Generalization of Shannon’s theorem for Tsallis entropy

By using the assumptions that the entropy must (i) be a continuous function of the probabilities {p i }(p i ∈(0,1)∀i), only; (ii) be a monotonic increasing function of the number of states W, in the case of equiprobability; (iii) satisfy the pseudoadditivity relation S q (A+B)/k=S q (A)/k+S q (B)/k+(1−q)S q (A)S q (B)/k 2 ( A and B being two independent systems, q∈ R and k a positive constant), and (iv) satisfy the relation S q ({p i })=S q (p L ,p M )+p L q S q ({p i /p L })+p M q S q ({p i /p M }), where p L +p M =1(p L =∑ i=1 W L p i and p M =∑ i=W L +1 W p i ), we prove, along Shannon’s lines, that the unique function that satisfies all these properties is the generalized Tsallis entropy S q =k(1−∑ i=1 W p i q )/(q−1).