  Quantum autoencoders which aim at compressing quantum information in a
low-dimensional latent space lie in the heart of automatic data compression in
the field of quantum information. In this paper, we establish an upper bound of
the compression rate for a given quantum autoencoder and present a learning
control approach for training the autoencoder to achieve the maximal
compression rate. The upper bound of the compression rate is theoretically
proven using eigen-decomposition and matrix differentiation, which is
determined by the eigenvalues of the density matrix representation of the input
states. Numerical results on 2-qubit and 3-qubit systems are presented to
demonstrate how to train the quantum autoencoder to achieve the theoretically
maximal compression, and the training performance using different machine
learning algorithms is compared. Experimental results of a quantum autoencoder
using quantum optical systems are illustrated for compressing two 2-qubit
states into two 1-qubit states.
