# Proof of conservation of information [duplicate]

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?

• It's not provable -- it's an axiom, aka "law" . Like the "parallel postulate" in geometry, it's something which is consistent w/ all we know and thus rather useful. – Carl Witthoft Dec 11 '13 at 14:20
• What is the precise statement of that axiom? – Valter Moretti Dec 11 '13 at 17:35
• possible duplicate of Why is information indestructable? – user10851 Dec 11 '13 at 18:50

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

• A trivial consequence of unitary evolution? It's not at all obvious to me how that's the case. If it's so trivial why not give a definition of "information" in the context of quantum mechanics and a proof that information is conserved under unitary evolution? – joshphysics Dec 11 '13 at 18:15
• @joshphysics: If a system has unitary evolution, then it's reversible, right? And if it's reversible then it can always get back to a prior state if the evolution is reversed. Any definition of information I know of (which may not be much) says that the only way you can get back to a prior state is if you don't forget what it was. That's putting it colloquially, I know, but when I worked on quantum algorithms, that was basic understanding. – Mike Dunlavey Dec 11 '13 at 18:37
• If we're satisfied with the intuition that reversibility implies no information loss for any reasonable definition of information, then that's certainly fair. Actually, when you put it in those terms, I am personally satisfied, but I'd still like to see a more precise definition of information. – joshphysics Dec 11 '13 at 18:45
• @joshphysics: being a CS guy myself, I'm most familiar with Shannon information, and to a lesser degree Kolmogorov information. I know there are others. – Mike Dunlavey Dec 11 '13 at 19:00

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.