Why is quantum mechanics reversible?

"Quantum mechanics is reversible" this statement is everywhere, some even said it's just an observed fact about the universe. I can't find a layman explanation or example why is it reversible?

• What does you mean by "reversible"? – Name YYY Sep 28 '17 at 15:32
• "Quantum gates have to be reversible because quantum mechanics is reversible (and even more specifically it is unitary)" what this really means – Consy Sep 28 '17 at 15:44

Quantum mechanics comprises a way of looking at the world where we represent the world with matrices[1] instead of numbers. Matrix multiplication has the quirk that $AB\ne BA$ in general, so that the order of multiplication usually matters: we use this to understand things like the Heisenberg uncertainty principle and the like.
The general principle is that the state of the world is some complex column vector[2] that we call $|\psi\rangle,$ and it corresponds to some complex-conjugate row vector that we call $\langle \psi|$. We predict only averages of measurements, and we predict them by associating to that measurable quantity some matrix $M.$ The average quantity for this measurement $\langle M \rangle$ in the state $|\psi\rangle$ is then given by the matrix multiplication,$$\langle M \rangle = \langle \psi|M|\psi \rangle.$$ This leaves a 1x1 matrix on the right hand side which we just interpret as a number. In order to guarantee that it is a real number, we must restrict $M$ to be a special sort of matrix: it must be equal to its "adjoint," which is its matrix-transpose with all of the elements complex-conjugated. We call these matrices "Hermitian" matrices, and the key point is that they make all of the bad imaginary numbers disappear for all of our predictions about averages, which is good because most of the numbers that we deal with in this world are not complex numbers.
The key question you have asked is, what happens over time? The answer is that we are chasing the Einstein-Planck equation, that the energy of a photon is given proportional to its frequency $f$ by some constant $\hbar$ as: $$E = 2\pi~\hbar~f.$$ This sort of wave equation can be chased by assuming that the total energy of a system is now given by some Hermitian matrix that we call $H$, for the "Hamiltonian" of the system. (Mr. Hamilton created a way to analyze the whole world where all of the laws of physics were contained in an expression for the total energy of a system. Quantum mechanics was designed to reduce to Mr. Hamilton's equations in a "classical limit.") This expression says that over a very short time scale $\delta t$ the state vector $|\psi(t)\rangle$ changes according to: $$|\psi(t+\delta t)\rangle \approx |\psi(t)\rangle - i~\frac{\delta t}{\hbar}~H|\psi(t)\rangle,\\ \langle\psi(t+\delta t)| \approx \langle\psi(t)| + i~\frac{\delta t}{\hbar}\langle\psi(t)| H.$$ To see why this has the associated property probably requires more than a lay education, but I will summarize very briefly to say that it is possible to find a bunch of "basis vectors" $|n\rangle$ where $H|n\rangle = E_n|n\rangle$ for some real numbers $E_n$ called the "eigenvalues" of the Hamiltonian; the $|n\rangle$ vectors are called its "eigenvectors." When we look at the above equations for these, it suggests that these eigenvectors oscillate in time as $e^{-iE_n t/\hbar}|n\rangle,$ which returns to itself after a period $T = 2\pi~\hbar/E_n.$ Since $f=1/T$ we get the Einstein-Planck equation.
What we mean when we say that the physics is entirely reversible is that the above expression for $|\psi(t + \delta t)\rangle$ does not lose any information: it cannot produce the same output given two different input $|\psi(t)\rangle$ terms. There is always a different Hamiltonian, usually $H\mapsto -H$, which will undo any time-evolution that you originally did.
There is one aspect that is not "unitary" like this, and it is the not-well-understood process by which we do not actually see averages: quantum mechanics predicts the averages $\langle M \rangle$ really well but what we actually see are the eigenvalues of $M$, so that the world always appears to be in an eigenvector state of whatever we just measured. But there are proposals to try and derive these things from the unitary evolution for $|\psi(t+\delta t)\rangle$ that we saw before: most notably, the many-worlds interpretation for quantum mechanics does this. They are explicitly not talking about this aspect that we do not understand very well because we do not understand it very well.