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arXiv:quant-ph/0012031 · Journal of Physics A Mathematical and General

Remarks on the extended characteristic uncertainty relations

D. A. Trifonov

Published Feb 26, 2001• 14 citations
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Abstract

Three remarks concerning the form and the range of validity of the state-extended characteristic uncertainty relations (URs) are presented. A more general definition of the uncertainty matrix for pure and mixed states is suggested. Some new URs are provided.
DOI
10.1088/0305-4470/34/9/101

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Quantum theory predicts an inherent joint unpredictability for some pairs of measurements. For example, Heisenberg showed that the more precisely the position of a quantum particle is known, the less precisely its momentum can be known and vice versa. Uncertainty relations (URs) are mathematical expressions quantifying the constraints between the output probability distributions of the given sets of measurements. Typically, URs are expressed in terms of uncertainty quantifiers such as entropies. Based on an information-theoretic approach, we discovered a characterization that unifies all uncertainty quantifiers and thus, generalizes a large class of URs into a single framework. We also prove new URs that are fundamentally different from typical URs in that they are fine-grained; i.e. they set restrictions directly on the output probability distributions without using any particular uncertainty quantifiers. We used Majorization theory and other techniques such as matrix analysis to prove our fine-grained uncertainty relations.