Information-entropy inequalities for pure states of quantum systems in the probabilistic representation of quantum mechanics

Due to the rapid development of technologies, more and more experiments are being carried out every year in the field of quantum optics, quantum computing and cryptography, therefore the relevance of works on quantum information theory is beyond doubt. In this paper, we studied information-entropy inequalities for pure states of quantum systems, and, in particular, considered inequalities for a one-dimensional harmonic quantum oscillator. A symplectic quantum tomogram of pure states in the probabilistic representation of quantum mechanics is obtained. The tomographic Shannon entropy is introduced and new expressions for the tomographic entropy of pure states of a one-dimensional quantum harmonic oscillator are obtained, as well as exact expressions for the ground and first excited states depending on the actual parameters, a shift in the first excited state by the Euler - Mascheroni constant is obtained. Numerical methods were also considered for obtaining approximate values ​​of the tomographic entropy of the second and third excited states, at fixed parameters. The expected growth of the tomographic entropy with an increase in the level was obtained depending on the actual parameter. The approach presented in the course of this work can be extended to the question of the entropy of arbitrary Fock states, as well as coherent states of a parametric quantum oscillator. This direction of research is of interest and will be developed in future works.