Proof of conservation of information

Question

After listening of some lectures of Leonard Susskind about black holes, he mentioned that conservation of information is one of the foundations of physics. After searching the web I cannot seem to find how we came up with this theory. Could someone explain how we know this is true and/or how did we come to this conclusion?

Answer 1

Maybe I'm wrong, but it seems to me a trivial consequence of quantum system evolution by means of unitary transforms and, thus, reversibility.

Answer 2

In a quantum context, or more generally in a statistical context, one may say that conservation of information is related to the fact that the sum of probabilities is $1$

For instance, suppose that the interactions of 2 particles $A$ and $A'$ could only produce these same particles $A$ and $A'$, but with different characteristics (momenta, polarizations, etc...), so a interaction $A_1+A'_1 \to A_2+A'_2$

We may consider that the initial state is $|i\rangle = |i_1\rangle |i'_1\rangle$, while the final state could be written : $|f\rangle = \sum\limits_{f_2,f'_2} A (i_1,i'_1, f_2, f'_2) |f_2\rangle |f'_2\rangle$.

Here, $A (i_1,i'_1, f_2, f'_2)$ represents some complex probability amplitude, but which one exactly ?

Conservation of information, means that the initial particles cannot disappear (by hypothesis, we said that the final state is always composed of a 2-particle state, so the final state cannot be "nothing" or zero), the laws of probability tell us that the sum of the probabilities is equal to $1$, that is :

$\sum\limits_{f_2,f'_2} |A (i_1,i'_1, f_2, f'_2)|^2=1$

So $A (i_1,i'_1, f_2, f'_2)$ really represents the probability amplitude to find the final system in the state $|f_2\rangle |f'_2\rangle$

If the sum of the probabilities were not equal to $1$, you will not be able to predict anything, physics will not be predictive, and so would not be a science.