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arXiv:2604.12410

Notes on some inequalities, resulting uncertainty relations and correlations

Krzysztof Urbanowski

quant-phmath-phmath.FASubmitted Apr 14, 2026• Announced Apr 15, 2026• 0 citations• Revised Aug 6, 2026 (v3)

v1 · Apr 14, 2026 v2 · Apr 15, 2026 v3 · Aug 6, 2026

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.

Comments: 33 pages, 2 figures, some typos and title corrected

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The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.

We analyze the weak and critical points of various uncertainty relations that follow from the inequalities for the norms of vectors in the Hilbert space of states of a quantum system. There are studied uncertainty relations for sums of standard deviations, for sums of variances, and other relations between standard deviations or variances. The obtained results are compared with the conclusions obtained in similar cases using the standard Heisenberg-Robertson uncertainty relation. We also show that there exists an upper bound on the product of standard deviations that appears in the Heisenberg-Robertson uncertainty relation.

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It is well known that almost all uncertainty relations including Heisenberg uncertainty relation and Schrödinger uncertainty relation were given by product types of trace inequalities.This is why these results were proved by Schwarz's inequality.These product types of uncertainty relations were extended to the case of not necessarily hermitian quantum mechanical observables and positive operators representing quantum states.On the other hand sum types of uncertainty relations were given for arbitrary finite N not necessarily hermitian quantum mechanical observables.Some uncertainty relations are presented by generalized quasi-metric adjusted skew informations for two different generalized states.These uncertainty relations are nontrivial as long as the observables are mutually noncommutative.The relations among these new and existing uncertainty inequalities have been investigated.Finally, the reverse inequalities of the sum types of uncertainty relations are obtained.

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