  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 one common form of the uncertainty principle. We
discuss some of the properties of the uncertainty relation relevant to
decoherence.
