Universal quantum uncertainty relations between nonergodicity and loss of information

TLDR: In this article, uncertainty relations between information loss in general open quantum systems and the amount of nonergodicity of the corresponding dynamics are established, and the elements of the uncertainty relations are quantified via distance measures on the space of quantum density matrices.

Abstract: We establish uncertainty relations between information loss in general open quantum systems and the amount of nonergodicity 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 nonergodicity of the dynamics increases, the lower bound on information loss decreases, which validates the belief that nonergodicity 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.