Mutual information between reflected and transmitted speckle images

We study theoretically the mutual information between reflected and transmitted speckle patterns produced by wave scattering from disordered media. The mutual information between the two speckle images recorded on an array of N detection points (pixels) takes the form of long-range intensity correlation loops, that we evaluate explicitly as a function of the disorder strength and the Thouless number g. Our analysis, supported by extensive numerical simulations, reveals a competing effect of cross-sample and surface spatial correlations. An optimal distance between pixels is proven to exist, that enhances the mutual information by a factor Ng compared to the single-pixel scenario.