The emerging field of quantum machine learning has the potential of revolutionizing our perspectives of quantum computing and artificial intelligence. In the predominantly empirical realm of quantum machine learning, a theoretical void persists. This paper addresses the gap by highlighting the quantum cross entropy, a pivotal counterpart to the classical cross entropy. We establish quantum cross entropy's role in quantum data compression, a fundamental machine learning task, by demonstrating that it acts as the compression rate for sub-optimal quantum source coding. Our approach involves a novel, universal quantum data compression protocol based on the quantum generalization of variable-length coding and the principle of quantum strong typicality. This reveals that quantum cross entropy can effectively serve as a loss function in quantum machine learning algorithms. Furthermore, we illustrate that the minimum of quantum cross entropy aligns with the von Neumann entropy, reinforcing its role as the optimal compression rate and underscoring its significance in advancing our understanding of quantum machine learning's theoretical framework.

翻译：量子机器学习这一新兴领域有潜力彻底改变我们对量子计算和人工智能的认知。在以经验为主导的量子机器学习领域中，理论空白依然存在。本文通过强调量子交叉熵（经典交叉熵的关键对应概念）来填补这一空白。我们论证了量子交叉熵在量子数据压缩（一项基础机器学习任务）中的作用，证明其作为次优量子信源编码的压缩率。我们的方法基于可变长编码的量子推广和量子强典型性原理，提出了一种新型通用量子数据压缩协议。该协议揭示了量子交叉熵可有效充当量子机器学习算法中的损失函数。此外，我们证明量子交叉熵的最小值与冯·诺依曼熵一致，这强化了其作为最优压缩率的角色，并凸显了其在推进对量子机器学习理论框架理解方面的重要意义。