Learning Density Operator Latent Variable Models via Quantum Information Projection

Latent variable models based on density operators, the mathematical foundation of quantum mechanics, remain far less capable than their probabilistic counterparts. This is primarily due to the lack of an Expectation-Maximization (EM) framework for learning density operator models, since the absence of conditional probability for density operators prevents a direct extension of the classical derivation. This paper addresses this challenge by proposing Density Operator Expectation Maximization (DO-EM), using tenets from quantum information theory. We first derive an operator-theoretic analog of the evidence lower bound from the data processing inequality and study its maximization as a projection onto the data manifold. We solve this information projection problem while also improving existing information recovery results. The DO-EM algorithm guarantees monotone likelihood ascent for a rich class of models and is simpler than direct likelihood gradient ascent. The Expectation step recovers the Petz Recovery Map when the model class can be aligned with the target operator. We specialize these models to Classical-Quantum LVMs which scale to standard image datasets via a per-datapoint decomposition of the variational bound and outperform their classical counterparts under matched computational resources.