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 (
A theoretical physics FAQ