Carlo Marconi, Guillem Müller-Rigat, Jordi Romero-Pallejà, Jordi Tura, Anna Sanpera (Jun 13 2025).

Abstract: Symmetric quantum states are fascinating objects. They correspond to multipartite systems that remain invariant under particle permutations. This symmetry is reflected in their compact mathematical characterisation and also in their unique physical properties: they exhibit genuine multipartite entanglement and notable robustness against noise and perturbations. These features make such states particularly well-suited for a wide range of quantum information tasks. Here, we provide a pedagogic analysis of the mathematical structure and relevant physical properties of this class of states. Beyond the theoretical framework, it is essential to develop robust tools for certifying and verifying the properties of symmetric states in experimental settings. In this regard, we explore how standard techniques -- such as quantum state tomography, Bell tests, and entanglement witnesses -- can be specifically adapted for symmetric systems. Next, we provide an up-to-date overview of the most relevant applications in which these states outperform other classes of states in specific tasks. Specifically, we address their central role in quantum metrology, highlight their use in quantum error correction codes, and examine their contribution in computation and communication tasks. Finally, we present the current state of the art in their experimental generation, ranging from systems of cold atoms to implementations via quantum algorithms. We also review the most recent breakthroughs obtained in the different experimental realizations. Despite the significant progress made in recent years regarding the characterization and application of symmetric quantum states, several intriguing questions remain unsolved. We conclude this review discussing some of these open problems and outlining promising directions for future research.

Arxiv: https://arxiv.org/abs/2506.10185