On the conservation of information

Is there a physics law that dictates information cannot be destroyed? No. But some physicists invoke the 2nd law of thermodynamics and the unitarity principle of Quantum Mechanics to support the claim that information is conserved.

It is easy to destroy information. I can delete this blogpost. There may be copies of this article in various places on the internet but those copies may disappear someday. When that happens no physics law is violated.

What information loss means

In this context, the term “loss” or “destruction” means “irrecoverable”. If information is scrambled and there is no algorithm or key to unscramble (decode, restore) then information is lost because it is irrecoverable.

Conservation laws of physics

There are conservation laws in physics (especially in the realm of elementary particles). These conservation laws can perhaps be expressed in the language of information. But the conservation of information as a universal law is not yet established. One can, of course, suggest that there may be laws that we do not yet know about. And those unknown laws may guarantee the conservation of information. Or one can resort to multiverses and claim that information is conserved in the totality of multiverses.

Information loss in Black Holes

In discussions of the black hole information paradox, the unitarity principle of Quantum Mechanics is invoked in support of the view that information is conserved. Unitarity refers to the conservation of probability. I wrote about this here. Can we equate conservation of probability with conservation of information? No. Besides, unitary (probability preserving) evolution of the quantum mechanical wavefunction is disturbed every time it interacts with the environment.

If the unitarity is broken even in a simple measurement (interaction with the environment) how can we speak of the unitary evolution of the wavefunction in the violent environment of a black hole? Let’s assume for a moment that the unitarity is not broken and the probabilities are conserved, I would still object to equating “preservation of probability” with “conservation of information”.

The other argument (in support of the conservation of information) is based on the 2nd law of thermodynamics. Argument goes like this. A black hole has huge entropy proportional to the area of its event horizon sphere. Black hole evaporates in billions of years due to Hawking radiation. Entropy of an isolated system never decreases (2nd law of thermodynamics). The isolated system here is black hole + Hawking radiation. As the black hole evaporates its entropy decreases because the evaporation shrinks the black hole. Smaller surface area of the event horizon sphere means smaller entropy. Hawking radiation is a black-body radiation. Does the increasing entropy of the Hawking radiation compensate the decreasing entropy of the black hole to satisfy the condition that the total entropy of the (black hole + Hawking radiation) system never decreases? Physicists do not have a satisfactory answer for this. But the majority of physicists think that the combined entropy cannot decrease. That’s a fine argument but where is “information” in this argument?

Well…physicists conflate entropy with information. Black hole entropy which is based on Shannon entropy should be interpreted as “information capacity” rather than “information” per se. If the total entropy of (black hole + Hawking radiation) cannot decrease then the information capacity cannot decrease either, this is their argument. My point: why confuse people by using the term “information”. Just argue using the entropy concept.

The book example is misleading and confusing

Most science journalists use the book example when they discuss the black hole information paradox. They say that when we burn a book the information contained in the book is recoverable in principle because burning process involves classical physics. They contrast this to a book falling into a black hole. When the black hole evaporates due to Hawking radiation which is a quantum mechanical process the book information is truly lost because the entropy of the Hawking radiation is much smaller than the original entropy of the black hole. This differentiation is unnecessary. Let’s be honest, information is lost in both cases. We cannot recover the information of that book in either case unless there are copies of the same book elsewhere.

Copies of the same book elsewhere! A clue for the solution of the black hole information paradox, perhaps. Yes, this is exactly the approach physicists are taking. There are efforts to show that the information of the book falling into a black hole is painted (encoded) on the surface of the black hole event horizon somehow. The key word here is “somehow”. So far, there is no satisfactory explanation of a mechanism for this kind of encoding.

Quantification of information is problematic

Defining the information in terms of the negative log of the probability (Shannon 1948; Shannon & Weaver 1949, Rényi 1961) is great progress but we should never forget that information is interpreted meaning. We assign a meaning to perception and call it information. It will always be difficult to demonstrate a universal law about conservation of information.

Information creation

Now, let’s examine the opposite of information loss, the creation of information which is evident. As the universe evolves, more complex structures (atoms, molecules, organic life) appear; that means more information. Any structure, including elementary particles carry information. Any wave formation carries information (frequency spectrum, amplitude variations). A quantum field has many characteristics; that’s information. The information creation accelerates after the emergence of agency (unit minds). Organized structures with minds create more structures therefore more information. Unit minds also write articles, books, scientific papers, software; make films and videos. Information creation is accelerating.

We don’t gain any insights about information creation from Quantum Mechanics. Why do we expect Quantum Mechanics to say anything meaningful about information loss?

Few comments on quantum information

Quantum Mechanics insists on the conservation of probability. This is known as the unitarity principle of Quantum Mechanics. This is essential for the internal consistency of the theory. This is all good. I don’t question the unitarity principle. I object to the interpretation of the unitarity principle as conservation of information.

Description of quantum states (information) is contained in the “amplitude” (another name for quantum wavefunction). The description of quantum states depends on the experimental setup or the natural environment that initiates the interaction. Quantum information is context dependent. Conservation of a context dependent quantity is problematic.

Last word

One has to consider the possibility that Quantum Mechanics may not be the final theory.

Reading material

Reading material on this subject is very confusing. I listed few below. Be prepared to be more confused. Such is life. Sometimes more information brings more confusion.

Sabine Hossenfelder, The Black Hole information loss problem is unsolved. Because it’s unsolvable.

Charlie Wood, Physicists Rewrite a Quantum Rule That Clashes With Our Universe.

http://qpt.physics.harvard.edu/simons/Polchinski.pdf

https://www.physicsforums.com/threads/is-information-lost-in-wavefunction-collapse.948993/