Relativistic quantum-speed limit for Gaussian systems and prospective experimental verification

Timing and phase resolution in satellite QKD, kilometre-scale gravitational-wave detectors, and space-borne clock networks hinge on quantum-speed limits (QSLs), yet benchmarks omit relativistic effects for coherent and squeezed probes. We derive first-order relativistic corrections to the Mandelstam-Tamm and Margolus-Levitin bounds. Starting from the Foldy-Wouthuysen expansion and treating-p4/(8m3c2) as a harmonic-oscillator perturbation, we propagate Gaussian states to obtain closed-form QSLs and the quantum Cram & eacute;r-Rao bound. Relativistic kinematics slow evolution in an amplitude-and squeezing-dependent way, increase both bounds, and introduce an e2t2 phase drift that weakens timing sensitivity while modestly increasing the squeeze factor. A single electron (e approximate to 1.5 x 10-10) in a 5.4 T Penning trap, read out with 149 GHz quantum-limited balanced homodyne, should reveal this drift within similar to 15 min - within known hold times. These results benchmark relativistic corrections in continuous-variable systems and point to an accessible test of the quantum speed limit in high-velocity or strong-field regimes.