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Non-Equilibrium Fluctuation-Dissipation Inequality and Non-Equilibrium Uncertainty Principle

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.