What experiment supports the axiom that quantum operations are reversible? Among the axioms of quantum mechanics there is one axiom that says transformations of a quantum state need to be continuous, linear, and reversible (and this together with the other axioms results in unitary operators). 
What is the simplest experiment that confirms the need for this axiom, in particular reversibility of the operations?
Perhaps another way to state this is: what experiment definitively shows that we can't simply use classical probability (and probabilistic operations) to model quantum mechanics?
 A: One place to look for relatively direct evidence is in the cross sections of time-reversed nuclear and particle physics reactions. For instance comparing
$$A + n \to B + \alpha $$
with 
$$B + \alpha \to A + n$$
consistently shows the same (energy dependent) cross-sections for both directions where these reaction can be done between ground states of the nuclei (or including long-lived meta-stable excited states).
Likewise with proton-alpha and proton-neutron reaction pairs.
A: Time evolution is governed by a one to one relationship. The Schrödinger equation says $$i\hbar \frac{\partial}{\partial t} \Psi=\hat H \Psi.$$
The fact that $\hat H$ is self adjoint and there is an $i$ on the left hand side means that norm doesn't change so two orthonormal states states stay orthonormal as they evolve. Thus it can't send two different states to the same state and so time evolution is reversible.
Every single experiment is about testing the predictions of the Schrödinger equation. The double slit has a free particle $\hat H$ so the phase changes at a fixed rate depending on the path distance. Which leads to interference, which is what we observe when we do the experiment. Other situations have different Hamiltonians, but each one shows the Schrödinger equation working.
In fact every aspect of how we use it shows it doesn't change the norm. For instance when we get frequencies of different results, the fact the frequencies sum to one at two different times is because the evolution was norm preserving.
So we would need a completely different theory if it wasn't reversible, since so many aspects are based on it being reversible.
A: Quantum mechanics conceives 'physical' transformations on physical systems represented in unitary operators. Such transformations range from simple operations like a global rotation, to complex microscopic operations like the Schrödinger operator that gives time evolution
But unitary transformations do not exhaust the list of available transformations that you can perform on a physical quantum system. Measurements, for instance, are neither unitary nor deterministic, and despite some special cases like the delayed quantum-eraser experiments, they are in general irreversible
So, to answer your question, any axiom that pretends to impose only unitary operations on quantum system, is just flagrantly ignoring measurements
Certain textbooks will try to avoid the conclusion of measurements being intrinsically non-unitary operations by confusing them with decoherence. The problem with equating measurement with decoherence is that decoherence can only remove the interference terms from a quantum density matrix, but the distribution of probabilities are still there. When you make measurements, you don't see the probability distribution, but individual eigenvalues, and each such measurement is intrinsically irreversible
