Why is information indestructible in quantum mechanics? Why is information indestructible in quantum mechanics?
 A: We can paraphrase the question to a statement like this: "Entropy of any isolated quantum system with density operator remains constant with time". 
$$S(ρ) = −\mathrm{Tr}(ρ \log_2 ρ)$$
This is because ρ is not time-dependent and the eigenvalues ρ does not change with time (preserves spectrum of ρ). Here there is no consideration of interacting quantum systems!
[When there are two interacting initially pure states (let's say), does not remain pure with time. The interaction correlation functions indeed add to the information. 
That is, 
$$S(ρ_{AB}) ≤ S(ρ_A) + S(ρ_B)$$
Hence the argument is entropy is not the right argument to account for conserved information. Hence there are formalisms like coherent entropy etc (Ref: https://arxiv.org/abs/1708.05727) to account for such complex systems.]
Any classical system would have these correlation contributions to the information! Hence we say the information is conserved in isolated quantum systems and not in classical systems.
A: The indestructibility of information is stated by saying that time evolution is unitary. This is typically taken as an axiom of quantum mechanics, e.g., in the pretty standard axiomatization by Carroll. If you use such an axiomatization, then you can't really give a theoretical justification. You can try to test experimentally whether time evolution is unitary, but an experiment doesn't work as a test of such a principle unless you have some other test theory that violates the principle. This is what is often done in relativity, for example, where the PPN test theory gives a way to quantify possible violations of general relativity.
But quantum mechanics seems to be a very "brittle" theory, in the sense that when people try to construct test theories more general than standard q.m., they keep failing. For example, Weinberg tried to produce a nonlinear generalization of quantum mechanics, but Gisin proved a no-go theorem showing that it would have unacceptable implications. Hawking tried to create a generalization of quantum mechanics that would allow for small violations of unitarity, but Banks then published a paper claiming to show that this would violate conservation of energy or causality. However, this seems to be controversial, and Unruh published what he claims is a counterexample. What I think is true is that we still don't have a useful test theory for small violations of unitarity.
Here's a simple heuristic example of why it's hard to create a reasonable theory in which unitarity is violated. In the photoelectric effect, conservation of energy works out only because of long-range correlations that prevent the same photon from being absorbed by two different atoms. So in general, q.m. doesn't give conservation laws unless we have such correlations. Preservation of such correlations is essentially what we mean by unitary time evolution.
Unitarity is also what we need in order to have conservation of probability. It's not clear how we could test conservation of probability, even in principle. If you do an experiment, something will happen with 100% probability.
References
Banks, Susskind, and Peskin, "Difficulties for the evolution of pure states into mixed states," Nuclear Physics B, Volume 244, Issue 1, 24 September 1984, Pages 125-134, copy hoarded in ~/Papers/banks_1984.pdf
Carroll and Sebens, "Many Worlds, the Born Rule, and Self-Locating Uncertainty," https://arxiv.org/abs/1405.7907
Gisin, "Weinberg's non-linear quantum mechanics and supraluminal communications," http://dx.doi.org/10.1016/0375-9601(90)90786-N , Physics Letters A 143(1-2):1-2 
Unruh and Wald, https://arxiv.org/abs/hep-th/9503024
