Almost all quantum states have low entropy rates for any coupling to the environment

The joint state of a system that is in contact with an environment is called lazy, if the entropy rate of the system under any coupling to the environment is zero. Necessary and sufficient conditions have recently been established for a state to be lazy [Phys. Rev. Lett. 106, 050403 (2011)], and it was shown that almost all states of the system and the environment do not have this property [Phys. Rev. A 81, 052318 (2010)]. At first glance, this may lead us to believe that low entropy rates themselves form an exception, in the sense that most states are far from being lazy and have high entropy rates. Here, we show that in fact the opposite is true if the environment is sufficiently large. Almost all states of the system and the environment are pretty lazy-their entropy rates are low for any coupling to the environment.