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arXiv:2504.09122

Closer look at sum uncertainty relations and related relations

K. Urbanowski

Published Apr 12, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2504.09122

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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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We analyze the uncertainty relation for the sum of variances, which is called, in some papers, the stronger uncertainty relation for all incompatible observables. We show that this uncertainty relation for the sum of variances of the observables A and B calculated for the eigenstate of one of these observables (say of B), contrary to the suggestions presented in some papers, leads to the same results as the Heisenberg–Robertson uncertainty relation; that is, it does not give any bounds on the variance of A.
We formulate uncertainty relations for arbitrary N observables. Two uncertainty inequalities are presented in terms of the sum of variances and standard deviations, respectively. The lower bounds of the corresponding sum uncertainty relations are explicitly derived. These bounds are shown to be tighter than the ones such as derived from the uncertainty inequality for two observables [Phys. Rev. Lett. 113, 260401 (2014)]. Detailed examples are presented to compare among our results with some existing ones.
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.
The Heisenberg-Robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial; i.e., the lower bound can be null even for two noncompatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.