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Geometric lower bound for a quantum coherence measure

Nowadays, geometric tools are being used to treat a huge class of problems of quantum information science. By understanding the interplay between the geometry of the state space and information-theoretic quantities, it is possible to obtain less trivial and more robust physical constraints on quantum systems. In this sense, here we establish a geometric lower bound for the Wigner-Yanase skew information (WYSI), a well-known information theoretic quantity recently recognized as a proper quantum coherence measure. Starting from a mixed state evolving under unitary dynamics, while WYSI is a constant of motion, the lower bound indicates the rate of change of quantum statistical distinguishability between initial and final states. Our result shows that, since WYSI fits in the class of Petz metrics, this lower bound is the change rate of its respective geodesic distance on quantum state space. The geometric approach is advantageous because raises several physical interpretations of this inequality under the same theoretical umbrella.