From the sandwiched quantum relative Tsallis entropy to its conditional form: Separability criterion beyond local and global spectra

The quantum relative Renyi entropy of two density matrices was recently extended when the two do not commute, from which a conditional entropy is identified. This is here extended to the corresponding Tsallis relative entropy and to its conditional form. This new expression of Tsallis conditional entropy is shown to witness entanglement beyond the method based on global and local spectra of composite quantum states. When the reduced density matrix happens to be a maximally mixed state, this conditional entropy coincides with the expression in terms of Tsallis entropies derived earlier by Abe and Rajagopal (Physica A 289, 157 (2001)). Separability range in one parameter family of W and GHZ states with 3 and 4 qubits is explored here and it is shown that the results inferred from negative Tsallis conditional entropy matches with that obtained through Peres partial transpose criteria for one-parameter family of W states, in one of its partitions. The criteria is shown to be non-spectral through its usefulness in identifying entanglement in isospectral density matrices.