Uncertainty quantification (UQ) plays a crucial role in data assimilation (DA) since it impacts both the quality of the reconstruction and near-future forecast. However, traditional UQ approaches are often limited in their ability to handle complex datasets and may have a large computational cost. In this paper, we present a new ensemble-based approach to extend the 4DVarNet framework, an end-to-end deep learning scheme backboned on variational DA used to estimate the mean of the state along a given DA window. We use conditional 4DVarNet simulations compliant with the available observations to estimate the 4DVarNet probability density function. Our approach enables to combine both the efficiency of 4DVarNet in terms of computational cost and validation performance with a fast and memory-saving Monte-Carlo based post-processing of the reconstruction, leading to the so-called En4DVarNet estimation of the state pdf. We demonstrate our approach in a case study involving the sea surface height: 4DVarNet is pretrained on an idealized Observation System Simulation Experiment (OSSE), then used on real-world dataset (OSE). The sampling of independent realizations of the state is made among the catalogue of model-based data used during training. To illustrate our approach, we use a nadir altimeter constellation in January 2017 and show how the uncertainties retrieved by combining 4DVarNet with the statistical properties of the training dataset lead to a relevant information providing in most cases a confidence interval compliant with the Cryosat-2 nadir alongtrack dataset kept for validation.