# 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.

• Entropy. It's not a symmetry, but there's the second law of thermodynamics.
– user14407
Oct 26, 2012 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. Oct 26, 2012 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 ? Oct 26, 2012 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. Oct 26, 2012 at 15:13
• Related: physics.stackexchange.com/q/2685 Aug 22, 2013 at 20:18

1) If you want a Noether theorem for information, there is no such thing.

Trying to obtain it from a symmetry law, by Noether's theorem can't work, simply because information is not a quantity that can be obtained for instance by the derivative of the Lagrangian with respect to some variable. Information is not scalar, vector, tensor, spinor etc.

2) Another way to obtain conservation laws can be found in quantum mechanics. The observables that commute with the Hamiltonian are conserved. Again, you don't have an observable, in the sense of quantum mechanic, for information.

Trying to obtain conservation of information from commutation with Hamiltonian can't work, because there is no observable (hermitian operator on the Hilbert space) associated to information. Information is not the eigenvalue of such an operator.

3) The only way, which also is the simplest and the most direct, is the following: to have information conservation, when you reverse the evolution laws, you have to obtain evolution laws that are deterministic. This ensures conservation of information, in fact, they are equivalent. In particular, most classical laws are deterministic and reversible. Also, in quantum mechanics, unitary evolution is reversible, giving you the conservation of information.

I don't say that the evolution laws have to be deterministic, or that they have to be invariant to time reversal. Just that, when you apply time reversal, the evolution equations you obtain (which are allowed to be different than the original ones) are deterministic. Simplest way to think about this is by using dynamical systems. Trajectories in phase space are not allowed to merge, because if they merge, the information about what trajectory was before merging is lost. They are allowed to branch, because you can still go back and see what any previous state was. Branching breaks determinism, but not preservation of information. Old information is preserved at branching, but, as WetSavannaAnimal mentioned, new information is added. Therefore, if we want strict conservation, we should forbid both merging and branching, and in this case determinism is required.

• +1 - quick question - wouldn't branching correspond to information increase? one needs to specify which branch a particular trajectory took at the point of branching. Aug 23, 2013 at 5:48
• This might be just the kind of explanation I'm looking for. But I'll see what else comes in. Aug 23, 2013 at 5:48
• @WetSavannaAnimal aka Rod Vance: The old information is preserved, but, as you mention, each branching adds new information. Good point, I will update. Aug 23, 2013 at 6:20
• +1 for the interesting answer. Some remarks. 1) Information of a given system is certainly an invariant, so it has to be a Lorentz scalar, I think 2) Yes, because we define observables as hermitian operators which are not depending on the density matrix, (otherwise we could define entropy, for instance, as an observable) 3)a) An unitary operator is inversible, yes, but the important thing is that the norm of the state is conserved, in the limit, we could imagine a possible non-invertible operator with this property. Aug 23, 2013 at 7:52
• @Trimok: What you mean by information being a scalar? Is the scalar $a\in\mathbb R$? In this case, conservation would imply that $a$ is constant. The phase space will be $\mathbb R$, and the universe will be static: his trajectory will reduce to $a\in\mathbb R$. Say information is a scalar field. Then, everything in the universe should be derivable from that scalar field. But the universe has other fields too, hence more degrees of freedom. Say the universe is discrete. In this case, we could encode all the information in a binary string, hence in a real scalar. Aug 23, 2013 at 9:05

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.

• 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. Oct 26, 2012 at 10:25
• Well, it doesn't have to be strictly CPT, any time reversal symmetry works. So Lorentzian relativity is not really required. Oct 26, 2012 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. Oct 26, 2012 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. Oct 26, 2012 at 11:25
• Well, TMS, have you some reference on this supposed CPT violations ? Oct 27, 2012 at 9:41

Conservation of information can be derived from Liouville theorem, which can be interpreted in terms of time-translation symmetries.

• 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 Oct 27, 2012 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" Oct 27, 2012 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? Oct 27, 2012 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... Oct 28, 2012 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. Oct 29, 2012 at 13:45

I realize I'm a bit late to this thread, but incase anyone stumbles upon this question, here's the answer:

The answer is that there is a symmetry associated with information conservation, but it doesn't come from the usual Lagrangian. Ordinarily for quantum or classical systems, we have a Lagrangian of the form $L=\frac{1}{2}m \dot x^2+V(x)$ and conserved quantities have to do with symmetries of this Lagrangian. For conservation of information, things are a little different. Instead of coming up with a Lagrangian that describes the motion of a particle, instead, we must treat the quantum wave function as a (classical) field. In this case the Action would look like $$S=\int dt\int dx \bigg[\frac{i\hbar}{2}[\dot \psi\psi^*-\psi\dot\psi^*]+\psi^*\hat H\psi \bigg],$$ where $\hat H$ is the Hamiltonian of the system. It is not hard to check that the Euler Lagrange equations coming from this action reproduce the Schrodinger equation.

Notice that under the transformation $$\psi\to e^{-i\alpha}\psi$$ $$\psi^*\to e^{i\alpha}\psi^*$$ for constant $\alpha$ leaves $S$ invariant, meaning that there is an associated conserved current. It turns out that this current is the probability current. (see https://en.wikipedia.org/wiki/Probability_current). As a result, $$\int dx~ \psi(x)\psi^*(x)= const.$$ for all time. In particular, for a normalized wave function, $$\int dx~ \psi(x)\psi^*(x)= 1,$$ meaning that the probability of $something$ happening is always 100%. So information (aka probability) is the conserved quantity corresponding the the fact that we can multiply wave functions by an overall complex phase without changing the physics.

In my answer, I focused on the case of ordinary non-relativistic quantum mechancis for the sake of clarity. But the above line of reasoning works (though in much more complicated ways) for any Unitary quantum theory eg QFT.

• this was very helpful, Mike L. I cannot grasp the derivation but your description of the process was exactly what I was after. thanks for posting it. Dec 19, 2017 at 5:14

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

• Unitarity is the quantum version of conservation of information. You have to show how this could be a symmetry, in which space, etc... Oct 27, 2012 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.

• 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. Oct 29, 2012 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, 2013 at 8:11
• Does the conservation of information not stem directly from the fact there exists equations of motion for a system? Not really. Classically, conservation of information is expressed by Liouville's theorem, and we have nonholonomic systems for which there are equations of motion, but Liouville's theorem fails.
– user4552
Aug 23, 2013 at 4:12

In quantum physics, information is not usually taken to be an observable. It does not make sense to ask that it be conserved, if we take conservation to have its usual mathematical meaning.

If you want to insist that information be an observable, you can imagine that it is the dimension of the Hilbert space, or alternately the identity operator. Conservation of information is then a poetic way of saying that time evolution does not transform the identity into a projection.

If you are willing to grant that information is the identity observable, then it is clear what symmetry group it generates: it is the trivial group which acts identically on all states.

• Or maybe one could just say "unitarity" ? :-)
– Siva
Sep 24, 2013 at 5:16

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")

• No, information is conserved, but entropy (the unknown information about some sytem, for some observer) always increase Oct 31, 2012 at 17:29

Information understood as number of configurations being conserved means, in fact, that probability is conserved, as the answer above told you. A more intuitive way to say that is that QM evolution conserves the number of stacks, even when the number of particles (or the particle+antiparticles) is NOT conserved in QFT, the probability to find some number of particles in certain microstates or stacks must be conserved for a given energy or for a given configuration. Of course, the real classical world is a bit different since we have dissipation, something that we can include in QM not without some concerns and care.

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.