Measuring non-linear functionals of quantum harmonic oscillator states

Using only linear interactions and a local parity measurement we show how entanglement can be detected between two harmonic oscillators. The scheme generalizes to measure both linear and non-linear functionals of an arbitrary oscillator state. This leads to many applications including purity tests, eigenvalue estimation, entropy and distance measures - all without the need for non-linear interactions or complete state reconstruction. Remarkably, experimental realization of the proposed scheme is already within the reach of current technology with linear optics.