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"Information is physical" carries zero information

Scott Aaronson wrote a long and tedious essay,

Is “information is physical” contentful?
which has ignited a long and tedious discussion. I remember that some 30 years ago, I was somewhat excited about the "idealism" that hides in statements such as "physics is all about the information". When you do physics – or any proper science, for that matter – you don't approach the world "practically". You don't need to "feel" the meat, scents, materials. Instead, you represent them by some abstract concepts, by information, and by real or complex numbers – that's the most typical attitude of the physicists who replace "intuitive objects" with coordinates, observables, positions, momenta, wave functions.

It doesn't really matter that "pork is real matter" and you remember its taste and consider it primary. Instead, in physics, pork is composed of some concepts and quantities that may be written on the paper and communicated as information. There's some sense in which "the ideas are primary" is preferred in physics over "the matter is primary" – sense in which idealism beats materialism in physics. And this point becomes particularly strong in quantum mechanics because quantum mechanics only calculates the probabilities that propositions about observations are correct – it doesn't allow you to assume that some "matter with objective properties" exists independently at every moment. So idealism wins.

Great. I think that already the two paragraphs above are too talkative and repetitive. These are really trivial matters and those 30 years ago, I decided that they're trivial indeed. So the excitement only lasted for hours or days. If you don't understand that physics ultimately deals with "ideas" or "information" or "numbers", not with "matter" (in the manual sense in which toolers and lathe operators deal with "matter"), you can't do physics. On the other hand, everyone who does physics in any meaningful way must understand, at least to some practical, intuitive extent, everything that is correct and that may be hiding behind the sentence "information is physical".

OK, so let's evaluate the proposition
Information is physical.
Is it correct? Is it deep? Is it important? Before we answer these three questions, we should know what the sentence actually means. There are at least three uncertainties that complicate the interpretation of the sentence "information is physical": What is "information"? What is "is"? In particular, Bill Clinton has asked: What is "is"? Can "is" be oral? And what is "physical"? Did Max Born's granddaughter really want to become physical or did she misinterpret the adjective? :-)

Again, you should better give a clear enough answer to these three questions before you try to decide whether "information is physical" is a correct, deep, or important sentence. And let me say that all these three "partial" questions are actually deeper, more meaningful, and more important than the question you started with, whether "information is physical".

Information is such an elementary concept that it cannot really be reduced to any simpler concepts. Various questions or sentences or events may have several or many or infinitely many options a priori. One doesn't know which one is true, which one actually happened, a priori. So he doesn't know some information. Once a particular choice, option, or answer is chosen, the information becomes known. My "definition" of information is circular and this circularity is unavoidable, at least to some extent. You simply can't build definitions of elementary concepts from nothing.

Instead, as in mathematical axiomatic systems, you need to postulate the existence of some basic objects or relations and you need to postulate some axioms that these objects obey. You may say lots of deep and important things about the information but "information is physical" isn't really one of them.

Information may be expressed in nats, natural digits. When \(N\) options are equally likely, \(p_i=1/N\), the information about which option is right corresponds to \(\ln N\) nats (dimensionless units) of information. Some people like to express information in bits. One bit is \(\ln 2\) nats. So to distinguish \(N\) options, you need \(\ln N / \ln 2 = \log_2 N \) bits of information. Other bases are possible as well. The base \(e\approx 2.718\) is the natural one. The previous sentence itself is arguably more important a lesson for a physicist than the vague philosophical "information is physical".

When the \(N\) options aren't equally likely but they have the probabilities \(p_i\) obeying \(\sum_i p_i = 1\), the information is carried by the Shannon entropy\[

S = -\sum p_i \ln p_i.

\] This quantity measures how much time you need to reserve when you want to transmit the information through a channel. We just saw that the information is related to the entropy. And entropy is a concept that we know from thermodynamics – the science about "heat" that may be measured, felt, and studied even if you can't see the atoms or other elementary building blocks. Because of the usual relationships between energy, temperature, and entropy, the change of the information or entropy may be manifested as heat. If a system is capable of carrying one extra nat of information, its heat capacity increases by \(k\) or so, the Boltzmann constant, some \(1.38 \times 10^{-23}\) joules per kelvin. Whenever you measure how hard is to heat an object by work, you measure how much information the object may carry in its "atomic storage of information".

Those insights are important and every physicist should learn at least basics of thermodynamics and statistical physics. In some sense, these more specific statements can be interpreted as "refinements" of the proposition that "information is physical". But I think it's right to say that the sentence
Information is physical.
is just too vague, too humanities-oriented, and a person who likes to say such propositions shouldn't be assumed to understand thermodynamics or statistical physics. The vague sentence may "encode" some wisdom about thermodynamics and statistical physics but unless a person rediscovers or learns thermodynamics and statistical physics, he doesn't know the "code" so from his perspective, the sentence doesn't really teach him anything about thermodynamics and statistical physics.

The vague sentence above is very analogous to sentences such as
A virtual particle is physical.
A virtual particle is real.
The path integral is real.
Climate change is real.
The electromagnetic potential is physical.
Gluons exist.
and millions of others. The point is that many people like to pick a concept – examples are above – and attach the words "is real" or "is physical" or "exists" or similar, very general verbs and adjectives. And they pretend that the result is some deep wisdom or at least a deep question.

I don't think that such questions are deep. They carry a very little information because they're just combinations of some basic words. And they're rather random combinations. Random combinations don't necessarily lead to very meaningful sentences – "the number five is green" is a popular example. And to make things much worse, the verbs and adjectives that are used to build these sentences are very low-information words. In particular, "is" carries almost zero information. That's why e.g. the Russian language often omits it altogether. "Moi brat – vrag" is their "My brother is an enemy." As you can see, Slavs agree that the articles "the" and "an" carry zero information, too. They're fired. Words "exist", "real", and "physical" don't carry too much information, either.

Some of these sentences may convey something rather meaningful. I deliberately mentioned the example
The electromagnetic potential is physical.
Particle physicists would usually say that this sentence is false because the electromagnetic potential \(A_\mu\) isn't gauge-invariant. Therefore, we can't uniquely associate the values of the fields \(A_\mu(x,y,z,t)\) with a real-world situation. We can't do that because the gauge-transformed values of the potential\[

A'_\mu(x,y,z,t) = A_\mu (x,y,z,t) + \partial_\mu \lambda (x,y,z,t)

\] are equally good to describe the same physical situation. You may see that I clarified the adjective "physical". What I really meant by "is physical" was "is gauge-invariant" and the potential wasn't gauge invariant.

The gauge invariance – and similarly, independence on the inertial frame, coordinate system, choice of observer, whether observations are made etc. – are "subtleties" that make it harder to decide about sentences such as "something is physical". Before you learn physics up to a sufficient depth, you're not even aware that such subtleties could exist. The more you learn, the more subtleties you know, and the more you know that it was naive to worship sentences such as "information is physical".

It matters whether you actually want your sentences to be well-defined and talk about some specific thing; or whether you want them to be vague, pretending to be deep, and be useless. If you want the former, you are more likely to pick a more well-defined language and adjectives such as "gauge-invariant" instead of "physical". Well, "physical" may replace the more well-defined adjectives and it can be clear from the context but the problem is that some people deliberately want the meaning of the word "physical" to remain ambiguous, and it's a problem, not a virtue.

I also mentioned that the sentences of the form "something is physical" are analogous to the sentences "something is real". To a large extent, these two sentences are actually completely equivalent. And after all, the sentence "something exists" may be considered yet another synonym. The words "real" and "exist" are as ill-defined in general as the word "physical". Philosophers may earn their salary for postulating new dogmas about the validity of sentences such as "something is real" but I think that physicists agree that these sentences and the research of such sentences has basically no value.

"A virtual particle is real" or "a virtual particle is physical" are true to the extent that the concept of a virtual particle (and thousands of others that could be used as examples) are exploited in correct, legitimate physics research and physical analyses of some phenomena. These concepts play some role and they affect the predictions – and the outcomes of the actual experiments. Because the physical analyses using "virtual particles", "path integrals", "gluons", and lots of other concepts aren't wrong, we may say that "virtual particles are physical" in this basic sense. And so is the information.

On the other hand, this is a very general, tolerant interpretation of the phrase "something is physical". If we choose to interpret "something is physical" in a narrower, more well-defined way, we may decide that the sentence "a virtual particle is physical" is false. After all, the adjective "virtual" may be said to be the antonym of "real". We divide particles in Feynman diagrams to real and virtual – and with this interpretation of the words "real" and "virtual", the sentence "a virtual particle is not real" is tautologically true!

The case of the "information" is fully analogous at some level. The information may be said to be unphysical because it depends on the knowledge of the observer, sometimes on the reference frame, on the choice of ensembles in statistical physics, and other things. In general, the phrase "something is physical" is too vague and unrefined.

I also think that an extra discussion about a pair of propositions, "information is physical" and "physics is informational", only adds another layer of worthless babbling. Both sentences clearly try to say something about the role of the information in the real world. The precise content of that statement isn't really clear and well-defined, as discussed above. But the difference between these two ill-defined statements is even smaller than the well-definedness of each of them separately! "Information is physics" is mostly foggy and the expansion of the discussion by another proposition "physics is informational" makes the fog even thicker. The same is true about many other "clever ideas" that people were adding in Aaronson's tedious thread.

To summarize, I think that it's just wrong to get carried away with vague metaphysical sentences such as "information is physical" and build a whole religion on the worshiping of the alleged depth of such statements. I believe that a person who is learning to think as a physicist must understand rather early on that the actual deep physics is composed of much more well-defined and specific statements than "information is physical" and that this vague statement leads to lots of subtleties and nuances – in some sense, all of our important knowledge of physics is made of such subtleties and nuances. So the people who try to impress the laymen with "information is physical" are ultimately contributing to the laymen's misunderstanding of the difference between science and philosophy, science and religion, physics and an empty humanities talk where the truthfulness is derived from constant repetition and authorities.

The laymen should be honestly told that legitimate physicists generally consider the talk about propositions such as "information is physical" to be a waste of time and everyone who actually starts to think as a physicist – including you – should consider it a waste of time, too. I actually think that this lesson is rather easy for the laymen to learn.

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