This is about the best question I've seen.
The first part is easy. Even on a mid-scale, say with a bunch of plants and animals living in an isolated box, all the processes increase entropy. There is absolutely nothing that we know of that makes life any different from any other process in this regard. As for whether the net effect of life has been to increase or slow down the rate of entropy increase, I bet the answer is that it has slightly increased it. I’m guessing that because it seems that life has increased the light absorbancy of the Earth, and led to the thermalization of some light energy that might otherwise be traveling through space in a somewhat lower entropy form.
The second part is more challenging. There is a precise quantum definition of entropy as the trace of a particular matrix, as you say, There is an exact theorem that if that matrix changes in time in a unitary way, as one would get from anything with the general form of Schroedinger’s equation, the entropy doesn’t change! So what gives? This is one of the deepest remaining mysteries.
Some answers include versions of the ’coarse graining" idea: that entropy is only defined with respect to an observer and that one must consider some limitations of the observer in determining the entropy of a system. If there is no concise way to convey the full description of the quantum system, limited observers such as ourselves would have to resort to partial descriptions, which have entropy. By the way, essentially the same problem arose for classical statistical mechanics.
Other ideas are more purely quantum mechanical. If each local system becomes progressively more entangled with the rest of the universe, any purely local description which ignores the detailed state of the rest of the universe will acquire entropy even though in a global sense the whole thing is in a single pure state. In a sense this idea too relies on a limitation of the observer, i.e. the inability to use arbitrarily remote information.
My favorite speculation, not generally very popular, is that problems with the "measurement" process in quantum mechanics are already telling us that the full time-dependence operator is NOT strictly unitary. Attempts to introduce non-unitary processes are far from being coherent theories yet, but if successful they will probably give the Second Law as a free bonus. Don’t tell anybody I said so.
p,s, Some years later, I no longer find that last idea so appealing.
(republished on 07/21/06)