Given a representation of a compact Lie group and a state we define a probability measure on the coadjoint orbits of the dominant weights by considering the decomposition into irreducible components. For large tensor powers and independent copies of the state we show that the induced probability distributions converge to the value of the moment map. For faithful states we prove that the measures satisfy the large deviation principle with an explicitly given rate function.
We acknowledge financial support from the European Research Council (ERC Grant Agreement no. 337603 and 818761) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059) and from Universidad de los Andes, Faculty of Sciences project INV-2017-51-1445 (AB). This research was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and the National Research, Development and Innovation Fund of Hungary within the Quantum Technology National Excellence Program (Project Nr. 2017-1.2.1-NKP-2017-00001) and via the research grants K124152, KH129601 (PV).
We dedicate this work to the memory of Graeme Mitchison
"Large deviation principle for moment map estimation." Electron. J. Probab. 26 1 - 23, 2021. https://doi.org/10.1214/21-EJP636