Generalized Entropy with Clustering and Quantum Entangled States

We first show how a new definition of entropy, which is intuitively very simple, as a divergence in cluster-size space, leads to a generalized form that is nonextensive for correlated units, but coincides exactly with the conventional one for completely independent units. We comment on the relevance of such an approach for variable-size microsystems such as in a liquid. We then indicate how the entanglement and purity of a two-unit compound state can depend on their entanglement with the environment. We consider entropies of Tsallis, which is used in many different real- life contexts, and also our new generalization, which takes into account correlated clustering in a more transparent way, and is just as amenable mathematically as that of Tsallis, and show how both purity and entangle- ment can appear naturally together in a measure of mutual information in such a generalized picture of the entropy, with values differing from the Shannon type of entropy. This opens up the possibility of using such an entropy in a quantum context for relevant systems, where interactions between microsystems makes clustering and correlations a non-ignorable characteristic.