Scientists have fully resolved the problem of characterizing the conversion of mixed quantum states under compact Lie group symmetries, a crucial advance in the quantum resource theory of asymmetry. They have established a single-letter formula for the optimal conversion rate between arbitrary states, with vanishing trace-distance error, in the i.i.d. asymptotic regime. This rate is determined by a one-parameter family of quantum Fisher information (QFI) matrices that interpolates between the symmetric and right-logarithmic-derivative QFIs. This result contrasts with pure-state conversion, where a state-independent finite subset is generally insufficient, even for U(1) symmetry.

The new formula further yields exact pure-state distillation rates in terms of the generalized quantum geometric tensor, characterizes asymptotically reversible interconversion, and identifies bound asymmetry for quantum clocks. Complementarity among different members of the QFI family also uncovers an activation mechanism for quantum clocks, suggesting new possibilities for their design and operation. This work addresses a fundamental limitation in understanding how quantum states can be manipulated and transformed in the presence of broken symmetries.

The proof relies on two key developments. First, quantum local asymptotic normality has been extended to unitary models with arbitrary rank and spectral degeneracy. Second, convertibility between quantum Gaussian shift models has been characterized in terms of the same one-parameter family of QFIs. Together, these results provide an operational characterization of symmetry breaking for general quantum states in the i.i.d. asymptotic regime and reveal how distinct QFI constraints give rise to irreversibility and activation in quantum systems.