Notes on some inequalities, resulting uncertainty relations and correlations. 1. General mathematical formalism

Krzysztof Urbanowski

arxiv: 2604.12410 · v2 · submitted 2026-04-14 · 🪐 quant-ph · math-ph· math.FA· math.MP

Notes on some inequalities, resulting uncertainty relations and correlations. 1. General mathematical formalism

Krzysztof Urbanowski This is my paper

Pith reviewed 2026-05-10 15:07 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.FAmath.MP

keywords uncertainty relationsSchwarz inequalityJensen inequalityPearson coefficientsquantum correlationsgeneralized uncertainty relationsHeisenberg-RobertsonSchrödinger-Robertson

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The pith

The Schrödinger-Robertson uncertainty relation for non-commuting observables can be rewritten equivalently using Pearson correlation coefficients.

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

This paper applies the Schwarz inequality and its generalizations, together with Jensen's inequality, to derive uncertainty relations in quantum mechanics. It extends the Heisenberg-Robertson and Schrödinger-Robertson relations from two observables to three or more non-commuting ones and studies the resulting bounds. The work focuses on how these relations connect to the correlations between the observables in a given quantum state. A key step is showing that the Schrödinger-Robertson relation, including its generalizations, takes an equivalent form when expressed through a matrix whose entries are the quantum versions of the Pearson coefficient. This link matters because it recasts measurement limits as statements about observable correlations rather than variances alone.

Core claim

Using these inequalities, we derive various types of generalized uncertainty relations for more than two non-commuting observables and analyze their properties and critical points. We also study the connections between the generalizations of the HR and SR uncertainty relations for two and more observables and the correlations of these observables in the state of the quantum system under study. We show also that the SR uncertainty relation (including the generalized ones) can be written in an equivalent way using these Pearson coefficients.

What carries the argument

The correlation matrix whose entries are the quantum versions of the Pearson coefficient, which reformulates the Schrödinger-Robertson uncertainty relations in equivalent form.

Load-bearing premise

The standard Schwarz and Jensen inequalities can be applied directly to commutators and expectation values involving three or more non-commuting quantum observables in an arbitrary state without further restrictions on the state or operators.

What would settle it

Three non-commuting observables and a quantum state where the product of uncertainties fails to match the bound given by the corresponding Pearson coefficient matrix, or where the generalized relation is violated.

Figures

Figures reproduced from arXiv: 2604.12410 by Krzysztof Urbanowski.

**Figure 2.** Figure 2: Allowable values rφ(A1, A3) and rφ(A2, A3) in a quantum system for given values rφ(A1, A2). 24 [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

read the original abstract

We analyze the Schwarz inequality and its generalizations, as well as inequalities resulting from the Jensen inequality. They are used in quantum theory to derive the Heisenberg-Robertson (HR) and Schroedinger-Robertson (SR) uncertainty relation for two non-commuting observables and their generalizations to three or more non-commuting observables. Jensen's inequality, in turn, is helpful in deriving various the "sum uncertainty relations" for two or more observables. Using these inequalities, we derive various types of generalized uncertainty relations for more than two non--commuting observables and analyze their properties and critical points. We also study the connections between the generalizations of the HR and SR uncertainty relations for two and more observables and the correlations of these observables in the state of the quantum system under study. In this analysis, we pay special attention to the consequences of the generalized SR uncertainty relation for three non--commuting observables on their correlations in a given state of a quantum system and to the connections of this relation with the appropriate correlation matrix, whose matrix elements are the quantum versions of the Pearson coefficient. We show also that the SR uncertainty relation (including the generalized ones) can be written in an equivalent way using these Pearson coefficients.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper collects derivations of generalized uncertainty relations for three or more observables from Schwarz and Jensen inequalities and rewrites the Schrödinger-Robertson form using Pearson coefficients, but the quantum operator application needs explicit domain checks.

read the letter

The main thing to know is that Urbanowski starts from the classical Schwarz and Jensen inequalities, applies them to expectation values and commutators, and produces generalized Heisenberg-Robertson and Schrödinger-Robertson relations for more than two non-commuting observables. He then shows that the Schrödinger-Robertson version can be rewritten directly in terms of the quantum Pearson coefficients built from the covariance matrix of those observables. That equivalence is the clearest new step here, even if it builds on existing two-observable results.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a general mathematical formalism using the Schwarz inequality and its generalizations, along with Jensen's inequality, to derive Heisenberg-Robertson (HR) and Schrödinger-Robertson (SR) uncertainty relations for two and more non-commuting observables. It examines connections to correlations in the quantum state, with particular attention to rewriting the generalized SR relation for three observables in an equivalent form using a correlation matrix whose elements are quantum versions of Pearson coefficients derived from covariances.

Significance. If the central derivations hold rigorously, the reformulation linking SR relations to Pearson coefficient matrices offers a potentially useful perspective on multi-observable correlations without introducing free parameters. The approach builds directly from standard classical inequalities, which is a methodological strength for deriving sum-type and generalized relations.

major comments (2)

[Generalized SR for three observables] Derivation of the generalized SR relation for three observables (around the claim of equivalence to Pearson coefficients): the direct application of the classical Schwarz inequality to inner products constructed from non-commuting operators A, B, C and their commutators requires explicit domain and self-adjointness conditions to guarantee that the resulting quadratic form remains positive semi-definite for arbitrary states; without these, the correlation matrix may admit spurious negative eigenvalues, undermining the claimed equivalence.
[Properties and critical points] Section on properties and critical points of the generalized relations: the analysis of extremal values for the multi-observable SR and HR forms should verify whether the inequalities remain valid when the operators are unbounded, as the Jensen-based sum relations may require additional restrictions on the state to avoid domain issues not addressed in the current derivation.

minor comments (2)

[Abstract] The abstract contains a minor grammatical error: 'various the sum uncertainty relations' should be rephrased for clarity.
[Introduction to correlations] Notation for the quantum Pearson coefficients and the correlation matrix should be introduced with an explicit definition early in the text to improve readability when discussing the equivalence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding mathematical rigor, particularly domain and self-adjointness conditions, are important for strengthening the presentation. We respond to each major comment below and indicate planned revisions.

read point-by-point responses

Referee: [Generalized SR for three observables] Derivation of the generalized SR relation for three observables (around the claim of equivalence to Pearson coefficients): the direct application of the classical Schwarz inequality to inner products constructed from non-commuting operators A, B, C and their commutators requires explicit domain and self-adjointness conditions to guarantee that the resulting quadratic form remains positive semi-definite for arbitrary states; without these, the correlation matrix may admit spurious negative eigenvalues, undermining the claimed equivalence.

Authors: We agree that explicit conditions are necessary for full rigor. Our derivations assume standard quantum-mechanical settings where A, B, C are self-adjoint operators (densely defined on a common dense domain) and the state vector lies in the intersection of the relevant domains so that all expectation values and commutators exist and are finite. Under these conditions the quadratic form is positive semi-definite by the properties of the inner product, and the equivalence to the Pearson-coefficient matrix holds without spurious negative eigenvalues. To make this explicit, we will insert a short clarifying paragraph (or subsection) immediately after the statement of the three-observable SR relation, listing the domain and self-adjointness assumptions. This addition addresses the concern directly while leaving the core derivations unchanged. revision: partial
Referee: [Properties and critical points] Section on properties and critical points of the generalized relations: the analysis of extremal values for the multi-observable SR and HR forms should verify whether the inequalities remain valid when the operators are unbounded, as the Jensen-based sum relations may require additional restrictions on the state to avoid domain issues not addressed in the current derivation.

Authors: The referee correctly notes that unbounded operators introduce domain subtleties for Jensen-based sum relations. In the manuscript we implicitly restrict attention to states for which all appearing moments are finite; for bounded operators this is automatic, while for unbounded operators the state must belong to the intersection of the domains of the relevant powers. We will revise the section on properties and critical points to state these restrictions explicitly, add a remark that the extremal values are attained only for states satisfying the domain conditions, and note that the inequalities remain valid precisely when the expectations exist. This clarification does not alter any of the derived relations but removes the ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations from external inequalities

full rationale

The paper starts from the standard Schwarz and Jensen inequalities as external mathematical inputs and applies them to derive generalized HR and SR uncertainty relations for two or more non-commuting observables, along with their connections to correlations. The claimed equivalence of the SR relation (including generalizations) to a form using Pearson coefficients follows directly from algebraic rewriting of the covariance terms in the relation, without any reduction of outputs to inputs by construction or self-reference. No fitted parameters are presented as predictions, no uniqueness theorems are imported via self-citation, and no ansatzes are smuggled in. The derivation chain remains self-contained against the cited inequalities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rest on the standard Schwarz inequality and Jensen inequality applied to quantum operators and states; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

standard math Schwarz inequality
Used to derive Heisenberg-Robertson and Schrödinger-Robertson uncertainty relations for non-commuting observables.
standard math Jensen inequality
Used to derive sum uncertainty relations for two or more observables.

pith-pipeline@v0.9.0 · 5516 in / 1167 out tokens · 31525 ms · 2026-05-10T15:07:58.832441+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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