A Margolus-Levitin speed limit for observables: mean energy bounds expectation-value change quadratically
Abstract
The Mandelstam-Tamm and Margolus-Levitin quantum speed limits bound how fast a state evolves, using the energy variance and the mean energy above the ground state. Speed limits on observables -- the change of an expectation value -- have so far used the variance (the Mandelstam-Tamm / quantum-Fisher-information lineage); the mean energy has not been brought to bear on the change of in a fixed state. We close this branch. First, a no-go theorem: there is no state-independent linear mean-energy bound on the time to change an observable's expectation by Delta = | - |; the optimal state-independent exponent of Delta is exactly two. Second, the corresponding sharp quadratic bound, T ( - E_0) >= C_* Delta^2 / sigma_A^2, for every time-independent Hamiltonian H (ground energy E_0), every bounded observable A with spectral spread sigma_A = (lambda_max - lambda_min)/2, and every pure or mixed state, with the dimension-independent constant C_* = 1/(8 sin x_*) = 0.172506267461..., where x_* is the smallest positive root of tan(x/2) = x. The bound is tight, approached but not attained by a near-ground two-level family. We then give the exact energy-time/swing trade-off curve of which C_* is the small-swing slope -- tight at every swing, the observable analog of the Giovannetti-Lloyd-Maccone curve for states -- show the constant survives for mixed states via joint convexity of the trace distance, sharpen it for bandwidth-limited generators and several observables at once, and show the quadratic law degrades to a linear one when the initial state is an eigenvector of the observable. It completes the (mean-energy x observable) corner of the speed-limit landscape and, being quadratic, is most constraining where the linear variance bounds are weakest. Its cleanest physical home is the autonomous quantum clock, where it gives a coherent mean-energy resolution floor.
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BibTeX
@misc{nasr2026margoluslevitin,
title = {{A Margolus-Levitin speed limit for observables: mean energy bounds expectation-value change quadratically}},
author = {Bryan Nasr},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.22658},
eprint = {2608.22658},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22658},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Margolus-Levitin speed limit for observables: mean energy bounds expectation-value change quadratically
AU - Bryan Nasr
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.22658
UR - https://arxiv.org/abs/2608.22658
AB - The Mandelstam-Tamm and Margolus-Levitin quantum speed limits bound how fast a state evolves, using the energy variance and the mean energy above the ground state. Speed limits on observables -- the change of an expectation value -- have so far used the variance (the Mandelstam-Tamm / quantum-Fisher-information lineage); the mean energy has not been brought to bear on the change of in a fixed state. We close this branch. First, a no-go theorem: there is no state-independent linear mean-energy bound on the time to change an observable's expectation by Delta = | - |; the optimal state-independent exponent of Delta is exactly two. Second, the corresponding sharp quadratic bound, T ( - E_0) >= C_* Delta^2 / sigma_A^2, for every time-independent Hamiltonian H (ground energy E_0), every bounded observable A with spectral spread sigma_A = (lambda_max - lambda_min)/2, and every pure or mixed state, with the dimension-independent constant C_* = 1/(8 sin x_*) = 0.172506267461..., where x_* is the smallest positive root of tan(x/2) = x. The bound is tight, approached but not attained by a near-ground two-level family. We then give the exact energy-time/swing trade-off curve of which C_* is the small-swing slope -- tight at every swing, the observable analog of the Giovannetti-Lloyd-Maccone curve for states -- show the constant survives for mixed states via joint convexity of the trace distance, sharpen it for bandwidth-limited generators and several observables at once, and show the quadratic law degrades to a linear one when the initial state is an eigenvector of the observable. It completes the (mean-energy x observable) corner of the speed-limit landscape and, being quadratic, is most constraining where the linear variance bounds are weakest. Its cleanest physical home is the autonomous quantum clock, where it gives a coherent mean-energy resolution floor.
ER -