Shannon's idea of diffusion is fundamental to cryptography.
Besides being a descriptive idea, is there any work on measuring or expressing it? Saying something like "System A has more diffusion than B" or "The information in bit x is diffused 1/3 into bit y, 1/3 into bit z, and the remaining third among 11 more bits".
Is there a way to evaluate the diffusion of a transform? XOR does no diffusion, Add with carry does small diffusion, and a permutation may do a lot of diffusion.
In short: Diffusion is a great idea - how can we approach it mathematically?
UPDATE: For instance, we should be able to quantify the diffusion done by adding with carry. This breaks both of the suggested answers: Adding may update the value of every bit. Modifying one bit of the input may modify several bits of the output. But adding still doesn't diffuse very well.