Motivated by the Dirac idea that fundamental constant are dynamical variables and by conjectures on quantum structure of spacetime at small distances, we consider the possibility that Planck constant $hbar$ is a time depending quantity, undergoing ra
ndom gaussian fluctuations around its measured constant mean value, with variance $sigma^2$ and a typical correlation timescale $Delta t$. We consider the case of propagation of a free particle and a one--dimensional harmonic oscillator coherent state, and show that the time evolution in both cases is different from the standard behaviour. Finally, we discuss how interferometric experiments or exploiting coherent electromagnetic fields in a cavity may put effective bounds on the value of $tau= sigma^2 Delta t$.
We propose a quantum optics experiment where a single two-mode Gaussian entangled state is used for realizing the paradigm of an amendable Gaussian channel recently presented in Phys. Rev. A, textbf{87}, 062307 (2013). Depending on the choice of the
experimental parameters the entanglement of the probe state is preserved or not and the relative map belongs or not to the class of entanglement breaking channels. The scheme has been optimized to be as simple as possible: it requires only a single active non-linear operation followed by four passive beam-splitters. The effects of losses, detection inefficiencies and statistical errors are also taken into account, proving the feasibility of the experiment with current realistic resources.
