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To enhance the phase sensitivity of Mach-Zehnder interferometers,we use a tunable phase shift before the light beams are injected into the interferometer.The analytical result of the optimal phase shift is obtained,which only depends on the initial input states.For a non-zero optimal phase shift,the phase sensitivity of the interferometers in the output ports is always enhanced.We can achieve this enhancement for most states,including entangled and mixed states.The optimal phase shift is exhibited in three examples.Compared to previous methods,this scheme provides a general way to improve phase sensitivity and could find wide applications in optical phase estimations.