In this paper, we introduce the notions of conditional mutual information (CMI) and normalized conditional mutual information (NCMI) for classification deep neural networks (DNNs). In particular, CMI and the ratio between CMI and NCMI quantify the intra-class concentration and inter-class separation of a DNN in its output probability distribution space, respectively. Utilizing NCMI to assess widely recognized DNNs pre-trained on ImageNet reveals a notable inverse relationship between their validation accuracies and NCMI values on the ImageNet validation dataset. Building upon this insight, the conventional deep learning (DL) framework is modified by minimizing the standard cross-entropy function while imposing an NCMI constraint. This refinement results in a novel approach known as CMI-constrained deep learning (CMIC-DL). Comprehensive experimental findings demonstrate that DNNs trained using CMIC-DL outperform state-of-the-art models trained within standard DL and other loss functions in the literature. In addition, some semantic meaning of CMI is also discovered.