Quantum geometric tensor determines the i.i.d. conversion rate in the resource theory of asymmetry for any compact Lie group

Abstract: Symmetry is one of the most significant foundational principles underlying nature. The resource theory of asymmetry (RTA) is a resource-theoretic framework for investigating asymmetry as a resource to break constraints imposed by symmetries. It has recently undergone significant developments, resulting in applications in a variety of research areas since symmetry and its breaking are ubiquitous in physics. Nevertheless, the resource conversion theory at the core of RTA remains incomplete. In the independent and identically distributed (i.i.d.) setup, where identical copies of a state are converted to identical copies of another state, conversion theory among pure states has been completed only for $U(1)$ group and finite groups. Here, we establish an i.i.d. conversion theory among any pure states in RTA for any continuous symmetry described by a compact Lie group, which includes the cases where multiple conserved quantities are involved. We show that the quantum geometric tensor is an asymmetry monotone for pure states that determines the optimal approximate asymptotic conversion rate. Our formulation achieves a unified understanding of conversion rates in prior studies for different symmetries. As a corollary of the formula, we also affirmatively prove the Marvian-Spekkens conjecture on reversible asymptotic convertibility in RTA, which has remained unproven for a decade.