Symmetric quantum states: a review of recent progress

Year: 2026

Authors: Marconi C., Muller-Rigat G., Romero-Palleja J., Tura J., Sanpera A.

Autors Affiliation: Ist Nazl Ottica Consiglio Nazl Ric INO CNR, Largo Enr Fermi 6, I-50125 Florence, Italy; Barcelona Inst Sci & Technol, ICFO Inst Ciencies Foton, Castelldefels 08860, Barcelona, Spain; Univ Autonoma Barcelona, Dept Fis, Fis Teor Informacio & Fenomens Quant, Bellaterra 08193, Spain; Leiden Univ, Inst Lorentz, POB 9506, NL-2300 RA Leiden, Netherlands; Leiden Univ, Appl Quantum Algorithms aQa, NL-2300 RA Leiden, Netherlands; ICREA, Pg Lluis Companys 23, Barcelona 08010, Spain.

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 but 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, robust tools for certifying and verifying the properties of symmetric states in experimental settings are essential. 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 significant results obtained in the different experimental realizations. Despite the notable progress made in recent years with regard to the characterisation and application of symmetric quantum states, several intriguing questions remain unsolved. We conclude this review by discussing some of these open problems and outlining promising directions for future research.

Journal/Review: REPORTS ON PROGRESS IN PHYSICS

Volume: 89 (2)      Pages from: 24001-1  to: 24001-39

More Information: Carlo Marconi acknowledges support from the European Union-NextGeneration EU, ’Integrated infrastructure initiative in Photonic and Quantum Sciences’-I-PHOQS [IR0000016, ID D2B8D520, CUP B53PGC22001750006] Guillem Muller-Rigat acknowledges support from: European Research Council AdG NOQIA; MCIN/AEI (PGC2018-0910.13039/501100011033, CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB, I00, project funded by MCIN/AEI/10.13039/501100011033 and by the ’European Union NextGenerationEU/PRTR’ (PRTR-C17.I1), FPI); QUANTERA DYNAMITE PCI2022-132919, QuantERA II Programme co-funded by European Union’s Horizon 2020 program under Grant Agreement No. 101017733; Ministry for Digital Transformation and of Civil Service of the Spanish Government through the QUANTUM ENIA project call-Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan-NextGenerationEU within the framework of the Digital Spain 2026 Agenda; Fundacio Cellex; Fundacio Mir-Puig; Generalitat de Catalunya (European Social Fund FEDER and CERCA program; Barcelona Supercomputing Center MareNostrum (FI-2023-3-0024); Funded by the European Union (HORIZON-CL4-2022-QUANTUM-02-SGA PASQuanS2.1, 101113690, EU Horizon 2020 FET-OPEN OPTOlogic, Grant No. 899794, QU-ATTO, 101168628), EU Horizon Europe Program (This project has received funding from the European Union’s Horizon Europe research and innovation program under Grant Agreement No. 101080086 NeQSTGrant Agreement 101080086-NeQST); ICFO Internal ’QuantumGaudi’ project; European Union-NextGeneration EU (PRTR-C17,l1), PPCC-MCIN-L5.
KeyWords: symmetry; permutational invariant systems; quantum correlations
DOI: 10.1088/1361-6633/ae440a