Here we present information theoretic measures based on the data diffusion operator as characterisations of the representations learned by neural networks. Specifically, we define diffusion spectral entropy (DSE), i.e., entropy of the diffusion operator computed on the neural representation of a dataset as well as diffusion spectral mutual information (DSMI), which assesses the relationship between different sets of variables representing data. First, we show that these definitions form robust measures of intrinsic dimensionality and relationship strength respectively on toy data, outperforming binned Shannon entropy in terms of accuracy. Then we study the evolution of representations within classification networks and networks with self-supervised losses. In both cases, we see that generalizable training results in decrease in DSE over epochs --- starting from a random initialization. We also see that there is an increase in DSMI with the class label over time. On the other hand, training with corrupt labels results in a maintenance or increase in entropy and near-zero DSMI with labels. We also assess DSMI with the input and observe differing trends. On MNIST it grows until plateaus, whereas on CIFAR it increases and then decreases. Overall results show that these measures can elucidate characteristics of network performance as well as data complexity. Code is available at https://github.com/ChenLiu-1996/DiffusionSpectralEntropy.