Quantum Data Compression and Quantum Cross Entropy

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.