Aspect-based sentiment analysis (ABSA) has attracted much attention due to its wide application scenarios. Most previous studies have focused solely on monolingual ABSA, posing a formidable challenge when extending ABSA applications to multilingual scenarios. In this paper, we study upgrading monolingual ABSA to cross-lingual ABSA. Existing methods usually exploit pre-trained cross-lingual language to model cross-lingual ABSA, and enhance the model with translation data. However, the low-resource languages might be under-represented during the pre-training phase, and the translation-enhanced methods heavily rely on the quality of the translation and label projection. Inspired by the observation that quantum entanglement can correlate multiple single systems, we map the monolingual expression to the quantum Hilbert space as a single quantum system, and then utilize quantum entanglement and quantum measurement to achieve cross-lingual ABSA. Specifically, we propose a novel quantum neural model named QPEN (short for quantum projection and quantum entanglement enhanced network). It is equipped with a proposed quantum projection module that projects aspects as quantum superposition on a complex-valued Hilbert space. Furthermore, a quantum entanglement module is proposed in QPEN to share language-specific features between different languages without transmission. We conducted simulation experiments on the classical computer, and experimental results on SemEval-2016 dataset demonstrate that our method achieves state-of-the-art performance in terms of F1-scores for five languages.