Non-Abelian entanglement asymmetry in random states

The entanglement asymmetry measures the extent to which a symmetry is broken within a subsystem of an extended quantum system. Here, we analyse this quantity in Haar random states for arbitrary compact, semi-simple Lie groups, building on and generalising recent results obtained for the $U(1)$ symmetric case. We find that, for any symmetry group, the average entanglement asymmetry vanishes in the thermodynamic limit when the subsystem is smaller than its complement. When the subsystem and its complement are of equal size, the entanglement asymmetry jumps to a finite value, indicating a sudden transition of the subsystem from a fully symmetric state to one devoid of any symmetry. For larger subsystem sizes, the entanglement asymmetry displays a logarithmic scaling with a coefficient fixed by the dimension of the group. We also investigate the fluctuations of the entanglement asymmetry, which tend to zero in the thermodynamic limit. We check our findings against exact numerical calculations for the $SU(2)$ and $SU(3)$ groups. We further discuss their implications for the thermalisation of isolated quantum systems and black hole evaporation.