Trace-to-Hilbert-Schmidt Speed Ratio in Quantum Dynamics: Universal Bounds and Effective Rank

Abstract

We address the ratio between the trace speed and the Hilbert-Schmidt speed for differentiable finite-dimensional quantum states, $\mathcal{R}=\Vert {\partial }_{\varphi }\rho {\Vert }_1/\Vert {\partial }_{\varphi }\rho {\Vert }_2$. Since the tangent ${\partial }_{\varphi }\rho$ is Hermitian and traceless, $\mathcal{R}$ obeys stronger bounds than those for generic operators. For any nonzero tangent of rank $r$, we prove the sharp bounds $\sqrt{2}\le \mathcal{R}\le \sqrt{r}$ for even $r$ and $\sqrt{2}\le \mathcal{R}\le \sqrt{r-1/r}$ for odd $r$, characterizing all equality cases. Nonstationary pure-state and qubit families saturate the lower bound $\mathcal{R}=\sqrt{2}$. For odd Hilbert-space dimension $d$, we further prove the sharp global maximum $\mathcal{R}\le \sqrt{d-1/d}$. Interpreting ${\mathcal{R}}^2$ as the participation ratio of the singular-value distribution yields an effective-rank picture, $r_{\mathrm{eff}}={\mathcal{R}}^2$. We decompose the effective rank into classical eigenvalue and quantum eigenvector contributions and obtain the bound $r_{\mathrm{eff}}\le r_C+r_Q$, with equality when either component vanishes. Linking the effective rank to the quantum Fisher information $F$ gives $r_{\mathrm{eff}}\ge 8{\mathrm{TS}}^2/F$, showing that a large effective rank is required when this lower bound substantially exceeds the universal minimum value $2$. Finally, a hierarchy of quantum speed limits shows how the effective rank controls the tightness of bounds expressed through the Hilbert-Schmidt speed.