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Conditional q-Entropies and Quantum Separability: A Numerical Exploration

We revisit the relationship between quantum separability and the sign of the relative q-entropies of composite quantum systems. The q-entropies depend on the density matrix eigenvalues p_i through the quantity omega_q = sum_i p_i^q. Renyi's and Tsallis' measures constitute particular instances of these entropies. We perform a systematic numerical survey of the space of mixed states of two-qubit systems in order to determine, as a function of the degree of mixture, and for different values of the entropic parameter q, the volume in state space occupied by those states characterized by positive values of the relative entropy. Similar calculations are performed for qubit-qutrit systems and for composite systems described by Hilbert spaces of larger dimensionality. We pay particular attention to the limit case q --> infinity. Our numerical results indicate that, as the dimensionalities of both subsystems increase, composite quantum systems tend, as far as their relative q-entropies are concerned, to behave in a classical way.