Two approaches to measurability conception and quantum theory

James L Park, Henry Margenau

Teiko Heinosaari, Mário Ziman

Guide to mathematical concepts of quantum theory Quantum Theory is one of the pillars of modern science developed over the last hundred years. In this review paper we introduce, step by step, the quantum theory understood as a mathematical model describing quantum experiments. We start with splitting the experiment into two parts: a preparation process and a measurement process leading to a registration of a particular outcome. These two ingredients of the experiment are represented by states and effects, respectively. Further, the whole picture of quantum measurement will be developed and concepts of observables, instruments and measurement models representing the three different descriptions on experiments will be introduced. In the second stage, we enrich the model of the experiment by introducing the concept of quantum channel describing the system changes between preparations and measurements. At the very end we review the elementary properties of quantum entanglement. The text contains many examples and exercise covering also many topics from quantum information theory and quantum measurement theory. The goal is to give a mathematically clear and self-containing explanation of the main concepts of the modern language of quantum theory.

Jiří Souček

We present the axiomatization of quantum mechanics which does not contain axioms concerning the measurement. Instead of the concept of measurement this axiomatization uses the concept of the observation of the individual state of the measuring system after the run of the experiment. It is proved that the resulting theory is empirically equivalent to the standard quantum mechanics but it is also shown that these two theories are (theoretically) different.

Herman Erlichson

Niels Arley

Alexander Shalyt-Margolin

In the present paper, in terms of the measurability concept introduced in the previous works of the author, a quantum theory is studied. Within the framework of this concept, several examples are considered using the Schrodinger picture; analogs of Fourier transformations from the momentum representation to the coordinate one and vice versa are constructed. It is shown how to produce a measurable analog of the Heisenberg picture. At the end of this paper the obtained results are used to substantiate another (more general) definition of the measurability concept that is not based on the Heisenberg Uncertainty Principle and its generalization, as it has been in the previous works of the author, and may be equally suitable for both non-relativistic and relativistic cases.