Quantum state change in light of changes in valuational entropies

Abstract

In the statement "The vector is an element of the closed linear subspace of the Hilbert space H", the predicate "... is an element of ..." might be not only determined, that is, either true or false (depending on whether set membership is applicable or inapplicable to the specified vector and subspace) but also undetermined, that is, neither true nor false. To evaluate the vagueness of set membership among arbitrary vectors and closed linear subspaces of H, the notion of the entropy of the predicate "... is an element of ..." is introduced in the present paper. Since each closed linear subspace in H uniquely represents the atomic proposition P about a quantum system, the entropy of this predicate can also be considered as the valuational entropy that measures the uncertainty about the assignment of truth values to the proposition P. As it is demonstrated in the paper, in the Hilbert space H of the dimension greater than or equal to 2, there always exists a nonempty set S of the closed linear subspaces in H, such that the entropy of the predicate "... is an element of ..." on the given vector of H and all the subspaces of S cannot be zero. This implies the existence of two different processes of the pure quantum state change: the process which yields no changes in the valuational entropies of the propositions (corresponding to the deterministic and reversible evolution) and the process which brings forth changes in the valuational entropies (corresponding to the gain or loss of information in a quantum measurement).