Complexity of Bose-Einstein condensates at finite temperature

Abstract: We investigate the geometric quantum complexity of Bose-Einstein condensate (BEC) at finite temperature. Specifically, we use the Bures and Sj\"oqvist metrics -- generalizations of the Fubini-Study metric for mixed quantum states, as well as the Nielsen geometric complexity approach based on purification of mixed states. Starting from the Bogoliubov Hamiltonian of BEC, which exhibits an $SU(1,1)$ symmetry, we explicitly derive and compare the complexities arising from these three distinct measures. For the Bures and Sj\"oqvist metrics, analytical and numerical evaluations of the corresponding geodesics are provided, revealing characteristic scaling behaviors with respect to temperature. In the Nielsen complexity approach, we rigorously handle the gauge freedoms associated with mixed state purification and non-uniqueness unitary operations, demonstrating that the resulting complexity aligns precisely with the Bures metric. Our work provides a comparative study of the geometric complexity of finite-temperature Bose-Einstein condensates, revealing its intimate connections to symmetry structures and temperature effects in BEC systems.