Abstract
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.
Highlights
It is shown that the additivity relation of Kolmogorov probabilities, is not valid in the Birkhoffvon Neumann orthocomplemented modular lattice of subspaces L(d)
Quantum mechanics is a probabilistic theory, but quantum probabilities are more general than the standard (Kolmogorov) probabilities
The Dempster-Shafer theory has been used extensively in Artificial Intelligence, Operations Research, Economics, etc[14, 15, 16, 17], and here we use it for quantum probabilities and link it to quantum logic
Summary
Quantum mechanics is a probabilistic theory, but quantum probabilities are more general than the standard (Kolmogorov) probabilities. Somebody might conjecture that the lack of distributivity is the only factor responsible for the non-additivity of quantum probabilities For this reason we considered the second lattice which is distributive. Even in this case, quantum probabilities are non-additive. The upper probability combines the ‘belong to A’ and the ‘don’t know’ In his original work Dempster[14] considered Kolmogorov probabilities associated to subsets of a space X. Within the lattice L(d), there are sublattices which are Boolean algebras, and there the projectors commute, the D(H1, H2) = 0 and quantum probabilities obey the additivity relation Tr[ρD(H1, H2)] = 0 which is analogous to δ(A, B) = 0 for Kolmogorov probabilities.
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