What kind of discovery would prove the impossibility of time travel? Bearing in mind that:


*

*"Time travel" does not have one fixed definition.

*"Proof" in physics is never absolute or final as in mathematics.
Except logical arguments which lead to paradoxes, what kind of discovery would prove the impossibility of time travel? (Even if the discovery requires far too advanced technology.)
 A: As some commenter have pointed out you can't prove a negative or impossibility. In fact you can't prove a theory true, you can support theories with evidence and you can show a theory is inconsistent with new data and thus potentially wrong outside some domain of observations. 
We can though say something is impossible in the same way we can say a perpetual motion machine is impossible or that traveling faster than light is impossible. The prospect for traveling back in time is minimal. I can give an informal argument for this. Suppose you have a quantum state that you sent on a closed timelike curve. By doing this there will be a point in spacetime where the path of the quantum state closes or comes close. The observer has then duplicated the quantum state. This is not possible by quantum physics.
The duplication of a quantum state $|\psi\rangle~\rightarrow~|\psi\rangle|\psi\rangle$ for $|\psi\rangle~=~a|+\rangle~+~b|-\rangle$ leads to
$$
\psi\rangle~\rightarrow~|\psi\rangle|\psi\rangle~=~a^2|+\rangle|+\rangle~+~b^2|-\rangle|-\rangle~+~ab(|+\rangle|-\rangle~+~|-\rangle|+\rangle).
$$
However, something funny is going on. For we could think of duplicating on the basis $|\pm\rangle$, which will eliminate the cross term above. This then means so called quantum cloning is not a unitary quantum process and is then not permitted.
One could say that maybe something with gravity permits these violations, so this might not be a proper proof of no time travel. Maybe gravitation and quantum mechanics do not mesh the way we think they should and that maybe spacetime physics overrules quantum mechanics. However, this no-cloning argument is reasonable though, just as arguments of causality violation. 
