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.

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关于量子自编码器的压缩率：控制设计、数值和实验实现

旨在在低维潜在空间中压缩量子信息的量子自编码器是量子信息领域自动数据压缩的核心。在本文中，我们为给定的量子自动编码器建立了压缩率的上限，并提出了一种用于训练自动编码器以实现最大压缩率的学习控制方法。使用特征分解和矩阵微分从理论上证明了压缩率的上限，这是由输入状态的密度矩阵表示的特征值确定的。给出了 2-qubit 和 3-qubit 系统的数值结果，以演示如何训练量子自动编码器以实现理论上的最大压缩，并比较使用不同机器学习算法的训练性能。