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The mutual relationship of the Heisenberg uncertainty relations, two recently derived uncertainty relations, the Robertson-Schrödinger uncertainty relation and the inequality for the Fisher information is discussed. [ABSTRACT FROM AUTHOR]

Taking its historical point of departure in Heisenberg’s work, this article offers a view of quantum mechanics as, arguably, the first truly experimental and truly mathematical physical theory, that is, a theory concerned with experimenting with nature and mathematics alike. It is truly experimental because it is not, as in classical physics, merely the independent behavior of the system considered, in other words, what happens in any event, that we track, but what kind of experiments we perform that defines what happens. By the same token, the theory is also truly mathematical because, at least in the interpretation adopted here, its mathematical formalism does not stand in the service of a mathematical description of (quantum) physical processes in space and time in the way the formalism of classical physics does, but is only used to predict the outcomes of relevant experiments. It also follows that quantum theories experiment more freely with mathematics itself, since we invent predictive mathematical schemes, rather than proceed by refining mathematically our phenomenal representations of nature, which process constrains us in classical mechanics. [ABSTRACT FROM AUTHOR]

Such entropies as Shannon entropy, Renyi entropy, and von Neumann entropy are discussed for quantum states. New inequalities for quantum tomographic-probability densities are considered. [ABSTRACT FROM AUTHOR]

Quantum mechanical uncertainty relations for the position and the momentum and for the angle and the angular momentum are expressed in the form of inequalities involving the Rényi entropies. These uncertainty relations hold not only for pure but also for mixed states. Analogous uncertainty relations are valid also for a pair of complementary observables (the analogs of x and p) in N-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized Rényi entropies. The mathematical proofs of all the inequalities discussed in this paper can be found in Phys. Rev. A 74, No. 5 (2006); arXiv:quant-ph/0608116. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]