Classical-quantum semigroups

22nd International Conference on Integrable Systems and Quantum Symmetries

Aniello, Paolo^1

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

80126, Italy^1

关键词: Classical-quantum;    Positive definite function;    Probability measures;    Quantum dynamical semigroups;    Quantum information science;    Quasiprobability distributions;    Semigroups of operators;    Theoretical framework;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/563/1/012002/pdf
DOI  :  10.1088/1742-6596/563/1/012002

来源: IOP

【 摘 要 】

An intriguing and deep analogy between classical and quantum states is revealed using the notion of positive definite function. By Bochner's theorem, (classical) positive definite functions on phase space can be obtained by taking the Fourier transform of probability measures; similarly, quantum positive definite functions are the image via the Fourier-Plancherel operator of Wigner quasi-probability distributions. Considering the basic properties of positive definite functions - classical and quantum - one is led to define a class of semigroups of operators, the so-called classical-quantum semigroups. It is then natural to wonder whether they have any physical meaning. It turns out that the classical-quantum semigroups can also be obtained by dequantizing a certain class of quantum dynamical semigroups, namely, the classical noise semigroups. This correspondence fits in a more general group-theoretical framework in which a larger class of quantum dynamical semigroups, the twirling semigroups, can be suitably dequantized. Connections with quantum information science will be briefly discussed.