# If a wave function represents knowledge, what does a density matrix mean, then?

I'm curious about this. I've heard of interpretations of quantum theory in which the wave function $\psi$ is taken as representing knowledge, or information (e.g. "Quantum Bayesianism"), about the physical parameters of a particle like its position and momentum, and thus things like the famous "wave function collapse" are taken as simply an update of knowledge due to new information received about the system. Moreover, with the Bayesian interpretation you can say a prior wave function before a measurement could also incorporate assumed knowledge as opposed to that actually received from the environment.

However, what I don't get is this. I also notice there's a thing called a "mixed state" or "density matrix" where you end up with a classical (i.e. not complex number-based, or norm-1 real, not norm-2 complex, probability) probability distribution among a wide panoply of wave functions, representing uncertain knowledge in what wave function it actually is, such as in the case of describing one particle of an entangled system (which has a composite wave function $\psi(x_1, x_2)$ and so the wave function $\psi_1(x_1)$ depends on specifically what value attains for $x_2$ which is unknown, and conversely), or where that a system is interacting with the macroscopic environment, or where there is uncertainty in how the system was prepared.

Yet it seems if we interpret the first wave function as knowledge, then the density matrix becomes some kind of redundant-seeming "knowledge about knowledge", which makes no sense. Does this mean the knowledge interpretation of a wave function itself is problematic, or is there some way to reconcile these two and explain why the density matrix is not redundant? It makes sense to consider it non-redundant in a "realist" interpretation where that $\psi$ is a physical wave or something like that, since then the DM straightfowardly becomes lack of knowledge about what the real physical wave is, but not where $\psi$ itself is still knowledge.

So what's up with this, anyways?

• and that's how you know that complex (relative) phase amplitudes of states are not merely "knowledge" but a distinct ontological phase quantity associated with every state – lurscher Dec 16 '19 at 16:04

A mixed state $$\Psi$$ is a probability distribution on the state space. The density matrix tells you how likely it is to be in some state. Knowledge of a mixed state is not the same as knowledge of a state; you may be confounding a mixed state with its expected value $$\mathbb E[\Psi]$$, which is a pure state with the property that for all measurable sets $$A$$, $$(\mathbb E(\Psi))(A) = \mathbb E[\mathbb P(A)]$$ where the expected value on the right is taken over the state space w.r.t. the probability measure $$\Psi$$.