A counterexample against the Lesche stability of a generic entropy functional

We provide a counterexample to show that the generic form of entropy ∑ = i i p g p S ) ( ) ( is not always stable against small variation of probability distribution (Lesche stability) even if g is concave function on [0,1] and analytic on ]0,1]. Our conclusion is that the stability of such a generic functional needs more hypotheses on the property of the function g, or in other words, the stability of entropy cannot be discussed at this formal stage. If a physical quantity observable is continuous function of the characteristic variables of motion such as time, configuration, velocity, energy, probability distribution etc., this quantity, and of course its mathematical definition, should undergo smooth variation for the system in smooth motion. Such a condition can be referred to as experimental robustness or observability and can be used to examine the validity of mathematical. An example of such quantity is the entropy which is characteristic of probabilistic uncertainty in stochastic dynamics and considered as continuous function of probability distribution. From this consideration, the mathematical definition of Shannon entropy, Renyi entropy and Tsallis entropy has been reviewed in [1] and [2], in which the robustness was called stability against small perturbation of probability or subsequently Lesche stability after Lesche who initialized the discussion by defining a restrictive uniform continuity criterion [1]. That stability criterion was afterwards used to examine many other quantities including the kappa-entropy [3], the stretched exponential entropy [3], the quantum group entropy [3], the incomplete entropy ha l-0 03 64 27 8, v er si on 1 24 M ar 2 00 9