Can one talk of the wave function of a subsystem?

Every physical system we can experiment with is in fact a subsystem of a bigger system that includes the environment. Nevertheless, one often considers the subsystem for itself, and talks about its wave function. How does this square with the formalism of quantum mechanics? Can one really talk of the wave function of a subsystem without inconsistency?

One can talk about the **density matrix** of a subsystem, which is
obtained by tracing out the rest of the universe. In general, the
density matrix is a mixture, and indeed, in quantum optics one needs
to consider this mixture to get accurate predictions.

For elementary textbook examples, one can idealize and pretend that the mixture has rank 1 and hence is a pure state - then and only then one can talk about the wave function of the subsystem.

In some cases, namely when the subsystem has only very few degrees of
freedom (such as a few spins or a few polarizations), one can prepare
pure states to a good approximation, and thus implement good
approximations to the ideal textbook scenarios. This is indeed done to
test fundamental questions of quantum mechanics, as in tests of Bell
inequalities and the like. But even in this case one needs the more
accurate description by density matrices to assess the accuracy of the
results.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ