  In recent years, there has been active research toward understanding the
connection between symmetry and physics from the viewpoint of quantum
information theory. This approach stems from the resource theory of asymmetry
(RTA), a general framework treating quantum dynamics with symmetry, and scopes
various fields ranging from the fundamentals of physics, such as thermodynamics
and black hole physics, to the limitations of information processing, such as
quantum computation, quantum measurement, and error-correcting codes. Despite
its importance, in RTA, the resource measures characterizing the asymptotic
conversion rate between i.i.d. states are not known except for $U(1)$ and
$\mathbb Z_2$. In this letter, we solve this problem for the finite group
symmetry and partially solve for the compact Lie group symmetry. For finite
groups, we clarify that (1) a set of resource measures characterizes the
optimal rate of the exact conversion between i.i.d. states in arbitrary finite
groups, and (2) when we consider the approximate conversion with vanishingly
small error, we can realize arbitrary conversion rate between almost arbitrary
resource states. For Lie group symmetry, we show that the optimal rate of the
i.i.d. state conversion with vanishingly small error is bounded by the ratio of
the Fisher information matrices. We give a conjecture that the Fisher
information matrices also characterize the optimal conversion rate, and
illustrate the reasoning. These results are expected to significantly broaden
the scope of the application of RTA.
