Information never gets lost theory - formalization From a video I say about information theory, the sentence "Information never gets lost, it is there but in different 'arrangement'"
Meaning that if we burn a toilet paper into ash, we could theoretically recreate that toilet paper if we had enough computing powers and an algorithm that would do so.
What I am trying to say is that even if it is may seem super hard - it is theoretically possible.

*

*My question is simple, what do we have about this theory? Is it even true? because there are counter-theories that information does get lost.

Also, how do we give "information does not get lost" a mathematical figure?
I would assume it is something along the lines of
$$ \Delta I = 0$$
But I've never seen this kind of equation before nor when I searched the internet.
It sounds reasonable for me, however I am not a scientist, so I would appreciate your help! Thanks in advance.
 A: The conservation of information is based on the postulates of quantum mechanics as they currently stand - that is, assuming no 'objective collapse' mechanism. It can be thought of as a consequence of three theorems in quantum information theory: the no-cloning, no-deletion and no-hiding theorems. However, it can be more simply encapsulated in terms of the von Neumann entropy (entropy is a measure of information)
\begin{equation}
S = - Tr[\rho\log\rho]
\end{equation}
where $Tr$ is the trace operator and $\rho$ is the density matrix of the system. For a pure state - i.e. a state described entirely by quantum mechanics - it can be shown that the entropy is zero. Moreover, no unitary evolution of the density matrix can change this fact. Hence
\begin{equation}
\frac{dS}{dt} = 0.
\end{equation}
Of course, this relies on the assumption that there is no 'objective collapse' of a quantum system. If it turns out that quantum mechanics needs to be modified by including a collapse mechanism, then the conservation of information will no longer hold.
The transition to classical physics is modelled by taking the reduced density matrix of a system. This is obtained by tracing out the environmental degrees of freedom of a system. The von Neumann entropy is then, in general, no longer zero.
However, the finite entropy of the reduced density matrix is a consequence of the fact that a system has become entangled with its environment. This may be viewed as a transfer of information between system and environment.
To know anything about the world, we need to have information about it. This involves entanglement and a consequent increase in entropy of the reduced density matrix. Whilst reversing this increase at the Universal level may be mathematically possible, it is not likely to be something that we could ever observe, since this would require a transfer of information.
