Density Matrix vs Wavefunction Formalism

Question

What is the main motivation and advantages of the density matrix formalism compared to the wavefunction formalism? From what I understand, the density matrix is more commonly used when you want to include mixed states and to trace out contributions you don't want to focus on in your problem. But is there actually anything that you cannot do in the wavefunction formalism that you can do with the density matrix, or is it just it is more convenient?

Answer 1

You gave the answer yourself. Mixed states cannot be described as wavefunctions in the way we usually do, so we need density matrices in order to deal with these sorts of states. The reason is pretty much because density matrices can deal with classical probabilities (in the sense you can say there is a 50% chance the state is this pure state, and 50% chance it is another pure state), but wavefunctions only deal with quantum probabilities (you can't do this description I just gave, you can only do superpositions).

Similarly, tracing out degrees of freedom can only be done with density matrices. As a quick example, notice that you can start with a pure state, trace out a subsystem, and get a mixed state as a result. This is actually the standard case. Since wavefunctions can't deal with mixed states, they can't deal with partial traces either.

Answer 2

The density matrix is convenient in the cases you mention, but what you can do with it can also be done with a pure state (or wave function, as you are asking) but the way to do it is perhaps a bit awkward...

You have to assume the existence of another Hilbert space of (at least) the same dimension as the rank of your density matrix. And then you can construct a pure state in the (bigger) tensor product space, say $\mathcal{H}_\text{sub} \otimes \mathcal{H}_\text{ext}$, where your subsystem's Hilbert space is combined with the external space $\mathcal{H}_\text{ext}$. Just start with the observation that the original density matrix in the subsystem can always be diagonalized: $$ \rho_\text{sub} = \sum_i r_i^2\,|\psi_i\rangle_\text{sub}\ {_\text{sub}\langle} \psi_i| $$ by appropriate choice of $|\psi_i\rangle_\text{sub}$. And then combine those with vectors from the other Hilbert space to get a pure state $$ |\psi\rangle_\text{tot} = \sum_i r_i\,|\psi_i\rangle_\text{sub} \otimes |\psi_i\rangle_\text{ext} $$ Clearly this construction works because:

1. Taking the partial trace over $\mathcal{H}_\text{ext}$ immediately gives you back $\rho_\text{sub}$.
2. Any time evolution $U_\text{tot} = U_\text{sub} \otimes U_\text{ext}$ which has no interaction between the two parts of the system will after partial tracing out $\mathcal{H}_\text{ext}$ have the same effect as only applying $U_\text{sub}$ on $\rho_\text{sub}$
3. Time evolution which does have interactions between $\mathcal{H}_\text{ext}$ and $\mathcal{H}_\text{sub}$ would not have this property but that's OK, because in case of external interactions the original density matrix description restricted to $\mathcal{H}_\text{sub}$ would break down as well.

If you take a simple example (of two qubits for instance) this can all be easily verified.

So mathematically this works, and physically one might say that it describes a system that has become entangled with the environment by interactions with it in the past. One might even speculate that pure states are all we need and density matrices are just derived quantities, but that is going a bit beyond your question whether we can do everything with a pure state. It can be done!

EDIT PS: I now see that this answer actually makes your question related to question 424007