Tsallis Entropy of Partitions in Quantum Logics

Abstract

The purpose of the present paper is to study the Tsallis entropy Sq,t(P) of a partition P, and the Tsallis conditional entropy Sq,t(P|Q) of finite partitions P,Q in a quantum logic (L,t), where L is a quantum logic and t is a Bayessian state. We extend some results of Furuichi (J. Math. Phys. 47: 023302, 2006) to the partitions in (L,t). We study the concavity and subadditivity properties for Tsallis entropy of partitions in (L,t). We show that the subadditivity property in the case of q 1,Sq,t(P|Q) ≤ Sq,t(P), but in the case of $q<1, S_{q,t} (P|Q) \nleq S_{q,t} (P)$ , in general. We also study the chain rules for the information measures in (L,t); and present a numerical example to illustrate the results. We study Tsallis entropy and Tsallis conditional entropy of t-independent partitions. Finally, we show that, Tsallis entropy and Tsallis conditional entropy of partitions of a couple (L,t), are invariant under the relation of $\circeq _{t}$ .