Rigorous Quantum Limits on Monitoring Free Masses and Harmonic Oscillators

There are heuristic arguments proposing that the accuracy of monitoring position of a free mass $m$ is limited by the standard quantum limit (SQL):$\sigma^2 (X(t)) \geq \sigma^2 (X(0)) +(t^2/m^2) \sigma^2 (P(0))\geq \hbar t/m$, where $\sigma^2 (X(t))$ and $\sigma^2 (P(t))$ denote variances of the Heisenberg representation position and momentum operators. Yuen discovered that there are contractive states for which this result is incorrect. Here I prove universally valid rigorous quantum limits (RQL) viz. rigorous upper and lower bounds on $\sigma^2 (X(t))$ in terms of $\sigma^2 (X(0))$ and $\sigma^2 (P(0))$ for a free mass, and for an oscillator. I also obtain the `maximally contractive' and `maximally expanding' states which saturate the RQL, and use the contractive states to set up an Ozawa-type measurement theory with accuracies respecting the RQL but beating the standard quantum limit. The Contractive states for oscillators improve on the Schr\"odinger coherent states of constant variance and may be useful for gravitational wave detection and optical communication.