Fluctuation-dissipation inequality for quadratic open systems

For open systems derived from quadratic total Hamiltonians, we derive a dynamic fluctuation-dissipation (FD) inequality valid for any total initial state and without regard to the sign of the dissipation. With the added constraint that this state be factorized with the reservoir in thermal equilibrium, an uncertainty relation arises naturally from the FD inequality that can be stronger than the usual uncertainty principle in the form 〈q2〉〈p2〉⩾ℏ2/4. We discuss some of the properties of the uncertainty relation relevant to decoherence.