Aiming at the limitation of existing methods for the medical ultrasound images despeckling, a novel quantum-inspired despeckling method based on the local entropy is proposed for the medical ultrasound images. Firstly, the log-transformed images are decomposed by the Dual-tree Complex Wavelet Transform (DTCWT), and the signal and speckle noise are modeled separately. Then, considering the normalized products of the local entropy of the real components extracted from coefficients and their parents, the adjustable parameter is obtained by the quantum inspired theory to adjust the probability of signal and noise. Finally, the modified bivariate shrinkage function is exploited to obtain the despeckled image. The experimental results show that the proposed method can preserve detail information effectively and reduce the speckle noise of medical ultrasound image at the same time.