Information Symmetry
Full Title or Meme
Conservation of information seems to be a deep physical principle. Per Noether's theorm, there must be an underlying symmetry, in some space, which may explain this conservation of information
Context
Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory.
Entropy. It's not a symmetry, but there's the second law of thermodynamics. I am not talking about entropy, which is the unknown information about some system, for a particular observer. I talk about information. –
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 don't conserve entropy and information when not in equilibrium ? –
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.
Problems
Information is not an observed quantity like position, momentum or angular momentum, which are conserved.
It isn't that 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 new information is added. Therefore, if we want strict conservation, we should forbid both merging and branching, and in this case determinism is required.
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.
Solutions
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=12mx˙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=∫dt∫dx[iℏ2[ψ˙ψ∗−ψψ˙∗]+ψ∗H^ψ], where H^
is the Hamiltonian of the system. It is not hard to check that the Euler Lagrange equations coming from this action reproduce the Schrödinger equation.
Notice that under the transformation
ψ→e−iαψ ψ∗→eiαψ∗
for constant α
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, ∫dx ψ(x)ψ∗(x)=const.
for all time. In particular, for a normalized wave function,
∫dx ψ(x)ψ∗(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.
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=I . I did not say "time symmetry" I said "time-translation symmetries"
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 I
depends of the kind of information that you are considering: Shannon, Rényi, Fisher.
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.
Unitarity is the quantum version of conservation of information. You have to show how this could be a symmetry, in which space, etc..
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.
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.
Information is conserved in a reversible, deterministic process because the original state of the system can be retrieved by reversing the flow of time (or, equivalently, by reversing the momenta of all the components of the system). This information (and, indeed, the whole history of the system) is stored in the state of the system at any point in time - indeed, it is the state of the system. There is no separate "storage" required. Note, however, that just because information is conserved does not mean it is easily accessible, as it may be distributed across the state of the system. There may be no quicker or simpler way of determining the original state of the system other than running the whole system backwards. In quantum mechanics the evolution of a quantum system through time in accordance with the Schrödinger equation is a reversible process and so conserves information. Whether wave function collapse leads to an actual loss of information or only an apparent loss depends on which interpretation of quantum mechanics you follow.
as long as the quantum state evolves according to the Schrodinger equation, information is conserved. If we adopt an interpretation of quantum mechanics in which collapse happens upon measurement (the Copenhagen interpretation), then even in the simplest case we can see that information would be lost upon collapse. For example, suppose your system is in a superposition of spin up and spin down states. If you measure it to be spin up, there is no way for you to find out whether it was in a pure spin up state, or in a superposition. Hence, information is lost. Clarification: in the above scenario, it's even "worse" than just you not being able to find out the initial state. The state of the whole universe (you, the system, the measuring device, etc.) will be the same whether or not the initial state was a pure spin up state or a superposition
The "conservation of information" follows from the unitarity property of quantum mechanics.
Whether it is actually conserved is a long and dramatic history with a rather a twisted plot. Steven Hawking with many other theorist accepted the possibility of irreversibility of certain physical laws and loss of information - " if irreversibility flouted the laws of physics as they were then understood, so much the worse for those laws". Another group of physicists, led by Don Page are sure, that the unitarity principle has to be true and information is necesarilly preserved. For the recent results and discussion I recommend to read this article https://www.quantamagazine.org/the-black-hole-information-paradox-comes-to-an-end-20201029/. If we believe, that QM evolution is unitary, that the time reversal holds, and one can in principle, although not always techically backtrace the history of a system under consideration. About the measurement and the wavefunction collapse, the terminology is rather abuse, and may lead one to conclusion that something is broken down, but in fact, the measurement replaces the intial apriori probability distribution, by the conditional distribution, aposteriori. Here you can find useful the answer of Lubos Motl https://physics.stackexchange.com/a/3163/261877 and the discussion below.
An alternate way to approach this is to use an interpretation which does not require collapse nor non-determinism. All of the interpretations are simply ways to reconcile the mathematics of a quantum reality with the mathematics of a classical reality as we observe it. There is no wave function collapse in quantum mechanics proper -- it is something which appears in the most common interpretation, the Copenhagen interpretation. We could use other interpretations to explore this answer. Pilot wave comes to mind as an excellent example. In the pilot wave interpretation, we can measure the state of particles that are constantly being affected by a "pilot wave," a wave function which jostles the particles, changing their state. Like all interpretations of QM, this view is perfectly consistent with the fundamental equations of QM. However, instead of a wave function collapse, like the Copenhagen Interpretation has, we have a pilot wave. The tricky bit about this pilot wave is it's equation at every moment in time is dependent on the state of all particles, at that moment, even those which are remote. This weirdness is how pilot wave gets around classical behaviors -- it has a wave that propagates infinitely fast. It can be shown that this yields the same statistical results that we get from the Copenhagen interpretation, with its wave function collapse, but no collapse is required. In this, we find it trivial to show that information is conserved for all actions, even "measurements," because the pilot wave gets defined with respect to the unital operators we see in quantum mechanics. However, that information has been dispersed across every particle in the known universe. So it shows that, by that interpretation, information is conserved across the entire universe, but any sub-system within the universe will lose information as it is scattered to all of the particles in existence.
References
- See the wiki page AI for Quantum Physics
