  The fluctuation-dissipation relation is usually formulated for a system
interacting with a heat bath at finite temperature in the context of linear
response theory, where only small deviations from the mean are considered. We
show that for an open quantum system interacting with a non-equilibrium
environment, where temperature is no longer a valid notion, a
fluctuation-dissipation inequality exists. Clearly stated, quantum fluctuations
are bounded below by quantum dissipation, whereas classically the fluctuations
can be made to vanish. The lower bound of this inequality is exactly satisfied
by (zero-temperature) quantum noise and is in accord with the Heisenberg
uncertainty principle, both in its microscopic origins and its influence upon
systems. Moreover, it is shown that the non-equilibrium fluctuation-dissipation
relation determines the non-equilibrium uncertainty relation in the
weak-damping limit.
