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理论物理通讯：英文版

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

Mona Mona, Khare, Shraddha Shraddha, Roy Roy

Published Jan 1, 2008• 0 citations
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The purpose of the present paper is to study the entropy h s (Φ) of a quantum dynamical systems Φ = (L,s,ϕ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h s (ϕ, ) of partition of a Boolean algebra B with respect to a state s and a state preserving homomorphism ϕ, we prove a few results on that, define the entropy of a dynamical system h s (Φ), and show its invariance. The concept of sufficient families is also given and we establish that h s (Φ) comes out to be equal to the supremum of h s (ϕ, ), where varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,ϕ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B,s 0 ,ϕ), where B is a Boolean algebra and s 0 is a state on B.

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We present several known and one new description of orthomodular structures (orthomodular lattices, orthomodular posets and orthoalgebras). Originally, orthomodular structures were viewed as pasted families of Boolean algebras. Here we introduce semipasted families of Boolean algebras as an alternative description which is not as detailed, but substantially simplex. Semipasted families of Boolean algebras correspond to orthomodular structures in such a way that states and evaluation functionals are preserved. As semipasted families of Boolean algebras are quite general, they allow an easy construction of orthomodular structures with given state space properties. Based on this technique, we give a simplified proof of Shultz's Theorem on characterization of spaces of finitely additive states on orthomodular lattices. We also put some other results into the new context. We give a detailed exposition of the construction techniques as a tool for further applications, especially for finding counterexamples to questions about states on orthomodular structures.

•Jan 1, 2000•12 citations
The purpose of the present paper is to study the entropy h s (Φ) of a quantum dynamical systems Φ = (L,s,ϕ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h s (ϕ, ) of partition of a Boolean algebra B with respect to a state s and a state preserving homomorphism ϕ, we prove a few results on that, define the entropy of a dynamical system h s (Φ), and show its invariance. The concept of sufficient families is also given and we establish that h s (Φ) comes out to be equal to the supremum of h s (ϕ, ), where varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,ϕ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B,s 0 ,ϕ), where B is a Boolean algebra and s 0 is a state on B.
We present several known and one new description of orthomodular structures (orthomodular lattices, orthomodular posets and orthoalgebras). Originally, orthomodular structures were viewed as pasted families of Boolean algebras. Here we introduce semipasted families of Boolean algebras as an alternative description which is not as detailed, but substantially simplex. Semipasted families of Boolean algebras correspond to orthomodular structures in such a way that states and evaluation functionals are preserved. As semipasted families of Boolean algebras are quite general, they allow an easy construction of orthomodular structures with given state space properties. Based on this technique, we give a simplified proof of Shultz's Theorem on characterization of spaces of finitely additive states on orthomodular lattices. We also put some other results into the new context. We give a detailed exposition of the construction techniques as a tool for further applications, especially for finding counterexamples to questions about states on orthomodular structures.