The purpose of the present paper is to study the entropy hs（Ф） of a quantum dynamical systems Ф = （ L, s, Ф）, where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h...The purpose of the present paper is to study the entropy hs（Ф） of a quantum dynamical systems Ф = （ L, s, Ф）, where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy hs（ Ф, A） of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism Ф, we prove a few results on that, define the entropy of a dynamical system hs（Ф）, and show its invariance. The concept of sufficient families is also given and we establish that hs （Ф） comes out to be equal to the supremum of hs （Ф,A）, where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system （ L, s, Ф）, which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system （B, s0, Ф）, where B is a Boolean algebra and so is a state on B.展开更多
The main results are as follows： （1） it deals with a number of basic operations （concatenation, Kleene closure, homomorphism, complement）; （2） due to a condition imposed on the implication operator for discussi...The main results are as follows： （1） it deals with a number of basic operations （concatenation, Kleene closure, homomorphism, complement）; （2） due to a condition imposed on the implication operator for discussing some basic issues in orthomodular lattice-valued automata, this condition is investigated in detail, and it is discovered that all the relatively reasonable five implication operators in quantum logic do not satisfy this condition, and that one of the five implications satisfies such a condition iff the truth-value lattice is indeed a Boolean algebra; （3） it deals further with orthomodular lattice-valued successor and source operators; （4） an example is provided, implying that some negative results obtained in the literature may still hold in some typical orthomodular lattice-valued automata.展开更多
文摘The purpose of the present paper is to study the entropy hs（Ф） of a quantum dynamical systems Ф = （ L, s, Ф）, where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy hs（ Ф, A） of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism Ф, we prove a few results on that, define the entropy of a dynamical system hs（Ф）, and show its invariance. The concept of sufficient families is also given and we establish that hs （Ф） comes out to be equal to the supremum of hs （Ф,A）, where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system （ L, s, Ф）, which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system （B, s0, Ф）, where B is a Boolean algebra and so is a state on B.
基金National Natural Science Foundation of China (Grant Nos. 90303024 and 60573006)the Research Foundation for the Doctorial Program of Higher School of Ministry of Education (Grant No. 20050558015)the Natural Science Foundation of Guangdong Province (Grant Nos. 020146 and 031541)
文摘The main results are as follows： （1） it deals with a number of basic operations （concatenation, Kleene closure, homomorphism, complement）; （2） due to a condition imposed on the implication operator for discussing some basic issues in orthomodular lattice-valued automata, this condition is investigated in detail, and it is discovered that all the relatively reasonable five implication operators in quantum logic do not satisfy this condition, and that one of the five implications satisfies such a condition iff the truth-value lattice is indeed a Boolean algebra; （3） it deals further with orthomodular lattice-valued successor and source operators; （4） an example is provided, implying that some negative results obtained in the literature may still hold in some typical orthomodular lattice-valued automata.
