^2 = 1/2 sum_{i!=j} (v_i w_j - v_j w_i)^2
    Used in Section 3.1 as the starting point for the sum-of-squares decomposition.
  • domain assumption Realignment criterion: if X is separable then ||R(X)||_tr <= tr(X)
    Cited from [CW03,Rud03] and used in Section 4.1 to re-derive Inequality (1.1).
  • domain assumption Multipartite realignment criterion [HHH06]: for separable X and any sigma in S_{2p}, ||R_sigma(X)||_tr <= tr(X)
    Used in Section 4.2 to prove Theorem 4.1 and Corollary 4.2.
  • standard math Von Neumann trace inequality: <X,Y>_F <= <sigma_X,sigma_Y> for singular values sorted in the same order
    Used in Corollary 2.3.
  • standard math For symmetric X,Y, <X,Y>_F >= <lambda_X,lambda_Y> when eigenvalues are sorted in opposite order
    Cited [Bha97, Problem III.6.14], used in Corollary 2.2.
  • standard math Binomial series for non-integer exponents converges for |x|<1
    Used in Example 1 (Section 5) for the asymptotic expansions.
  • domain assumption Entries of vectors are restricted to non-negative reals when non-integer exponents are discussed
    Stated in Section 5; needed so that entrywise powers are real and inequalities make sense.

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Pith. "Pith review of Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products." pith.science (2026). https://pith.science/paper/PV7F3FCR

@misc{pith2026250710327,
  author       = {Pith},
  title        = {Pith review of: Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PV7F3FCR}},
  note         = {Machine review of arXiv:2507.10327}
}
abstract

We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\|\mathbf{v}^2\|\|\mathbf{w}^2\| - \langle\mathbf{v}^2,\mathbf{w}^2\rangle \leq \|\mathbf{v}\|^2\|\mathbf{w}\|^2 - \langle\mathbf{v},\mathbf{w}\rangle^2$ for all $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$. We present three new proofs of this inequality that better illustrate "why" it is true and generalize it in several different ways: we generalize from vectors to matrices, we explore which exponents other than 2 result in the inequality holding, and we derive a version of the inequality involving three or more vectors.

Figures

Figures reproduced from arXiv: 2507.10327 by the authors.

Figure 1
Figure 1. An illustration of how the functions f and g from Equation (2.1) may be interpreted as areas of parallelograms. The angle between x and y is half of the angle between v and w, and the lengths of x and y are the square roots of the lengths of v and w, respectively. For example, if we choose P = O (the all-zeros matrix) in Fact 1, then we recover exactly the Cauchy– Schwarz inequality itself, since the left-hand side … view at source ↗
Figure 2
Figure 2. Plots of p versus the difference between the right-hand and left-hand sides of Inequality (5.2). 104 randomly-generated values of p ∈ (0, 5] and pairs of unit vectors v, w ∈ R n are used, with (a) showing the n = 2 case and (b) showing the n = 3 case. We note that the top curve of these figures appears to be RHS − LHS = 1 − n + 1 2  ·  n + 1 2 (1−p)/2 , which is attained by the normalizations of the vectors v … view at source ↗

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Works this paper leans on

5 extracted references · 5 canonical work pages

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    ↑1, 7 [LL VM24] D

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