### Is “information is physical” contentful?

Thursday, July 20th, 2017“Information is physical.”

This slogan seems to have originated around 1991 with Rolf Landauer. It’s ricocheted around quantum information for the entire time I’ve been in the field, incanted in funding agency reports and popular articles and at the beginnings and ends of talks.

But what the hell does it mean?

There are many things it’s *taken* to mean, in my experience, that don’t make a lot of sense when you think about them—or else they’re vacuously true, or purely a matter of perspective, or not faithful readings of the slogan’s words.

For example, some people seem to use the slogan to mean something more like its converse: “physics is informational.” That is, the laws of physics are ultimately not about mass or energy or pressure, but about bits and computations on them. As I’ve often said, my problem with that view is less its audacity than its timidity! It’s like, what would the universe have to do in order *not* to be informational in this sense? “Information” is just a name we give to whatever picks out one element from a set of possibilities, with the “amount” of information given by the log of the set’s cardinality (and with suitable generalizations to infinite sets, nonuniform probability distributions, yadda yadda). So, as long as the laws of physics take the form of telling us that some observations or configurations of the world are possible and others are not, or of giving us probabilities for each configuration, *no duh* they’re about information!

Other people use “information is physical” to pour scorn on the idea that “information” could mean anything without some actual physical instantiation of the abstract 0’s and 1’s, such as voltage differences in a loop of wire. Here I certainly agree with the tautology that in order to *exist physically*—that is, be embodied in the physical world—a piece of information (like a song, video, or computer program) does need to be embodied in the physical world. But my inner Platonist slumps in his armchair when people go on to assert that, for example, it’s meaningless to discuss the first prime number larger than 10^{10^125}, because according to post-1998 cosmology, one couldn’t fit its digits inside the observable universe.

If the cosmologists revise their models next week, will this prime suddenly burst into existence, with all the mathematical properties that one could’ve predicted for it on general grounds—only to fade back into the netherworld if the cosmologists revise their models again? Why would *anyone* want to use language in such a tortured way?

Yes, brains, computers, yellow books, and so on that encode mathematical knowledge comprise only a tiny sliver of the physical world. But it’s equally true that the physical world we observe comprises only a tiny sliver of mathematical possibility-space.

Still other people use “information is physical” simply to express their enthusiasm for the modern merger of physical and information sciences, as exemplified by quantum computing. Far be it from me to temper that enthusiasm: rock on, dudes!

Yet others use “information is physical” to mean that the *rules* governing information processing and transmission in the physical world aren’t knowable *a priori*, but can only be learned from physics. This is clearest in the case of quantum information, which has its own internal logic that generalizes the logic of classical information. But in some sense, we didn’t need quantum mechanics to tell us this! *Of course* the laws of physics have ultimate jurisdiction over whatever occurs in the physical world, information processing included.

My biggest beef, with *all* these unpackings of the “information is physical” slogan, is that none of them really engage with any of the deep truths that we’ve learned about physics. That is, we could’ve had more-or-less the same debates about any of them, even in a hypothetical world where the laws of physics were completely different.

So then what *should* we mean by “information is physical”? In the rest of this post, I’d like to propose an answer to that question.

We get closer to the meat of the slogan if we consider some actual physical phenomena, say in quantum mechanics. The double-slit experiment will do fine.

Recall: you shoot photons, one by one, at a screen with two slits, then examine the probability distribution over where the photons end up on a second screen. You ask: does that distribution contain alternating “light” and “dark” regions, the signature of interference between positive and negative amplitudes? And the answer, predicted by the math and confirmed by experiment, is: *yes, but only if the information about which slit the photon went through failed to get recorded anywhere else in the universe, other than the photon location itself.*

Here a skeptic interjects: but that *has* to be wrong! The criterion for where a physical particle lands on a physical screen can’t possibly depend on anything as airy as whether “information” got “recorded” or not. For what counts as “information,” anyway? As an extreme example: what if God, unbeknownst to us mortals, took divine note of which slit the photon went through? Would *that* destroy the interference pattern? If so, then every time we do the experiment, are we collecting data about the existence or nonexistence of an all-knowing God?

It seems to me that the answer is: *insofar as the mind of God can be modeled as a tensor factor in Hilbert space, yes, we are.* And crucially, if quantum mechanics is universally true, then the mind of God would *have* to be such a tensor factor, in order for its state to play any role in the prediction of observed phenomena.

To say this another way: it’s obvious and unexceptionable that, by observing a physical system, you can often learn something about what information must be in it. For example, you need never have heard of DNA to deduce that chickens must somehow contain information about making more chickens. What’s much more surprising is that, in quantum mechanics, you can often deduce things about what information *can’t* be present, anywhere in the physical world—because if such information existed, even a billion light-years away, it would necessarily have a physical effect that you don’t see.

Another famous example here concerns identical particles. You may have heard the slogan that “if you’ve seen one electron, you’ve seen them all”: that is, apart from position, momentum, and spin, every two electrons have *exactly* the same mass, same charge, same every other property, including even any properties yet to be discovered. Again the skeptic interjects: but that *has* to be wrong. Logically, you could only ever confirm that two electrons were *different*, by observing a difference in their behavior. Even if the electrons had behaved identically for a billion years, you couldn’t rule out the possibility that they were actually different, for example because of tiny nametags (“Hi, I’m Emily the Electron!” “Hi, I’m Ernie!”) that had no effect on any experiment you’d thought to perform, but were visible to God.

You can probably guess where this is going. Quantum mechanics says that, no, you *can* verify that two particles are perfectly identical by doing an experiment where you swap them and see what happens. If the particles are identical in all respects, then you’ll see quantum interference between the swapped and un-swapped states. If they aren’t, you won’t. The *kind* of interference you’ll see is different for fermions (like electrons) than for bosons (like photons), but the basic principle is the same in both cases. Once again, quantum mechanics lets you verify that a specific type of information—in this case, information that distinguishes one particle from another—was *not* present anywhere in the physical world, because if it were, it would’ve destroyed an interference effect that you in fact saw.

This, I think, already provides a meatier sense in which “information is physical” than any of the senses discussed previously.

But we haven’t gotten to the filet mignon yet. The late, great Jacob Bekenstein will forever be associated with the discovery that information, wherever and whenever it occurs in the physical world, *takes up a minimum amount of space*. The most precise form of this statement, called the covariant entropy bound, was worked out in detail by Raphael Bousso. Here I’ll be discussing a looser version of the bound, which holds in “non-pathological” cases, and which states that a bounded physical system can store at most A/(4 ln 2) bits of information, where A is the area in Planck units of any surface that encloses the system—so, about 10^{69} bits per square meter. (Actually it’s 10^{69} *qubits* per square meter, but because of Holevo’s theorem, an upper bound on the number of qubits is also an upper bound on the number of classical bits that can be reliably stored in a system and then retrieved later.)

You might have heard of the famous way Nature enforces this bound. Namely, if you tried to create a hard drive that stored more than 10^{69} bits per square meter of surface area, the hard drive would necessarily collapse to a black hole. And from that point on, the information storage capacity would scale “only” with the area of the black hole’s event horizon—a black hole itself being the densest possible hard drive allowed by physics.

Let’s hear once more from our skeptic. “Nonsense! *Matter* can take up space. *Energy* can take up space. But information? Bah! That’s just a category mistake. For a proof, suppose God took one of your black holes, with a 1-square-meter event horizon, which already had its supposed maximum of ~10^{69} bits of information. And suppose She then created a bunch of new fundamental fields, which didn’t interact with gravity, electromagnetism, or any of the other fields that we know from observation, but which had the effect of encoding 10^{300} new bits in the region of the black hole. Presto! An unlimited amount of additional information, exactly where Bekenstein said it couldn’t exist.”

We’d like to pinpoint what’s wrong with the skeptic’s argument—and do so in a self-contained, non-question-begging way, a way that doesn’t pull any rabbits out of hats, other than the general principles of relativity and quantum mechanics. I was confused myself about how to do this, until a month ago, when Daniel Harlow helped set me straight (any remaining howlers in my exposition are 100% mine, not his).

I believe the logic goes like this:

- Relativity—even just Galilean relativity—demands that, in flat space, the laws of physics must have the same form for all inertial observers (i.e., all observers who move through space at constant speed).
- Anything in the physical world that varies in space—say, a field that encodes different bits of information at different locations—also varies in
*time*, from the perspective of an observer who moves through the field at a constant speed. - Combining 1 and 2, we conclude that
*anything that can vary in space can also vary in time*. Or to say it better, there’s only one kind of varying: varying in spacetime. - More strongly, special relativity tells us that there’s a specific numerical conversion factor between units of space and units of time: namely the speed of light, c. Loosely speaking, this means that if we know the
*rate*at which a field varies across space, we can also calculate the rate at which it varies across time, and vice versa. - Anything that varies across time carries energy. Why? Because this is essentially the
*definition*of energy in quantum mechanics! Up to a constant multiple (namely, Planck’s constant), energy is the expected speed of rotation of the global phase of the wavefunction, when you apply your Hamiltonian. If the global phase rotates at the slowest possible speed, then we take the energy to be zero, and say you’re in a vacuum state. If it rotates at the next highest speed, we say you’re in a first excited state, and so on. Indeed, assuming a time-independent Hamiltonian, the evolution of any quantum system can be fully described by simply decomposing the wavefunction into a superposition of energy eigenstates, then tracking of the phase of each eigenstate’s amplitude as it loops around and around the unit circle. No energy means no looping around means nothing ever changes. - Combining 3 and 5, any field that varies across space carries energy.
- More strongly, combining 4 and 5, if we know how
*quickly*a field varies across space, we can lower-bound how much energy it has to contain. - In general relativity, anything that carries energy couples to the gravitational field. This means that anything that carries energy necessarily has an observable effect: if nothing else, its effect on the warping of spacetime. (This is dramatically illustrated by dark matter, which is currently observable via its spacetime warping effect
*and nothing else*.) - Combining 6 and 8, any field that varies across space couples to the gravitational field.
- More strongly, combining 7 and 8, if we know how quickly a field varies across space, then we can lower-bound by how much it has to warp spacetime. This is so because of another famous (and distinctive) feature of gravity: namely, the fact that it’s universally attractive, so all the warping contributions add up.
- But in GR, spacetime can only be warped by so much before we create a black hole: this is the famous Schwarzschild bound.
- Combining 10 and 11, the information contained in a physical field can only vary so quickly across space, before it causes spacetime to collapse to a black hole.

Summarizing where we’ve gotten, we could say: *any information that’s spatially localized at all, can only be localized so precisely*. In our world, the more densely you try to pack 1’s and 0’s, the more energy you need, therefore the more you warp spacetime, until all you’ve gotten for your trouble is a black hole. Furthermore, if we rewrote the above conceptual argument in math—keeping track of all the G’s, c’s, h’s, and so on—we could derive a quantitative *bound* on how much information there can be in a bounded region of space. And if we were careful enough, that bound would be precisely the holographic entropy bound, which says that the number of (qu)bits is at most A/(4 ln 2), where A is the area of a bounding surface in Planck units.

Let’s pause to point out some interesting features of this argument.

Firstly, we pretty much needed the whole kitchen sink of basic physical principles: special relativity (both the equivalence of inertial frames and the finiteness of the speed of light), quantum mechanics (in the form of the universal relation between energy and frequency), and finally general relativity and gravity. All three of the fundamental constants G, c, and h made appearances, which is why all three show up in the detailed statement of the holographic bound.

But secondly, gravity only appeared from step 8 onwards. Up till then, everything could be said solely in the language of *quantum field theory*: that is, quantum mechanics plus special relativity. The result would be the so-called Bekenstein bound, which upper-bounds the number of bits in any spatial region by the *product* of the region’s radius and its energy content. I learned that there’s an interesting history here: Bekenstein originally deduced this bound using ingenious thought experiments involving black holes. Only later did people realize that the Bekenstein bound can be derived purely within QFT (see here and here for example)—in contrast to the holographic bound, which really *is* a statement about quantum gravity. (An early hint of this was that, while the holographic bound involves Newton’s gravitational constant G, the Bekenstein bound doesn’t.)

Thirdly, speaking of QFT, some readers might be struck by the fact that at no point in our 12-step program did we ever seem to need QFT machinery. Which is fortunate, because if we *had* needed it, I wouldn’t have been able to explain any of this! But here I have to confess that I cheated slightly. Recall step 4, which said that “if you know the rate at which a field varies across space, you can calculate the rate at which it varies across time.” It turns out that, in order to give that sentence a definite meaning, one uses the fact that in QFT, space and time derivatives in the Hamiltonian need to be related by a factor of c, since otherwise the Hamiltonian wouldn’t be Lorentz-invariant.

Fourthly, eagle-eyed readers might notice a loophole in the argument. Namely, *we never upper-bounded how much information God could add to the world, via fields that are constant across all of spacetime*. For example, there’s nothing to stop Her from creating a new scalar field that takes the same value everywhere in the universe—with that value, in suitable units, encoding 10^{50000} separate divine thoughts in its binary expansion. But OK, being constant, such a field would interact with nothing and affect no observations—so Occam’s Razor itches to slice it off, by rewriting the laws of physics in a simpler form where that field is absent. If you like, such a field would at most be a comment in the source code of the universe: it could be as long as the Great Programmer wanted it to be, but would have no observable effect on those of us living inside the program’s execution.

Of course, even before relativity and quantum mechanics, information had already been playing a surprisingly fleshy role in physics, through its appearance as *entropy* in 19^{th}-century thermodynamics. Which leads to another puzzle. To a computer scientist, the concept of entropy, as the log of the number of microstates compatible with a given macrostate, seems clear enough, as does the intuition for why it should increase monotonically with time. Or at least, to whatever extent we’re confused about these matters, we’re no *more* confused than the physicists are!

But then why should this information-theoretic concept be so closely connected to tangible quantities like temperature, and pressure, and energy? From the mere assumption that a black hole has a nonzero entropy—that is, that it takes many bits to describe—how could Bekenstein and Hawking have possibly deduced that it also has a nonzero temperature? Or: if you put your finger into a tub of hot water, does the heat that you feel somehow reflect *how many bits are needed to describe the water’s microstate*?

Once again our skeptic pipes up: “but surely God could stuff as many additional bits as She wanted into the microstate of the hot water—for example, in degrees of freedom that are still unknown to physics—without the new bits having any effect on the water’s temperature.”

But we should’ve learned by now to doubt this sort of argument. There’s no general principle, in our universe, saying that you can hide as many bits as you want in a physical object, without those bits influencing the object’s observable properties. On the contrary, in case after case, our laws of physics seem to be intolerant of “wallflower bits,” which hide in a corner without talking to anyone. If a bit is there, the laws of physics want it to affect other nearby bits and be affected by them in turn.

In the case of thermodynamics, the assumption that does all the real work here is that of *equidistribution*. That is, *whatever* degrees of freedom might be available to your thermal system, your gas in a box or whatever, we assume that they’re all already “as randomized as they could possibly be,” subject to a few observed properties like temperature and volume and pressure. (At least, we assume that in classical thermodynamics. Non-equilibrium thermodynamics is a whole different can of worms, worms that don’t stay in equilibrium.) Crucially, we assume this despite the fact that we might not even *know* all the relevant degrees of freedom.

Why is this assumption justified? “Because experiment bears it out,” the physics teacher explains—but we can do better. The assumption is justified because, as long as the degrees of freedom that we’re talking about all interact with each other, they’ve already had plenty of time to equilibrate. And conversely, if a degree of freedom *doesn’t* interact with the stuff we’re observing—or with anything that interacts with the stuff we’re observing, etc.—well then, who cares about it anyway?

But now, because the microscopic laws of physics have the fundamental property of *reversibility*—that is, they never destroy information—a new bit has to go *somewhere*, and it can’t overwrite degrees of freedom that are already fully randomized. This is why, if you pump more bits of information into a tub of hot water, while keeping it at the same volume, the new bits have nowhere to go except into pushing up the energy. Now, there are often ways to push up the energy other than by raising the temperature—the concept of specific heat, in chemistry, is precisely about this—but if you need to stuff more bits into a substance, at the cost of raising its energy, certainly one of the obvious ways to do it is to describe a greater range of possible speeds for the water molecules. So since that *can* happen, by equidistribution it typically *does* happen, which means that the molecules move faster on average, and your finger feels the water get hotter.

In summary, our laws of physics are structured in such a way that *even pure information often has “nowhere to hide”*: if the bits are there at all in the abstract machinery of the world, then they’re forced to pipe up and have a measurable effect. And this is not a tautology, but comes about only because of nontrivial facts about special and general relativity, quantum mechanics, quantum field theory, and thermodynamics. And this is what I think people should mean when they say “information is physical.”

Anyway, if this was all obvious to you, I apologize for having wasted your time! But in my defense, it was never explained to me quite this way, nor was it sorted out in my head until recently—even though it seems like one of the most basic and general things one can possibly say about physics.

**Endnotes.** Thanks again to Daniel Harlow, not only for explaining the logic of the holographic bound to me but for several suggestions that improved this post.

Some readers might suspect circularity in the arguments we’ve made: are we merely saying that “any information that has observable physical consequences, has observable physical consequences”? No, it’s more than that. In all the examples I discussed, the magic was that we inserted certain information into our *abstract mathematical description* of the world, taking no care to ensure that the information’s presence would have any observable consequences whatsoever. But then the principles of quantum mechanics, quantum gravity, or thermodynamics *forced* the information to be detectable in very specific ways (namely, via the destruction of quantum interference, the warping of spacetime, or the generation of heat respectively).