Non-additive probabilities and quantum logic in finite quantum systems

30th International Colloquium on Group Theoretical Methods in Physics

Vourdas, A.^1

Department of Computing, University of Bradford, Bradford

BD7 1DP, United Kingdom^1

关键词: Birkhoff-von neumann;    Dempster-shafer;    Finite quantum systems;    Probability theory;    Quantum logic;    Quantum probabilities;    Quantum system;    Sub-lattices;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012073/pdf
DOI  :  10.1088/1742-6596/597/1/012073

来源: IOP

【 摘 要 】

A quantum system Σ(d) with variables in Z(d) and with Hilbert space H(d), is considered. It is shown that the additivity relation of Kolmogorov probabilities, is not valid in the Birkhoff-von Neumann orthocomplemented modular lattice of subspaces L(d). A second lattice Λ(d) which is distributive and contains the subsystems of Σ(d) is also considered. It is shown that in this case also, the additivity relation of Kolmogorov probabilities is not valid. This suggests that a more general (than Kolmogorov) probability theory is needed, and here we adopt the Dempster-Shafer probability theory. In both of these lattices, there are sublattices which are Boolean algebras, and within these 'islands' quantum probabilities are additive.