Functions of positive type on phase space, between classical and quantum, and beyond

23rd International Conference on Integrable Systems and Quantum Symmetries

Aniello, Paolo^1

Dipartimento di Fisica dell'Universita di Napoli Federico II, INFN - Sezione di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, Napoli

I-80126, Italy^1

关键词: Classical statistical mechanics;    Generalized phase;    Open quantum systems;    Probability theory;    Quantum information science;    Quantum setting;    Quasiprobability distributions;    Stieltjes transform;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/670/1/012004/pdf
DOI  :  10.1088/1742-6596/670/1/012004

来源: IOP

【 摘 要 】

Functions of positive type on locally compact abelian groups can be defined as positive functionals on group algebras, and play a remarkable role in probability theory and in classical statistical mechanics. By Bochner's celebrated theorem, indeed, they are Fourier-Stieltjes transforms of finite positive measures. Hence, a properly normalized nonzero function of positive type on (the group of translations on) phase space provides a realization of a classical state, so it may be called a function of classical positive type. A similar result holds in the quantum setting as well, where a generalized kind of functions of positive type on phase space - the so-called functions of quantum positive type - are related, via the Fourier-Plancherel transform, to the Wigner quasi-probability distributions. In this paper, we will argue that, as in the classical setting, the notion of function of quantum positive type is of a group-theoretical nature. Exploiting an interesting interplay between functions of classical and quantum positive type, we will then provide an interesting characterization of a class of semi-groups of operators that describe the evolution of certain open quantum systems which are of interest in quantum information science. Finally, a suitable extension of this framework to generalized phase spaces that are relevant for current applications will be proposed.