Due to some ambiguity in defining mutual Tsallis entropy in the classical probability theory, its generalization to quantum theory is discussed and, as a consequence, two types of generalized quantum discord, called $q$-discords, are defined in terms of quantum Tsallis entropy. $q$-discords for two-qubit Werner and isotropic states are calculated and it is shown that one of them is positive, at least for states under investigation, for all $q>0$. Finally, an analytical expression for $q$-discord of certain family of two-qubit X states is presented.
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