Mixtures of high dimensional Gaussian distributions have been studied\nextensively in statistics and learning theory. While the total variation\ndistance appears naturally in the sample complexity of distribution learning,\nit is analytically difficult to obtain tight lower bounds for mixtures.\nExploiting a connection between total variation distance and the characteristic\nfunction of the mixture, we provide fairly tight functional approximations.\nThis enables us to derive new lower bounds on the total variation distance\nbetween pairs of two-component Gaussian mixtures that have a shared covariance\nmatrix.\n