  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.
