text: The paper makes a striking connection between two apparently unrelated problems: the problem of designing neural networks to handle a certain type of correlation and the problem of designing a structure to represent wave-function with quantum entanglement. In the wave-function context, the Schmidt decomposition of the wave function is an inner product of tensors. Thus, the mathematical glue connecting the neural networks and quantum entanglement is shown to be tensor networks, which can represent higher order tensors through inner product of lower-order tensors. The main technical contribution in the paper is to map convolutional networks with product pooling function (called ConvACs) to a tensor network. Given this mapping, the authors exploit results in tensor networks (in particular the quantum max-flow min-cut theorem) to calculate the rank of the matricized tensor between a pair of vertex sets using the (appropriately defined) min-cut. The connection has potential to yield fruitful new results, however, the potential is not manifested (yet) in the paper. The main application in deep convolutional networks proposed by the paper is to model how much correlation between certain partition of input variables can be captured by a given convolutional network design. However, it is unclear how to use Theorem 1 to design neural networks that capture a certain correlation. A simple example is given in the experiment where the wider layers can be either early in the the neural network or at the later stages; demonstrating that one does better than the other in a certain regime. It seems that there is an obvious intuition that explains this phenomenon: wider base networks with large filters are better suited to the global task and narrow base networks that have more parameters later down have more local early filters suited to the local task. The experiments do not quite reveal the power of the proposed approach, and it is unclear how, if at all, the proposed approach can be applied to more complicated networks. In summary, this paper is of high theoretical interest and has potential for future applications.