Do objective collapse equations actually collapse the state?

Question

Why are objective collapse theories stated to collapse the state from a superposition to a single eigenstate (corresponding to the measured eigenvalue)? For this discussion, we are focusing on the mathematics of the equations, not on any interpretation on quantum mechanics. I merely wish to know if objective collapse equations (the dynamics as described by the typically stochastic differential equations) yield something like: (a) \begin{equation} \langle x\mid \rho(t)\mid y\rangle \propto e^{-\lambda(x-y)^2 t}\langle x\mid \rho(0)\mid y\rangle \end{equation} or (b) \begin{equation} \langle x\mid \rho(t)\mid y\rangle \propto e^{-\lambda((x-x_0)^2+(y-x_0)^2) t}\langle x\mid \rho(0)\mid y\rangle \end{equation} where the system tends to the pure state $\mid x_0\rangle \langle x_0\mid $, where $x_0$ is the randomly measured eigenvalue.

In (a), we have a process which can be achieved via a Lindblad equation derived from an open quantum system. In this context the process would be termed "decoherence".If we find (a), then I find the name "objective collapse" to be inappropriate because I would define collapse as in (b). Hence, the second question is whether anyone actually uses the term "collapse" for the process in (a).

Answer 1

There are three ways to read this question, depending on your perspective about wavefunctions and density matrices. Here are three perspectives you might take, listed in order from most common to least common.

1. There is in fact one objective wavefunction, from which you can construct one objective pure-state density matrix. A mixed-state density matrix is not an objective state of reality, but rather a (subjective) state of knowledge or belief, indicating how one would assign probabilities given limited knowledge about the actual full wavefunction.

2. The wavefunction itself is subjective, and so is the associated pure-state density matrix. A mixed-state density matrix is evidently at least as subjective as a pure-state, if not more so.

3. There is one objectively correct density matrix in any situation, even if it is in a mixed state.

I'm not sure your question makes much sense from perspective 1). You are not really distinguishing between wavefunctions and density matrices, and you are talking about an "objective" collapse as ending up with a mixed state density matrix, so I suspect you're not thinking of this density matrix as being subjective. (?)

Perspective 2) also seems like there would be a disconnect. If everything in ordinary quantum theory is subjective, then evidently there would be a world of difference between an "objective collapse" and mere subjective uncertainty.

Perspective 3) is quite uncommon, but I suspect that's the best way to make sense of your question. This perspective grapples with the idea that even a mixed state could somehow be an objective state of affairs. I've never been able to wrap my head around this perspective very well. (What does probability even mean in such a situation? What does objectivity mean?) Still, I guess I can almost see how (given this viewpoint) an objective collapse could be said to land you into an "objectively mixed state".

But even if your perspective is 3), then there is still a huge difference between this and decoherence. For this discussion, please see the answers to this previous question. These answers might also help if you are taking perspective 1).