Specifically, why would we expect it to be conserved?
From my understanding, it means things like the spin of a particle. Let’s say two electrons are entangled. One would be spinning up, the other spinning down. If we then separate them, that doesn’t change. If we look at one of them and see that it’s spinning up, we know the other one is spinning down without having to look at it.
ETA. If I recall correctly electron spin comes in either 1/2 or -1/2. I’m not sure if up and down are technically correct terms or not.
ETA2. I think this would apply, for example, to the two electrons in a helium atom.
The point about information conservation stems from a fundamental postulate of quantum mechanics. It is assumed that evolution of a system is unitary. Which means that the sum of probabilities of the system does not change - and always sums to one (hence unity).
So a system, which has some ensemble of probabilities evolves into some other state that itself has an ensemble of probabilities. All the way along, as the system evolves, whilst it may change in all manner of ways, the total of the probabilities of what the system is, is always one. If there is subsets of the system, unitarity may be violated for a subset, but when you treat the entire quantum system as a whole, evolution is unitary.
This isn’t a huge difference from a more classical world of systems running like clockwork obeying Newton’s laws and their brethren. At least in terms of understanding causality, and reasoning about the arrow of time. QM of course differs in having that pesky ensemble of probabilities instead of a single nice clean progression. If QM wasn’t unitary, and thus information wasn’t conserved, causality would take a rather mortal hit.
There is no other special nature of information here specific to QM. Just the way it evolves remains sane.
Right. What makes it quantum is that the information comes in discrete packets rather than being a smooth continuum.
So it has nothing to do with time symmetry, correct? IIUC, you can’t determine the previous ensemble of properties from the prior ensemble, if it even makes sense to talk about them. An electron being spin-up has a 100% probability of being so, once we measure it and find that it is. As far as I know, we can only infer that beforehand, it had a 50/50 probability of being spin-up. I also don’t understand how maintaining that the probability of all particles in a system add to 100% is somehow violated when particles fall into a black hole.
This StackExchange answer seems to deal with both your questions.
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 Black hole information paradox - Wikipedia
So yes, quantum information conservation and quantum state determinism or quantum causality are the same things.
Unfortunately, I don’t really understand that. I think it means that you have initial conditions, i.e. the probabilities of the system are such and such and some states are disallowed. Therefore, the system can only move to another state which includes the allowed possibilities. If particles from the system fall into a black hole, some of the allowed states don’t exist anymore and therefore the sum of all the probabilities no longer adds to 100%.
In the example of Hawking radiation, part of the system is lost beyond the event horizon, so that state is lost.
Is that even close?
(I took Diff Eq over 30 years ago and don’t remember what eigenvalues are, so I would have to spend a good deal of time catching up on the math.)