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?

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

score 3 · Answer 1 · answered Dec 11 '13 at 14:20

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

answered Dec 11 '13 at 14:20

Mike Dunlavey

17,055

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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
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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

score 0 · Answer 2 · answered Dec 11 '13 at 19:45

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.

Proof of conservation of information

2 Answers2