to be $\sigma(T)=
\left({T_{\rm Planck}}\over{T_{\rm max}-T}\right)^{1/3}T_{\rm
  Planck}$. With this estimate of the absolute best accuracy of a clock, we can work out again the evolution of the density matrix for a physical system in the energy eigenbasis. One gets $$\rho(T)_{nm} = \rho_{nm}(0) 
e^{-i\omega_{nm} T} e^{-\omega_{nm}^2  T_{\rm Planck}^{4/3} T^{2/3}}.$$

So we conclude that [*any*]{} physical system that we study in the lab will suffer loss of quantum coherence at least at the rate given by the formula above. This is a fundamental inescapable limit. A pure state inevitably will become a mixed state due to the impossibility of having a perfect classical clock in nature.

Possible experimental implications
==================================

Given the conclusions of the previous section, one can ask what are the prospects for detecting the fundamental decoherence we propose. If one would like to observe the effect in the lab one would require that the decoherence manifest itself in times of the order of magnitude of hours, perhaps days at best. That requires energy differences of the order of $10^{10}eV$ in the Bohr frequencies of the system. Such energy differences can only be achieved in “Schrödinger cat” type experiments. Among the best candidates today are Bose–Einstein condensates, which can have 