Breakdown of the classical description of a local system

Abstract: We provide a straightforward demonstration of a fundamental difference between classical and quantum mechanics for a single local system; namely the absence of a joint probability distribution of the position $x$ and momentum $p$. Elaborating on a recently reported criterion by Bednorz and Belzig [Phys. Rev. A {\bf 83}, 52113] we derive a simple criterion that must be fulfilled for any joint probability distribution in classical physics. We demonstrate the violation of this criterion using homodyne measurement of a single photon state, thus proving a straightforward signature of the breakdown of a classical description of the underlying state. Most importantly, the criterion used does not rely on quantum mechanics and can thus be used to demonstrate non-classicality of systems not immediately apparent to exhibit quantum behavior. The criterion is directly applicable any system described by the continuous canonical variables x and p, such as a mechanical or an electrical oscillator and a collective spin of a large ensemble.