Privacy breaches of cyber-physical systems could expose vulnerabilities to an adversary. Here, privacy leaks of step inputs to linear time-invariant systems are mitigated through additive Gaussian noise. Fundamental lower bounds on the privacy are derived, which are based on the variance of any estimator that seeks to recreate the input. Fully private inputs are investigated and related to transmission zeros. Thereafter, a method to increase the privacy of optimal step inputs is presented and a privacy-utility trade-off bound is derived. Finally, these results are verified on data from the KTH Live-In Lab Testbed, showing good correspondence with theoretical results.