Is information conserved in quantum mechanics (after wave function collapse)?

The "conservation of information" follows from the unitarity property of quantum mechanics.

Whether it is actually conserved is a long and dramatic history with a rather a twisted plot. Steven Hawking with many other theorist accepted the possibility of irreversibility of certain physical laws and loss of information - " if irreversibility flouted the laws of physics as they were then understood, so much the worse for those laws".

Another group of physicists, led by Don Page are sure, that the unitarity principle has to be true and information is necesarilly preserved. For the recent results and discussion I recommend to read this article https://www.quantamagazine.org/the-black-hole-information-paradox-comes-to-an-end-20201029/.

If we believe, that QM evolution is unitary, that the time reversal holds, and one can in principle, although not always techically backtrace the history of a system under consideration.

About the measurement and the wavefunction collapse, the terminology is rather abuse, and may lead one to conclusion that something is broken down, but in fact, the measurement replaces the intial apriori probability distribution, by the conditional distribution, aposteriori. Here you can find useful the answer of Lubos Motl https://physics.stackexchange.com/a/3163/261877 and the discussion below.

Short answer: the collapse of a wavefunction destroys information.

As you correctly said, as long as the quantum state evolves according to the Schrodinger equation, information is conserved.

If we adopt an interpretation of quantum mechanics in which collapse happens upon measurement (the Copenhagen interpretation), then even in the simplest case we can see that information would be lost upon collapse.

For example, suppose your system is in a superposition of spin up and spin down states. If you measure it to be spin up, there is no way for you to find out whether it was in a pure spin up state, or in a superposition. Hence, information is lost.

Clarification: in the above scenario, it's even "worse" than just you not being able to find out the initial state. The state of the whole universe (you, the system, the measuring device, etc.) will be the same whether or not the initial state was a pure spin up state or a superposition.

Yes and no. One can study the loss of information in controlled experiments in two-slit like settings - using optical Mach-Zehnder interferometers or solid state which path? interferometers. E.g., in the latter case, one can cause the wave function collapse in a controlled way by coupling one of the arms of the interferometer to a nearby quantum wire or another such interferometer. This is equivalent to causing the collapse of a wave function by observing which path electrons take. It can then be shown theoretically and experimentally that the information lost in the first interferometer can be recovered by considering its correlation with the second. In other words, the information lost in one place, reappears in the other.

On the other hand, in an open system setting, where the collapse is caused by coupling to an infinite/uncontrolled number of the degrees of freedom, such a recovery would be impossible.

Two more points to note:

1. Yes, in the Copenhagen interpretation,information is lost upon wavefunction collapse. On the other hand in a many-worlds-interpretation of quantum mechanics, there is no collapse of the wave function. The true complete wavefunction of the universe always evolves in a unitary (i.e. information-preserving) way, just getting more and more entangled. Subjectively, you are just experiencing one part of the wave-function, and information in other parts of the wavefunction can become inaccesible to you, but nothing is really lost in a global sense.

2. Combining Quantum mechanics and general relativity leads to the "Black hole information paradox": General relativity suggests that:

a) The state of a black hole (as viewed from the outside) is exactly determined by three quantities (mass, angular momentum, electric charge). Any additional information about the stuff that has fallen into it is not accesible to the outside world anymore. (But that information might still be considered stored inside the black hole).

b) Black holes evaporate over time (by "Hawking radiation"). That means, after the black hole is gone, even the stored information seems to be gone. This paradox is an open question of current physics. There are some proposed solutions (see here: https://en.wikipedia.org/wiki/Black_hole_information_paradox#Postulated_solutions),but no consensus has been reched among physicists (and nothing has been experimentally verified).

An alternate way to approach this is to use an interpretation which does not require collapse nor non-determinism. All of the interpretations are simply ways to reconcile the mathematics of a quantum reality with the mathematics of a classical reality as we observe it. There is no wave function collapse in quantum mechanics proper -- it is something which appears in the most common interpretation, the Copenhagen interpretation.

We could use other interpretations to explore this answer. Pilot wave comes to mind as an excellent example. In the pilot wave interpretation, we can measure the state of particles that are constantly being affected by a "pilot wave," a wave function which jostles the particles, changing their state. Like all interpretations of QM, this view is perfectly consistent with the fundamental equations of QM. However, instead of a wave function collapse, like the Copenhagen Interpretation has, we have a pilot wave.

The tricky bit about this pilot wave is it's equation at every moment in time is dependent on the state of all particles, at that moment, even those which are remote. This weirdness is how pilot wave gets around classical behaviors -- it has a wave that propagates infinitely fast. It can be shown that this yields the same statistical results that we get from the Copenhagen interpretation, with its wave function collapse, but no collapse is required.

In this, we find it trivial to show that information is conserved for all actions, even "measurements," because the pilot wave gets defined with respect to the unital operators we see in quantum mechanics. However, that information has been dispersed across every particle in the known universe.

So it shows that, by that interpretation, information is conserved across the entire universe, but any sub-system within the universe will lose information as it is scattered to all of the particles in existence.