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Using the statistical inference method, a non-relativistic, spinless,non-linear quantum dynamical equation is derived with the Fisher informationmetric substituted by the Jensen-Shannon distance information. Among allpossible implications, it is shown that the non-linear Schr\"odinger equationpreserves the symplectic structure of the complex Hilbert space, hence aHamiltonian dynamics. The canonically projected dynamics is obtained on thecorresponding projective Hilbert space of pure state density operators.

Jensen-Shannon Divergence and Non-linear Quantum Dynamics