# Is there a symmetry associated to the conservation of information?

Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory.

We may wonder if there is an underlying symmetry, in some space, which may explain this conservation of information.

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 Entropy. It's not a symmetry, but there's the second law of thermodynamics. – user14407 Oct 26 '12 at 11:23 I am not talking about entropy, which is the unknown information about some system, for a particular observer. I talk about information. – Trimok Oct 26 '12 at 11:25 @Trimok Are the known information and the loss thereof (entropy) about a system not related by something like "as the entropy increases the information decreases" ...? Could it be that you are right when one talks about a fine grained microscopic description of the system which is reversible and therefore both, information and entropy are conserved (such that it is very interesting to ask for a symmetry corresponding to the conservation of information +1), and Lunge is right when talking about course grained systems that dont conserve entropy and information when not in equilibrium ? – Dilaton Oct 26 '12 at 12:32 Well, I am maybe wrong, but I think that information is always conserved, but entropy always increases. And I think also, that this applies to microscopic systems as well as to macroscopic systems. But I concede that all these questions are very subtle, because you have to decide what is subjective, what is objective, what is the role of the observer, and so on. – Trimok Oct 26 '12 at 15:13

CPT seems to imply it. You can reverse the system evolution by applying charge, parity and time conjugation, so the information about the past must be contained in the present state. That implies conservation of information by the evolution.

This may not be the answer you wanted, because it does not imply unitarity, but it is the only relationship between symmetry and information conservation that I can think of. Unitarity seems to be a very fundamental assumption though, and there is not much more fundamental mathematical structure you could use to argue about its necessity.

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You need to assume Lorentz invariance for CPT, which is not a problem for me, because Nature is relativist.. I need to think to your answer which seems interesting. – Trimok Oct 26 '12 at 10:25
Well, it doesn't have to be strictly CPT, any time reversal symmetry works. So Lorentzian relativity is not really required. – A.O.Tell Oct 26 '12 at 10:49
I think there could be a potential problem, because, in some interactions, you have a CP violation, and so a T violation. But information, I hope, is still conserved. – Trimok Oct 26 '12 at 11:16
CPT is always a symmetry. A CP violation does not invalidate the argument, you can still construct a time-reversed solution with charge and parity conjugation. – A.O.Tell Oct 26 '12 at 11:25
Well, TMS, have you some reference on this supposed CPT violations ? – Trimok Oct 27 '12 at 9:41
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Conservation of information can be derived from Liouville theorem, which can be interpreted in terms of time-translation symmetries.

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Well, Liouville theorem is about the conservation of phase space during time evolution. It is the classical version of the conservation of information. But you cannot consider, I think, this, as a time symmetry, it is only something which is constant in time, and this is very different. For instance, angular momentum is constant in time, but this not due to a time symmetry – Trimok Oct 27 '12 at 9:37
There is a quantum version of the Liouville theorem. It ensures conservation of quantum information. In general symmetries are of the kind $\{G,P\}=0$ where $G$ is the generator of the translation, $P$ the conserved property and the braces denote the quantum or classical brackets. The generator of time translations is the Hamiltonian, therefore any property conserved in time satisfies $\{H,P\}=0$. This is a consequence of the Liouville theorem. This is also satisfied for information $P = \mathcal{I}$. I did not say "time symmetry" I said "time-translation symmetries" – juanrga Oct 27 '12 at 15:34
I meant time-translation symmetries too. Sorry for having been not precise. But what is explicitely your operator $I$ ? Do you think to the density matrix/operator or is it another thing? – Trimok Oct 27 '12 at 18:26
$\mathcal{I}$ is information. It does not need to be an operator. It is a phase space function in the Wigner-Moyal formulation of QM, for instance. The explicit form for $\mathcal{I}$ depends of the kind of information that you are considering: Shannon, Rényi, Fisher... – juanrga Oct 28 '12 at 11:51
Well, if there is a symmetry, and if we take the quantum point of view, there should exist a infinitesimal operator $I$, such as $[H, I] = 0$. But I don't think that such an operator exists. And there should exist only one version of this operator, and not several versions. – Trimok Oct 29 '12 at 13:45

Unitarity is the symmetry you seek. Whats wrong with that?

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 Unitarity is the quantum version of conservation of information. You have to show how this could be a symmetry, in which space, etc... – Trimok Oct 27 '12 at 9:40

Does the conservation of information not stem directly from the fact there exists equations of motion for a system? So the fact we can actually form a Lagrangian for a system implies information conservation? At least in a classical perspective. Unitary evolution would be the quantum mechanical version. Sorry if that is a naive suggestion.

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 Well, I think you should precise your idea, and, mainly, you should give the explicit description of the symmetry, which, if it exists, implies information conservation. – Trimok Oct 29 '12 at 13:39 No, it doesn't. An equation of motion for a system a priori doesn't even have to be reversible, in which case it certainly doesn't preserve anything that deserves to be called information. – WIMP Apr 11 at 8:11

Entropy is used to quantify information and since disorder increases information decreases. I think you mean some specific information is conserved, not the general amount of information in the universe. In the same way, one can show that entropy doesn't always increase - only when you look at the entire universe ("generalized entropy")

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No, information is conserved, but entropy (the unknown information about some sytem, for some observer) always increase – Trimok Oct 31 '12 at 17:29

In classical statistical mechanics, information conservation, comes in the form of Liouville's theorem, simply says that the object will not disappear or created (Or, the phase space density is constant along its trajectory). This do not correspond to any symmetry.

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try improving the answer – 007 Apr 5 at 14:54

Yes, it is a deep principle, beyond physics orthodoxy, I'm afraid -- relating, as it does, to consciousness itself, and how it prevents the heat death or the entropy of the universe. But to understand this, you have to know that the scale of the universe means nothing in relation to consciousness.

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 Religion hates science. So please choose between the two. – Trimok Oct 31 '12 at 17:30 Think about this, when you observe a bunch of disorder, you have now put order on it with your observation. Like a snapshot of a camera arranges order onto film. – Mark J Oct 31 '12 at 19:32 The entropy of the observed sytem decreases after measurement (observation), OK, but the entropy of the set (observer + environnement) also increases during this observation, in such a way that the net result of the two terms is positive. – Trimok Nov 2 '12 at 9:00 Well, two things: the recording of an observation on silver gelatin, for example, could use less energy than the amount of disorder (out in front of the lens) that it is converting into order (in the re-ordered granules of silver). Secondly, no physicist really understands consciousness and its relationship to life, so you can't say for sure that there is an increase of entropy in the recording of memory -- there is no equation for this. – Mark J Nov 2 '12 at 18:27 I don't know precisely the details of your particular silver example, however a classical example is Maxwell's Demon (see Wikipedia), and it seems that the Demon, because he has only a limited memory, cannot store infinite information (about the molecules he decide to choose), and so have to discard, at some point, some information, and this provide dissipation of heat, and increase of entropy (it is irreversible). So it seems that there are equations for this (The Wikipedia article is interesting). – Trimok Nov 5 '12 at 10:28