  We uncover the quantum fluctuation-response inequality, which, in the most
general setting, establishes a bound for the mean difference of an observable
at two different quantum states, in terms of the quantum relative entropy. When
the spectrum of the observable is bounded, the sub-Gaussian property is used to
further our result by explicitly linking the bound with the sub-Gaussian norm
of the observable, based on which we derive a novel bound for the sum of
statistical errors in quantum hypothesis testing. This error bound holds
nonasymptotically and is stronger and more informative than that based on
quantum Pinsker's inequality. We also show the versatility of our results by
their applications in problems like thermodynamic inference and speed limit.
