  The success of deep convolutional neural network (CNN) in computer vision
especially image classification problems requests a new information theory for
function of image, instead of image itself. In this article, after establishing
a deep mathematical connection between image classification problem and quantum
spin model, we propose to use entanglement entropy, a generalization of
classical Boltzmann-Shannon entropy, as a powerful tool to characterize the
information needed for representation of general function of image. We prove
that there is a sub-volume-law bound for entanglement entropy of target
functions of reasonable image classification problems. Therefore target
functions of image classification only occupy a small subspace of the whole
Hilbert space. As a result, a neural network with polynomial number of
parameters is efficient for representation of such target functions of image.
The concept of entanglement entropy can also be useful to characterize the
expressive power of different neural networks. For example, we show that to
maintain the same expressive power, number of channels $D$ in a convolutional
neural network should scale with the number of convolution layers $n_c$ as
$D\sim D_0^{\frac{1}{n_c}}$. Therefore, deeper CNN with large $n_c$ is more
efficient than shallow ones.
