Derivatives of Mutual Information in Gaussian Channels

The I-MMSE formula connects two important quantities in information theory and estimation theory: the mutual information and the minimum mean-squared error (MMSE). It states that in a scalar Gaussian channel, the derivative of the mutual information with respect to the signal-to-noise ratio (SNR) is one-half of the MMSE. Although any derivative at a fixed order can be computed in principle, a general formula for all the derivatives is still unknown. In this paper, we derive this general formula for vector Gaussian channels. The obtained result is remarkably similar to the classic cumulant-moment relation in statistical theory.