  We establish uncertainty relations between information loss in general open
quantum systems and the amount of non-ergodicity of the corresponding dynamics.
The relations hold for arbitrary quantum systems interacting with an arbitrary
quantum environment. The elements of the uncertainty relations are quantified
via distance measures on the space of quantum density matrices. The relations
hold for arbitrary distance measures satisfying a set of intuitively
satisfactory axioms. The relations show that as the non-ergodicity of the
dynamics increases, the lower bound on information loss decreases, which
validates the belief that non-ergodicity plays an important role in preserving
information of quantum states undergoing lossy evolution. We also consider a
model of a central qubit interacting with a fermionic thermal bath and derive
its reduced dynamics, to subsequently investigate the information loss and
nonergodicity in such dynamics. We comment on the minimal situations that
saturate the uncertainty relations.
