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

• URL: http://arxiv.org/abs/2607.04488v2
• Date: Thu, 09 Jul 2026 21:01:03 GMT
• Title: Trace-to-Hilbert-Schmidt Speed Ratio in Quantum Dynamics: Universal Bounds and Effective Rank
• Abstract summary: We address the ratio between the trace speed and the Hilbert-Schmidt speed for differentiable finite-dimensional quantum states.<n>Since the tangent $partial_$ is Hermitian and traceless, $mathcal R$ obeys stronger bounds than those for generic operators.<n>For any nonzero tangent of rank $r$, we prove the sharp bounds $sqrt2lemathcal Rlesqrtr$ for even $r$ and $sqrt2lemathcal Rlesqrtr
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We address the ratio between the trace speed and the Hilbert-Schmidt speed for differentiable finite-dimensional quantum states, $\mathcal R=\|\partial_φρ\|_1/\|\partial_φρ\|_2$. Since the tangent $\partial_φρ$ 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 $\mathcal R=\sqrt2$. 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.

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