It's pretty simple, and there's been various questions on this site that have had this discussion. And it does get controversial, but the phsycis I straightforward.
The issue is: is the fact that the evolution of the wave function, or the state of a system, is determined uniquely by its initial conditions and the unitary operator that quantum mechanically (or equivalently for quantum field theory) is its time evolution operator? The answer is obviously yes, the wave function or system state evolves uniquely in time according to that operator. The evolution is deterministic as far as the quantum state is concerned.
This is labeled as unitary time evolution. It means that the quantum information that defines the initial state is not 'lost', but rather simply evolved into the information that defines the evolved state. Quantum states evolve deterministically if they are pure states.
In simple terms it means that quantum theory follows causal laws. Causality so not broken
There is nothing controversial there. Wave function or quantum state evolution is perfectly deterministic. What happens with statistical mixes of pure states is statistical mechanics, and does not contradict the determinism, only the practical limits on it due to the large numbers of states and interactions.
The issue comes up when you measure some observable of the state. It is then probabilistically determined exactly what you will get. It is this latter fact that has led to quantum theory be labeled as probabilistic. In doing a measurement you place that system in one of the eigenvalues of the observable operator. It is well known how to compute the probabilities of measuring any specific value. That is what is meant by saying quantum theory is not deterministic.
Note that even then, the quantum state had evolved deterministically, and it is only when you measure, or decohere the system, and interact with it with a lot of degrees of freedom, you get a classical average value with variance around it.
So if you want to determine classical observables, which means you have to measure and not simply let the quantum state go its own way, it produces the probabilistic results and has the quantum uncertainties given by the uncertainty principle for the different observable pairs. But it does not mean the state did not evolve In a perfectly unitary and causal way given by then laws of quantum theory. Sometimes it is loosely said that the wavefunction collapsed into its one observed classical value. And it could have been another. It was determined probabilistically.
That quantum information defined by the state of the system before you measure it, i.e. before you (or anything else) interacts with it, is the quantum information that cannot be lost or destroyed. It can be modified only by the deterministic time evolution operator (and of course by interactions with other particles or fields, which would be then represented in the unitary time evolution operator for it). That quantum information could be also quantum numbers that are conserved in various interactions - for instance total energy, spin, lepton number, fermion number, and others -- in those cases, given by what entities are conserved by the various SM forces.
Now, there is a Black Hole Information paradox that has surfaced because when the particles with specific quantum numbers or quantum states dissappear inside a Black Hole, you can never get them back and that equivalent information is lost. After Hawking radiation it just disappears. Quantum theory says that's impossible. Thus the paradox. There's been plenty of discussion and work on it, but no definitive resolution - probably it'll have to await a well accepted quantum gravity theory. Most physicists probably believe that there's a deeper solution, and that quantum theory causality or information will be preserved.
See the article at Wikipedia https://en.m.wikipedia.org/wiki/Black_hole_information_paradox
So yes, quantum information conservation and quantum state determinism or quantum causality are the same things.
Best Answer
No it doesn't. Given some set of initial conditions, the gravitational three-body problem can be solved for any time you like, to whatever degree of precision you like. It's true that the result can't be expressed in a neatly-packaged "closed form," but that's more a statement about how we write formulas down than it is a statement about the behavior of the system.
More interesting is the fact that the gravitational three-body problem is generically chaotic, meaning (among other things) that nearby initial conditions diverge from each other exponentially. From a practical point of view, this limits how far in the future we can reliably predict the motion of an observed three-body system. Because our measurements have finite resolution, we can't precisely pin down the true initial state of the system, and two initial conditions which are both compatible with our measurements will eventually evolve to completely different final states. Even if exact measurement of initial conditions were possible, our calculations necessarily have finite precision, which would give us the same problem.
However, this is not the same as non-deterministic evolution. Given exact initial conditions, the evolution of a chaotic system is completely deterministic - it's just that if the model is based on real-world observations, those predictions quickly become relatively meaningless because of uncertainty in the initial conditions and finite numerical precision.