“Information is physical.”<\/p>\n
This slogan seems to have originated around 1991<\/a> with Rolf Landauer. \u00a0It’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.<\/p>\n But what the hell does it mean?<\/p>\n There are many things it’s taken<\/em> 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.<\/p>\n For example, some people seem to use the slogan to mean something more like\u00a0its converse: “physics is informational.” \u00a0That is, the laws of physics are ultimately not about mass or energy or pressure, but about bits and computations\u00a0on them. \u00a0As I’ve often said, my problem\u00a0with that view is less its audacity than its timidity! \u00a0It’s like, what would the universe have to do in order not<\/em>\u00a0to be informational in this sense? \u00a0“Information”\u00a0is just\u00a0a name we give to whatever picks out one\u00a0element 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). \u00a0So, as long as the laws of physics take the form of telling us\u00a0that\u00a0some observations or configurations of the world are possible and others are not, or of giving us probabilities for each configuration,\u00a0no duh<\/em> they’re about information!<\/p>\n Other people use “information is physical” to pour\u00a0scorn on the idea that “information” could mean anything without some actual physical instantiation\u00a0of the abstract 0’s and 1’s, such as voltage differences in a loop of wire. \u00a0Here I certainly agree with the tautology that in order to exist physically<\/em>—that is, be embodied in the physical world—a piece of information (like a song, video, or computer program) does need\u00a0to be embodied in the physical world. \u00a0But my inner Platonist slumps in his armchair when people go on to assert\u00a0that, for example, it’s meaningless to discuss\u00a0the first prime number larger than 1010^125<\/sup>, because according to post-1998 cosmology, one couldn’t fit\u00a0its digits inside the observable universe.<\/p>\n If the cosmologists revise their models next week, will this prime suddenly burst\u00a0into existence, with all the mathematical properties that one\u00a0could’ve predicted for it on general grounds—only to fade back into the netherworld if the cosmologists revise their models again? \u00a0Why would anyone<\/em> want to use language in such a tortured way?<\/p>\n Yes, brains, computers, yellow books<\/a>, and so on that encode mathematical knowledge\u00a0comprise\u00a0only a tiny sliver\u00a0of the physical world. \u00a0But it’s equally true\u00a0that the physical world we observe comprises only\u00a0a tiny sliver of mathematical possibility-space.<\/p>\n 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. \u00a0Far be it from me to temper that enthusiasm: rock on, dudes!<\/p>\n Yet others use “information is physical” to mean that the rules<\/em>\u00a0governing information processing and transmission in the physical world aren’t knowable a priori<\/em>, but can only be learned from physics. \u00a0This is clearest\u00a0in the case of quantum information, which has its own internal logic that generalizes the logic of classical information. \u00a0But in some sense, we didn’t need quantum mechanics to tell us this! \u00a0Of course<\/em> the laws of physics have ultimate\u00a0jurisdiction over whatever occurs\u00a0in the physical world, information processing included.<\/p>\n My biggest beef, with all<\/em> these unpackings of the “information is physical” slogan, is that none of them really engage with any of the deep\u00a0truths that we’ve learned about physics. \u00a0That 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.<\/p>\n So then what should<\/em> we mean by “information is physical”? \u00a0In the rest of this post, I’d like to propose an answer to that question.<\/p>\n We get closer to the meat of the slogan if we consider some actual physical phenomena, say in quantum mechanics. \u00a0The double-slit experiment will do fine.<\/p>\n 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. \u00a0You ask: does that distribution contain alternating “light” and “dark” regions, the signature of interference between positive and negative amplitudes? \u00a0And the answer, predicted by the math and confirmed by experiment, is: yes, but\u00a0only if the information about which slit the photon went through failed to\u00a0get\u00a0recorded anywhere else in the universe, other than the photon location itself.<\/em><\/p>\n Here a skeptic interjects: but that has<\/em> to be wrong! \u00a0The criterion for where a physical particle lands on a physical screen can’t possibly depend on anything as airy\u00a0as whether “information” got “recorded” or not. \u00a0For what counts as\u00a0“information,” anyway? \u00a0As an extreme example:\u00a0what if God, unbeknownst to us mortals, took\u00a0divine note of which slit the photon went through? \u00a0Would that<\/em> destroy the interference pattern? \u00a0If so, then every time we do\u00a0the experiment, are we collecting data about the existence or nonexistence of an all-knowing\u00a0God?<\/p>\n 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.<\/em> \u00a0And crucially, if quantum mechanics is universally true, then the mind of God would\u00a0have<\/em>\u00a0to be such a tensor factor, in order for its state to play any role in\u00a0the prediction of observed phenomena.<\/p>\n To say this another way: it’s obvious and unexceptionable that, by observing\u00a0a physical system, you can often learn\u00a0something about what information must\u00a0be in it. \u00a0For example, you need never\u00a0have heard of DNA to deduce\u00a0that chickens must\u00a0somehow\u00a0contain information about\u00a0making more chickens. \u00a0What’s much more surprising is that, in quantum mechanics, you can often deduce things about what information can’t<\/em> 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.<\/p>\n Another famous example here concerns identical\u00a0particles. \u00a0You\u00a0may 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\u00a0two electrons have exactly<\/em> the same mass, same charge, same every other property, including even any properties yet to be discovered. \u00a0Again the skeptic interjects: but that has<\/em> to be wrong. \u00a0Logically, you could only ever confirm\u00a0that two electrons were different<\/em>, by observing a difference in their behavior. \u00a0Even 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\u00a0tiny 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.<\/p>\n You can probably guess where this is going. \u00a0Quantum mechanics says that, no, you can<\/em> verify that two particles\u00a0are perfectly\u00a0identical by doing an experiment where you swap them and see what happens. \u00a0If the particles\u00a0are identical in all respects, then you’ll see quantum interference between the swapped and un-swapped states.\u00a0 If they aren’t, you won’t. \u00a0The\u00a0kind<\/em> of interference you’ll see is different for fermions (like electrons) than\u00a0for bosons (like photons), but the basic principle is the same in both cases. \u00a0Once again, quantum mechanics lets you\u00a0verify that a specific type of information—in this case, information that distinguishes one particle\u00a0from another—was not<\/em> present anywhere in the physical world, because if it were, it would’ve destroyed an interference effect that you in fact saw.<\/p>\n This, I think, already provides a meatier sense in which “information is physical” than any of the senses\u00a0discussed previously.<\/p>\n But we haven’t gotten to the filet mignon yet. \u00a0The late, great Jacob Bekenstein<\/a> 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<\/em>. \u00a0The most precise form of this statement, called the\u00a0covariant entropy bound<\/a>, was worked out in detail by Raphael Bousso<\/a>. \u00a0Here 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\u00a0in Planck units of any surface that encloses the system—so, about 1069<\/sup> bits per square meter. \u00a0(Actually it’s 1069<\/sup>\u00a0qubits<\/em> per square meter, but because of Holevo’s theorem<\/a>, 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\u00a0later.)<\/p>\n You might have heard of the famous way Nature enforces this bound. \u00a0Namely, if you tried to create a hard drive that stored more than 1069<\/sup>\u00a0bits per square meter of surface area, the hard drive would necessarily collapse to a black hole. \u00a0And 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.<\/p>\n Let’s hear once more from our skeptic. \u00a0“Nonsense! \u00a0Matter<\/em> can take up space. \u00a0Energy<\/em> can take up space. \u00a0But information? \u00a0Bah! \u00a0That’s just a category mistake. \u00a0For a proof, suppose God took one of your black holes, with a 1-square-meter event horizon, which already had its supposed maximum of ~1069<\/sup>\u00a0bits of information. \u00a0And 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 10300<\/sup>\u00a0new bits in the region of the black hole. \u00a0Presto! \u00a0An unlimited amount of additional information, exactly where Bekenstein said it couldn’t exist.”<\/p>\n 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. \u00a0I was confused myself about how to do this, until a month ago, when Daniel Harlow<\/a> helped set me straight (any remaining howlers\u00a0in my\u00a0exposition are 100%\u00a0mine, not his).<\/p>\n I believe the logic goes like this:<\/p>\n Summarizing where we’ve gotten, we could say: any information that’s spatially localized at all, can only be localized so precisely<\/em>. \u00a0In our world, the more densely you try to pack 1’s and 0’s, the more energy you need, therefore\u00a0the more you warp spacetime, until all you’ve gotten for your trouble is a black hole. \u00a0Furthermore, if we rewrote the above conceptual argument in math—keeping track of all the\u00a0G’s, c’s, h’s, and so on—we could derive a quantitative\u00a0bound<\/em> on how much information there can be\u00a0in a bounded region of space. \u00a0And if we were\u00a0careful enough, that bound would be precisely the holographic entropy bound<\/a>, 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.<\/p>\n Let’s pause to point out some interesting features of this argument.<\/p>\n Firstly, we pretty much needed the whole kitchen sink of basic\u00a0physical 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. \u00a0All three of the fundamental constants G<\/a>, c<\/a>, and h<\/a> made appearances, which is why all three show up in the detailed statement of the holographic bound.<\/p>\n But secondly, gravity only appeared from step 8 onwards. \u00a0Up till then, everything could be said solely\u00a0in the language of quantum field theory<\/em>: that is, quantum mechanics plus special relativity. \u00a0The result would be\u00a0the so-called\u00a0Bekenstein bound<\/a>, which upper-bounds the number of bits\u00a0in any spatial region by\u00a0the product<\/em> of the region’s radius and its energy content. \u00a0I learned that there’s an interesting history here: Bekenstein originally deduced this bound using ingenious thought experiments involving black holes. \u00a0Only later did people realize that the Bekenstein bound can be derived\u00a0purely within\u00a0QFT (see here<\/a>\u00a0and here<\/a> for example)—in contrast to the holographic bound, which really is<\/em>\u00a0a statement about quantum gravity. \u00a0(An early hint of this was that, while the holographic bound involves Newton’s gravitational constant G, the Bekenstein bound doesn’t.)<\/p>\n Thirdly, speaking of QFT, some readers\u00a0might be struck by the fact that at no point in our 12-step program did we ever seem to need\u00a0QFT machinery. \u00a0Which is\u00a0fortunate, because if we had<\/em> needed it, I wouldn’t have been able to explain any of this! \u00a0But here I have to confess that I cheated slightly. \u00a0Recall 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.” \u00a0It turns out that, in order\u00a0to give that sentence a definite meaning, one\u00a0uses the fact that in QFT, space and time derivatives in the Hamiltonian need\u00a0to be related by a factor of c, since otherwise the Hamiltonian wouldn’t be Lorentz-invariant.<\/p>\n Fourthly, eagle-eyed readers might notice a loophole in the argument. \u00a0Namely, we never upper-bounded how much information God could add to the world, via fields that are constant across all of spacetime<\/em>. \u00a0For 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 1050000<\/sup>\u00a0separate divine thoughts in its binary expansion. \u00a0But OK, being constant, such a field would interact with nothing and affect\u00a0no observations—so Occam’s Razor itches\u00a0to slice it off,\u00a0by rewriting the laws of physics in a simpler form where that field is absent. \u00a0If you like, such a field would at most be\u00a0a comment in the source code of the universe: it could be as long as the Great Programmer wanted it to be, but would have\u00a0no observable effect on those of us living inside the program’s execution.<\/p>\n Of course, even before relativity and quantum mechanics, information had already been playing a surprisingly fleshy role in physics, through its appearance as entropy<\/em> in 19th<\/sup>-century thermodynamics. \u00a0Which leads to another puzzle. \u00a0To a computer scientist, the concept\u00a0of entropy<\/a>, 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. \u00a0Or at least, to whatever extent we’re confused about these matters, we’re no more<\/em> confused than the physicists are!<\/p>\n But then why should this information-theoretic concept be so closely connected to tangible quantities like temperature, and pressure, and energy? \u00a0From 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? \u00a0Or: 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<\/em>?<\/p>\n Once again our skeptic pipes up:\u00a0“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\u00a0having any effect on the water’s temperature.”<\/p>\n But we should’ve\u00a0learned by now to doubt this sort of\u00a0argument. \u00a0There’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. \u00a0On the contrary, in case after case,\u00a0our\u00a0laws of physics seem to be intolerant of “wallflower bits,” which\u00a0hide in a corner without talking to anyone. \u00a0If a bit is there, the laws of physics want it to affect other nearby bits and be affected by them in turn.<\/p>\n In the case of thermodynamics, the assumption that does all the real\u00a0work here is that of equidistribution<\/em>. \u00a0That is, whatever<\/em> degrees of freedom might be\u00a0available\u00a0to 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\u00a0properties like temperature and volume and pressure. \u00a0(At least, we assume that in classical thermodynamics. \u00a0Non-equilibrium thermodynamics<\/a> is a whole different can of worms, worms that don’t stay\u00a0in equilibrium.) \u00a0Crucially, we assume this despite the fact that\u00a0we might not even know<\/em> all the relevant degrees of freedom.<\/p>\n Why is this assumption justified? \u00a0“Because experiment bears it out,” the physics teacher explains—but we can do better. \u00a0The assumption is justified because, as long as the degrees of freedom that we’re talking about all\u00a0interact with each other, they’ve already had plenty\u00a0of time to equilibrate. \u00a0And conversely, if a degree of freedom\u00a0doesn’t<\/em> 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?<\/p>\n
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