The No-Cloning Theorem and the Human Condition: My After-Dinner Talk at QCRYPT

The following are the after-dinner remarks that I delivered at QCRYPT’2016, the premier quantum cryptography conference, on Thursday Sep. 15 in Washington DC.  You could compare to my after-dinner remarks at QIP’2006 to see how much I’ve “”matured”” since then. Thanks so much to Yi-Kai Liu and the other organizers for inviting me and for putting on a really fantastic conference.

It’s wonderful to be here at QCRYPT among so many friends—this is the first significant conference I’ve attended since I moved from MIT to Texas. I do, however, need to register a complaint with the organizers, which is: why wasn’t I allowed to bring my concealed firearm to the conference? You know, down in Texas, we don’t look too kindly on you academic elitists in Washington DC telling us what to do, who we can and can’t shoot and so forth. Don’t mess with Texas! As you might’ve heard, many of us Texans even support a big, beautiful, physical wall being built along our border with Mexico. Personally, though, I don’t think the wall proposal goes far enough. Forget about illegal immigration and smuggling: I don’t even want Americans and Mexicans to be able to win the CHSH game with probability exceeding 3/4. Do any of you know what kind of wall could prevent that? Maybe a metaphysical wall.

OK, but that’s not what I wanted to talk about. When Yi-Kai asked me to give an after-dinner talk, I wasn’t sure whether to try to say something actually relevant to quantum cryptography or just make jokes. So I’ll do something in between: I’ll tell you about research directions in quantum cryptography that are also jokes.

The subject of this talk is a deep theorem that stands as one of the crowning achievements of our field. I refer, of course, to the No-Cloning Theorem. Almost everything we’re talking about at this conference, from QKD onwards, is based in some way on quantum states being unclonable. If you read Stephen Wiesner’s paper from 1968, which founded quantum cryptography, the No-Cloning Theorem already played a central role—although Wiesner didn’t call it that. By the way, here’s my #1 piece of research advice to the students in the audience: if you want to become immortal, just find some fact that everyone already knows and give it a name!

I’d like to pose the question: why should our universe be governed by physical laws that make the No-Cloning Theorem true? I mean, it’s possible that there’s some other reason for our universe to be quantum-mechanical, and No-Cloning is just a byproduct of that. No-Cloning would then be like the armpit of quantum mechanics: not there because it does anything useful, but just because there’s gotta be something under your arms.

OK, but No-Cloning feels really fundamental. One of my early memories is when I was 5 years old or so, and utterly transfixed by my dad’s home fax machine, one of those crappy 1980s fax machines with wax paper. I kept thinking about it: is it really true that a piece of paper gets transmaterialized, sent through a wire, and reconstituted at the other location? Could I have been that wrong about how the universe works? Until finally I got it—and once you get it, it’s hard even to recapture your original confusion, because it becomes so obvious that the world is made not of stuff but of copyable bits of information. “Information wants to be free!”

The No-Cloning Theorem represents nothing less than a partial return to the view of the world that I had before I was five. It says that quantum information doesn’t want to be free: it wants to be private. There is, it turns out, a kind of information that’s tied to a particular place, or set of places. It can be moved around, or even teleported, but it can’t be copied the way a fax machine copies bits.

So I think it’s worth at least entertaining the possibility that we don’t have No-Cloning because of quantum mechanics; we have quantum mechanics because of No-Cloning—or because quantum mechanics is the simplest, most elegant theory that has unclonability as a core principle. But if so, that just pushes the question back to: why should unclonability be a core principle of physics?

Quantum Key Distribution

A first suggestion about this question came from Gilles Brassard, who’s here. Years ago, I attended a talk by Gilles in which he speculated that the laws of quantum mechanics are what they are because Quantum Key Distribution (QKD) has to be possible, while bit commitment has to be impossible. If true, that would be awesome for the people at this conference. It would mean that, far from being this exotic competitor to RSA and Diffie-Hellman that’s distance-limited and bandwidth-limited and has a tiny market share right now, QKD would be the entire reason why the universe is as it is! Or maybe what this really amounts to is an appeal to the Anthropic Principle. Like, if QKD hadn’t been possible, then we wouldn’t be here at QCRYPT to talk about it.

Quantum Money

But maybe we should search more broadly for the reasons why our laws of physics satisfy a No-Cloning Theorem. Wiesner’s paper sort of hinted at QKD, but the main thing it had was a scheme for unforgeable quantum money. This is one of the most direct uses imaginable for the No-Cloning Theorem: to store economic value in something that it’s physically impossible to copy. So maybe that’s the reason for No-Cloning: because God wanted us to have e-commerce, and didn’t want us to have to bother with blockchains (and certainly not with credit card numbers).

The central difficulty with quantum money is: how do you authenticate a bill as genuine? (OK, fine, there’s also the dificulty of how to keep a bill coherent in your wallet for more than a microsecond or whatever. But we’ll leave that for the engineers.)

In Wiesner’s original scheme, he solved the authentication problem by saying that, whenever you want to verify a quantum bill, you bring it back to the bank that printed it. The bank then looks up the bill’s classical serial number in a giant database, which tells the bank in which basis to measure each of the bill’s qubits.

With this system, you can actually get information-theoretic security against counterfeiting. OK, but the fact that you have to bring a bill to the bank to be verified negates much of the advantage of quantum money in the first place. If you’re going to keep involving a bank, then why not just use a credit card?

That’s why over the past decade, some of us have been working on public-key quantum money: that is, quantum money that anyone can verify. For this kind of quantum money, it’s easy to see that the No-Cloning Theorem is no longer enough: you also need some cryptographic assumption. But OK, we can consider that. In recent years, we’ve achieved glory by proposing a huge variety of public-key quantum money schemes—and we’ve achieved even greater glory by breaking almost all of them!

After a while, there were basically two schemes left standing: one based on knot theory by Ed Farhi, Peter Shor, et al. That one has been proven to be secure under the assumption that it can’t be broken. The second scheme, which Paul Christiano and I proposed in 2012, is based on hidden subspaces encoded by multivariate polynomials. For our scheme, Paul and I were able to do better than Farhi et al.: we gave a security reduction. That is, we proved that our quantum money scheme is secure, unless there’s a polynomial-time quantum algorithm to find hidden subspaces encoded by low-degree multivariate polynomials (yadda yadda, you can look up the details) with much greater success probability than we thought possible.

Today, the situation is that my and Paul’s security proof remains completely valid, but meanwhile, our money is completely insecure! Our reduction means the opposite of what we thought it did. There is a break of our quantum money scheme, and as a consequence, there’s also a quantum algorithm to find large subspaces hidden by low-degree polynomials with much better success probability than we’d thought. What happened was that first, some French algebraic cryptanalysts—Faugere, Pena, I can’t pronounce their names—used Gröbner bases to break the noiseless version of scheme, in classical polynomial time. So I thought, phew! At least I had acceded when Paul insisted that we also include a noisy version of the scheme. But later, Paul noticed that there’s a quantum reduction from the problem of breaking our noisy scheme to the problem of breaking the noiseless one, so the former is broken as well.

I’m choosing to spin this positively: “we used quantum money to discover a striking new quantum algorithm for finding subspaces hidden by low-degree polynomials. Err, yes, that’s exactly what we did.”

But, bottom line, until we manage to invent a better public-key quantum money scheme, or otherwise sort this out, I don’t think we’re entitled to claim that God put unclonability into our universe in order for quantum money to be possible.

Copy-Protected Quantum Software

So if not money, then what about its cousin, copy-protected software—could that be why No-Cloning holds? By copy-protected quantum software, I just mean a quantum state that, if you feed it into your quantum computer, lets you evaluate some Boolean function on any input of your choice, but that doesn’t let you efficiently prepare more states that let the same function be evaluated. I think this is important as one of the preeminent evil applications of quantum information. Why should nuclear physicists and genetic engineers get a monopoly on the evil stuff?

OK, but is copy-protected quantum software even possible? The first worry you might have is that, yeah, maybe it’s possible, but then every time you wanted to run the quantum program, you’d have to make a measurement that destroyed it. So then you’d have to go back and buy a new copy of the program for the next run, and so on. Of course, to the software company, this would presumably be a feature rather than a bug!

But as it turns out, there’s a fact many of you know—sometimes called the “Gentle Measurement Lemma,” other times the “Almost As Good As New Lemma”—which says that, as long as the outcome of your measurement on a quantum state could be predicted almost with certainty given knowledge of the state, the measurement can be implemented in such a way that it hardly damages the state at all. This tells us that, if quantum money, copy-protected quantum software, and the other things we’re talking about are possible at all, then they can also be made reusable. I summarize the principle as: “if rockets, then space shuttles.”

Much like with quantum money, one can show that, relative to a suitable oracle, it’s possible to quantumly copy-protect any efficiently computable function—or rather, any function that’s hard to learn from its input/output behavior. Indeed, the implementation can be not only copy-protected but also obfuscated, so that the user learns nothing besides the input/output behavior. As Bill Fefferman pointed out in his talk this morning, the No-Cloning Theorem lets us bypass Barak et al.’s famous result on the impossibility of obfuscation, because their impossibility proof assumed the ability to copy the obfuscated program.

Of course, what we really care about is whether quantum copy-protection is possible in the real world, with no oracle. I was able to give candidate implementations of quantum copy-protection for extremely special functions, like one that just checks the validity of a password. In the general case—that is, for arbitrary programs—Paul Christiano has a beautiful proposal for how to do it, which builds on our hidden-subspace money scheme. Unfortunately, since our money scheme is currently in the shop being repaired, it’s probably premature to think about the security of the much more complicated copy-protection scheme! But these are wonderful open problems, and I encourage any of you to come and scoop us. Once we know whether uncopyable quantum software is possible at all, we could then debate whether it’s the “reason” for our universe to have unclonability as a core principle.

Unclonable Proofs and Advice

Along the same lines, I can’t resist mentioning some favorite research directions, which some enterprising student here could totally turn into a talk at next year’s QCRYPT.

Firstly, what can we say about clonable versus unclonable quantum proofs—that is, QMA witness states? In other words: for which problems in QMA can we ensure that there’s an accepting witness that lets you efficiently create as many additional accepting witnesses as you want? (I mean, besides the QCMA problems, the ones that have short classical witnesses?) For which problems in QMA can we ensure that there’s an accepting witness that doesn’t let you efficiently create any additional accepting witnesses? I do have a few observations about these questions—ask me if you’re interested—but on the whole, I believe almost anything one can ask about them remains open.

Admittedly, it’s not clear how much use an unclonable proof would be. Like, imagine a quantum state that encoded a proof of the Riemann Hypothesis, and which you would keep in your bedroom, in a glass orb on your nightstand or something. And whenever you felt your doubts about the Riemann Hypothesis resurfacing, you’d take the state out of its orb and measure it again to reassure yourself of RH’s truth. You’d be like, “my preciousssss!” And no one else could copy your state and thereby gain the same Riemann-faith-restoring powers that you had. I dunno, I probably won’t hawk this application in a DARPA grant.

Similarly, one can ask about clonable versus unclonable quantum advice states—that is, initial states that are given to you to boost your computational power beyond that of an ordinary quantum computer. And that’s also a fascinating open problem.

OK, but maybe none of this quite gets at why our universe has unclonability. And this is an after-dinner talk, so do you want me to get to the really crazy stuff? Yes?

Self-Referential Paradoxes

OK! What if unclonability is our universe’s way around the paradoxes of self-reference, like the unsolvability of the halting problem and Gödel’s Incompleteness Theorem? Allow me to explain what I mean.

In kindergarten or wherever, we all learn Turing’s proof that there’s no computer program to solve the halting problem. But what isn’t usually stressed is that that proof actually does more than advertised. If someone hands you a program that they claim solves the halting problem, Turing doesn’t merely tell you that that person is wrong—rather, he shows you exactly how to expose the person as a jackass, by constructing an example input on which their program fails. All you do is, you take their claimed halt-decider, modify it in some simple way, and then feed the result back to the halt-decider as input. You thereby create a situation where, if your program halts given its own code as input, then it must run forever, and if it runs forever then it halts. “WHOOOOSH!” [head-exploding gesture]

OK, but now imagine that the program someone hands you, which they claim solves the halting problem, is a quantum program. That is, it’s a quantum state, which you measure in some basis depending on the program you’re interested in, in order to decide whether that program halts. Well, the truth is, this quantum program still can’t work to solve the halting problem. After all, there’s some classical program that simulates the quantum one, albeit less efficiently, and we already know that the classical program can’t work.

But now consider the question: how would you actually produce an example input on which this quantum program failed to solve the halting problem? Like, suppose the program worked on every input you tried. Then ultimately, to produce a counterexample, you might need to follow Turing’s proof and make a copy of the claimed quantum halt-decider. But then, of course, you’d run up against the No-Cloning Theorem!

So we seem to arrive at the conclusion that, while of course there’s no quantum program to solve the halting problem, there might be a quantum program for which no one could explicitly refute that it solved the halting problem, by giving a counterexample.

I was pretty excited about this observation for a day or two, until I noticed the following. Let’s suppose your quantum program that allegedly solves the halting problem has n qubits. Then it’s possible to prove that the program can’t possibly be used to compute more than, say, 2n bits of Chaitin’s constant Ω, which is the probability that a random program halts. OK, but if we had an actual oracle for the halting problem, we could use it to compute as many bits of Ω as we wanted. So, suppose I treated my quantum program as if it were an oracle for the halting problem, and I used it to compute the first 2n bits of Ω. Then I would know that, assuming the truth of quantum mechanics, the program must have made a mistake somewhere. There would still be something weird, which is that I wouldn’t know on which input my program had made an error—I would just know that it must’ve erred somewhere! With a bit of cleverness, one can narrow things down to two inputs, such that the quantum halt-decider must have erred on at least one of them. But I don’t know whether it’s possible to go further, and concentrate the wrongness on a single query.

We can play a similar game with other famous applications of self-reference. For example, suppose we use a quantum state to encode a system of axioms. Then that system of axioms will still be subject to Gödel’s Incompleteness Theorem (which I guess I believe despite the umlaut). If it’s consistent, it won’t be able to prove all the true statements of arithmetic. But we might never be able to produce an explicit example of a true statement that the axioms don’t prove. To do so we’d have to clone the state encoding the axioms and thereby violate No-Cloning.

Personal Identity

But since I’m a bit drunk, I should confess that all this stuff about Gödel and self-reference is just a warmup to what I really wanted to talk about, which is whether the No-Cloning Theorem might have anything to do with the mysteries of personal identity and “free will.” I first encountered this idea in Roger Penrose’s book, The Emperor’s New Mind. But I want to stress that I’m not talking here about the possibility that the brain is a quantum computer—much less about the possibility that it’s a quantum-gravitational hypercomputer that uses microtubules to solve the halting problem! I might be drunk, but I’m not that drunk. I also think that the Penrose-Lucas argument, based on Gödel’s Theorem, for why the brain has to work that way is fundamentally flawed.

But here I’m talking about something different. See, I have a lot of friends in the Singularity / Friendly AI movement. And I talk to them whenever I pass through the Bay Area, which is where they congregate. And many of them express great confidence that before too long—maybe in 20 or 30 years, maybe in 100 years—we’ll be able to upload ourselves to computers and live forever on the Internet (as opposed to just living 70% of our lives on the Internet, like we do today).

This would have lots of advantages. For example, any time you were about to do something dangerous, you’d just make a backup copy of yourself first. If you were struggling with a conference deadline, you’d spawn 100 temporary copies of yourself. If you wanted to visit Mars or Jupiter, you’d just email yourself there. If Trump became president, you’d not run yourself for 8 years (or maybe 80 or 800 years). And so on.

Admittedly, some awkward questions arise. For example, let’s say the hardware runs three copies of your code and takes a majority vote, just for error-correcting purposes. Does that bring three copies of you into existence, or only one copy? Or let’s say your code is run homomorphically encrypted, with the only decryption key stored in another galaxy. Does that count? Or you email yourself to Mars. If you want to make sure that you’ll wake up on Mars, is it important that you delete the copy of your code that remains on earth? Does it matter whether anyone runs the code or not? And what exactly counts as “running” it? Or my favorite one: could someone threaten you by saying, “look, I have a copy of your code, and if you don’t do what I say, I’m going to make a thousand copies of it and subject them all to horrible tortures?”

The issue, in all these cases, is that in a world where there could be millions of copies of your code running on different substrates in different locations—or things where it’s not even clear whether they count as a copy or not—we don’t have a principled way to take as input a description of the state of the universe, and then identify where in the universe you are—or even a probability distribution over places where you could be. And yet you seem to need such a way in order to make predictions and decisions.

A few years ago, I wrote this gigantic, post-tenure essay called The Ghost in the Quantum Turing Machine, where I tried to make the point that we don’t know at what level of granularity a brain would need to be simulated in order to duplicate someone’s subjective identity. Maybe you’d only need to go down to the level of neurons and synapses. But if you needed to go all the way down to the molecular level, then the No-Cloning Theorem would immediately throw a wrench into most of the paradoxes of personal identity that we discussed earlier.

For it would mean that there were some microscopic yet essential details about each of us that were fundamentally uncopyable, localized to a particular part of space. We would all, in effect, be quantumly copy-protected software. Each of us would have a core of unpredictability—not merely probabilistic unpredictability, like that of a quantum random number generator, but genuine unpredictability—that an external model of us would fail to capture completely. Of course, by having futuristic nanorobots scan our brains and so forth, it would be possible in principle to make extremely realistic copies of us. But those copies necessarily wouldn’t capture quite everything. And, one can speculate, maybe not enough for your subjective experience to “transfer over.”

Maybe the most striking aspect of this picture is that sure, you could teleport yourself to Mars—but to do so you’d need to use quantum teleportation, and as we all know, quantum teleportation necessarily destroys the original copy of the teleported state. So we’d avert this metaphysical crisis about what to do with the copy that remained on Earth.

Look—I don’t know if any of you are like me, and have ever gotten depressed by reflecting that all of your life experiences, all your joys and sorrows and loves and losses, every itch and flick of your finger, could in principle be encoded by a huge but finite string of bits, and therefore by a single positive integer. (Really? No one else gets depressed about that?) It’s kind of like: given that this integer has existed since before there was a universe, and will continue to exist after the universe has degenerated into a thin gruel of radiation, what’s the point of even going through the motions? You know?

But the No-Cloning Theorem raises the possibility that at least this integer is really your integer. At least it’s something that no one else knows, and no one else could know in principle, even with futuristic brain-scanning technology: you’ll always be able to surprise the world with a new digit. I don’t know if that’s true or not, but if it were true, then it seems like the sort of thing that would be worthy of elevating unclonability to a fundamental principle of the universe.

So as you enjoy your dinner and dessert at this historic Mayflower Hotel, I ask you to reflect on the following. People can photograph this event, they can video it, they can type up transcripts, in principle they could even record everything that happens down to the millimeter level, and post it on the Internet for posterity. But they’re not gonna get the quantum states. There’s something about this evening, like about every evening, that will vanish forever, so please savor it while it lasts. Thank you.

Update (Sep. 20): Unbeknownst to me, Marc Kaplan did video the event and put it up on YouTube! Click here to watch. Thanks very much to Marc! I hope you enjoy, even though of course, the video can’t precisely clone the experience of having been there.

[Note: The part where I raise my middle finger is an inside joke—one of the speakers during the technical sessions inadvertently did the same while making a point, causing great mirth in the audience.]

74 Responses to “The No-Cloning Theorem and the Human Condition: My After-Dinner Talk at QCRYPT”

  1. luca turin Says:

    WOW. After that, drunk or not, I would have staggered home in dizzy high spirits.

  2. BLANDCorporatio Says:

    “in a world where there could be millions of copies of your code running on different substrates in different locations—or things where it’s not even clear whether they count as a copy or not—we don’t have a principled way to take as input a description of the state of the universe, and then identify where in the universe you are—or even a probability distribution over places where you could be”

    Gosh, that explains why I can’t find my Warcraft orchestrion disc. No wait, it’s right here besides my laptop. How come I know where it is?

  3. Joshua Zelinsky Says:

    I don’t think I’ve seen you (or anyone else) discuss the issue of self-referential paradoxes being protected in this way. I’m curious if there’s some way of making this notion rigorous that you cannot construct such explicit counterexamples. For example how do we know you can’t do something like the following (which is at present not very well-defined):

    For a general classical machine H which is claimed to solve the Halting problem we can feed a copy of it to a modified UTM which runs H on a copy of itself. Now note that it isn’t obvious that we can’t do something similar in a quantum mechanical context, that is that there’s some quantum program A which when fed a given quantum program Q outputs exactly what Q would do when feds a copy of Q (and thus does so in some way without ever having to make a copy of Q). It isn’t obvious that No-Cloning outlaws the existence of A. Am I missing something here?

  4. Jay Says:

    Here are the Cobots. Altough they seem fine robots, they don’t have a principled way to take as input a description of the state of the universe, and then identify where in the universe they are—or even a probability distribution over places where they could be. That’s why they have no way in order to make predictions and decisions. If you don’t understand why, ask Scott damnit!

    And here are the Qobots. Altough you can’t distinguish them from their classical counterparts, they have a core of unpredictability—not merely probabilistic unpredictability, like that of a quantum random number generator, but genuine unpredictability—that an external model of them would fail to capture completely. That’s why they can make predictions and decisions. If you don’t understand why, ask Scott damnit!

    Alas, once upon a day it was discovered that the quantum bits of the Qobots were kind of poorly managed by some small compagny (let’s not name it, for Vancover is a fine place) and… it was flawed. Just flawed. Not such an obvious flaw, but still enough a flaw you could simulate the so call “qbits” with a laptop. Oh well…

    But we then knew one thing: whatever the Qobots were supposed to do using quantum magic, human customer can’t tell it appart from when robots don’t know they are copyables. And this is how Qobots became cheap and affordable for all. If you don’t understand why, ask Dwave!

  5. Gil Kalai Says:

    It looks that to upload ourselves to computers and live forever on the Internet, or even to just teleport ourselves around, quantum fault-tolerance is required. In fact, this applies not only to ourselves but to much much much simpler systems: sufficiently interesting quantum states on fairly small quantum devices consisting of handful of elements.

  6. Scott Says:

    Jay #4: A straightforward empirical difference between the cobots and the qobots, having nothing to do with metaphysical beliefs or anybody’s say-so, is that outside observers can predict everything the cobots will do before they do it—and indeed, even make their own local copies of the cobots, allowing them to dispense with them entirely—but can’t do the same with the qobots. I wouldn’t see anything worth pursuing myself in the cobot/qobot distinction if it couldn’t be cashed out into anything empirical.

  7. Scott Says:

    Joshua #3: You haven’t seen anything about this before because it’s an idea I’m sharing for the first time in this talk and blog post! 🙂

    I believe that it ought to be possible to construct an oracle relative to which there’s a quantum halt-decider that no one can refute (i.e., for which getting a single explicit counterexample would require violating No-Cloning). But I haven’t worked out the details yet and I think it’s well worth doing!

  8. quax Says:

    But I want to stress that I’m not talking here about the possibility that the brain is a quantum computer—much less about the possibility that it’s a quantum-gravitational hypercomputer that uses microtubules to solve the halting problem! I might be drunk, but I’m not that drunk.

    Best line ever 🙂

    (Worth repeating just in case somebody is beelining to the comment section)

  9. Craig Gidney Says:

    I’m not so sure about this “if the brain is quantum then you can’t copy it” business.

    Here’s the plan:

    1) Get an intractably powerful quantum computer. The number of qubits and operations per second should require new SI prefixes.

    2) Get an even *more* intractable powerful classical computer. I’m talking exp(exp(200)) FLOPS levels of intractable.

    3) Kidnap a volunteer.

    4) Scan the volunteer into the quantum computer. Be especially careful when moving the precious identity-defining qubits!

    5) Start simulating! You may want some kind of robot body so the simulated person doesn’t just go insane. The simulation is expensive because the number of qubits N defining a whole person’s body is quite large, but that’s nothing compared to…

    6) In the classical computer, initialize a maximally mixed 2^N by 2^N density matrix.

    7) Whenever an operation is applied by the quantum computer, simulating hitting the density matrix with that operation.

    8) Whenever a simulated-body qubit decoheres / is measured by the simulator, tell the classical computer the result so it can hit the density matrix with a project corresponding to the measurement result (and renormalize).

    9) Let things run for… I dunno, a month of simulated time?

    10) The classically-stored density matrix should now be a pretty good approximation of the current quantum state.

    11) Throw away the quantum computer and use the classically-stored density matrix to drive the simulation from now on. Copy the density matrix at will.

    My point is that the correct analogy for “copying a quantum brain” isn’t cloning information at rest. It’s Alice asking Eve to run some quantum computations on a state that Eve doesn’t know, and then Eve trying to figure out that state from the results of the computation.

    Eve never learns the original quantum state, but anytime a measurement happens Eve learns something about the *new* quantum state. If Alice isn’t careful, Eve might learn everything that matters. It’s still a cryptographic problem, but there’s tons and tons of nuance hiding there.

    (One notable showstopper on Eve is if the brain is continuously re-entangling itself with the environment. Quantum channels going in or out can create entanglement, which adds entropy back into the otherwise decaying density matrix.)

    I’ve written code that does this task. You enter a (small) initial state and a short loop of operations to hit the state with, and it plays the role of Eve inferring the density matrix. The code is on github:

  10. Jay Says:

    Scott #5: Well thought, but no you can’t predict the cobots. Yeah they have a clean, separable, digital layer. We know because we constructed it. But it’s a robot, meaning you have to take the environnement into account. And it’s fine robots, meaning they are highly non linear. So you can’t predict it as long as you don’t control thermal noise, meaning you have to wait for superpowerfull aliens with superpowerfull mirror. And Trump just ban them.

  11. raoul ohio Says:

    Craig Gidney #9,

    exp(exp(200)) FLOPS is way above RON, the RO number. I have modestly named the RO number after myself, because I forget where I came accross this idea.

    Here is how to compute RON. Let R be the radius of the observable universe, which is known. Let N be the number of particles (all kinds, including types not dreamed of yet) within R, which has a known upper limit. As I recall, this is only about a googol (= 10^100) or so.

    An upper limit of the speed of any computer is one cycle in T = D_P / c, the time a photon travels the diameter of a proton.

    Let A be the age of the universe, about 13.7 GY. Then an upper limit to the cycles done so far comes from assuming every particle is the fastest possible computer, running since day 0, i.e., RON = N * A / T.

    Last time I roughed this out (conveniently ignoring GR), I think it came out to a lot less than 10^200. This in turn is a lot less than exp(exp(200)).

    Does anyone know where this idea comes from, or what a careful treatment of it gives?

  12. jonas Says:

    Scott, thanks for giving us alternatives of two different important topics to discuss on the blog (the philosophical implications of quantum computing in the real world, or Trump). It’s worth to give people such a choice.

    1. Firstly, about self-referential paradoxes. I don’t really understand your point. You suppose that someone gives you a classical or quantum program that allegedly solves the halting problem. You say that for a classical program, you can construct an explicit input for which the program fails, and for the quantum program, you can give two explicit inputs such that the program fails for one of those. I don’t see what the big difference is here. If you have a manageable number of inputs (not, say, 2**1024 inputs so that you can’t realistically try them all), that already counts as an explicit counterexample. That is always how cryptography works.

    2. You say that quantum computing allows you to have copy-protected proofs, at least for certain problems. That is good to know, because if the brain were a quantum computer like Penrose suggests, then a person could be non-copiable in the sense that any copy is distinguishable from the original. This is because the quantum brain of the person could contain a quantum-proof that other people could verify, but no other person could replicate.

    But you said that even while you’re drunk, you don’t believe that the brain is a quantum computer in a meaningful way. In that case, how can it still be possible that a person can’t be copied for some underlying reason connected to quantum mechanics. Can you suggest what mechanism there can be what stops copying, even in principle?

    3. The problem of personal identity is an interesting and difficult one. It would really be worth to know whether a person can be copied in principle, and what would happen if you copy them, and how you should think about your subjective experiences in that case.

    Some of the examples you bring up don’t seem very useful to me though, even if people can be copied.

    You talk about a criminal threatening to copy a thousand instances of you and torture them. But criminals in the real world already take hostages, sometimes even a thousand of them, and threaten to torture or kill them. And when they do that, that’s an effective technique, and they can influence people with it even if those hostages aren’t copies of them.

    Then you talk about making a hundred temporary copies of yourself to meet a deadline. But even if you can’t copy yourself, couldn’t you just freeze the rest of the earth for a hundred days, or slow it down a hundred times? There are a lot of technical difficulties, but there doesn’t seem to be a theoretical reason why you couldn’t do that, even if the rest of the earth is also quantum mechanical in some meaningful sense.

    You even mention not running yourself for eight years if Trump becomes president. I really don’t understand what that would have to do with copying yourself or quantum mechanics. Even if quantum mechanics stops you from copying yourself, it can’t stop you from not running yourself for eight years.

    4. I personally think that there is no physical reason that would ensure that people can’t be copied. There are a lot of technical difficulties of course, such as that the human body is fragile, and if you want to do complicated operations on it, medicine first needs to figure out a lot of little things to just keep people alive during. In particular, I also think that I can be represented by a huge finite number, and that number can be extracted in principle, but that does not make me depressed. I also don’t believe that the brain is a quantum computer in a meaningful way. But I’m not completely sure about either of those things, so I still think this topic is worth to talk about.

  13. jonas Says:

    raoul ohio, re #11:

    In the context of computations, I’m not sure, but I think it’s folklore. I’d imagine that once people started to work with algorithms and estimate their time complexity, multiple people would soon figure out that some problems can be solved with algorithms that take an exponential time, and for large inputs, those often run for such a long time that it would be impossible to run them to completion even with very efficient future computers. There’s also some sort of arguments about a thermodynamic limitation why you can’t just cram too much information in a small space, but I don’t really understand how that works.

    If you’re not asking about computation or complexity, then the idea that the number of particles in the world is finite and, in modern terminology, only so big that its logarithm is a number we can imagine, that idea goes back to Archimedes. Archimedes has an essay where he does a specific calculation to obtain a limit on how many grains of sands can fit in a sphere whose radius is as large as the distance of Earth from the Sun. (He has to take that distance as a proxy for the size of the world, because at that time they didn’t yet know enough about the rest of the solar system, and especially not about the stars, to do a concrete calculation.) It was a very novel idea at that time, and still quite novel for a while, because people didn’t have a system of exponentials to even talk about numbers as big as that. David Madore’s writeup tells about this in detail.

  14. James Cross Says:

    If I had my own personal integer, would the integer be the same for me at age 5 or age 65?

    Would the integer be the same from moment to moment or constantly changing?

  15. wolfgang Says:

    What if no-cloning is the ultimate law to protect secret ballots ?

    Whoever runs The Matrix wants to make sure that vote-swappers of the future cannot verify with absolute certainty that
    their vote-swap buddy has voted as promised using iPhone 2345 with quantum-cloning cameras ?

  16. fred Says:

    “every time you wanted to run the quantum program, you’d have to make a measurement that destroyed it. So then you’d have to go back and buy a new copy of the program for the next run, and so on.”

    This seems to assume that the “originator” of the non-clonable program can regenerate the program “at will”.
    Therefore to clone the program, one would have to clone the system (brain+tools) that created it.
    So we reduced it to the non-clonability of the human mind?

  17. Scott Says:

    James #14: I said it’s an integer that encodes your entire life history, so clearly, the same from moment to moment.

  18. fred Says:

    I think that ultimately it’s about the fact that, yes, your mind’s trajectory in space/time is a big ass integer, but you just can’t take it out of the system it belongs to, the entire universe, which itself is described by a big ass integer.
    Or, you can’t simulate a system from within itself.
    So, the non-clonability is a bit like saying that a number can’t be irrational and rational at the same time.

  19. Scott Says:

    jonas #12:

      But you said that even while you’re drunk, you don’t believe that the brain is a quantum computer in a meaningful way. In that case, how can it still be possible that a person can’t be copied for some underlying reason connected to quantum mechanics. Can you suggest what mechanism there can be what stops copying, even in principle?

    A system wouldn’t have to behave as a quantum computer in any interesting sense for it to be buffeted around by chaotically-amplified microscopic fluctuations that are ultimately traceable to quantum-mechanical degrees of freedom. (E.g., it wouldn’t have to maintain any long-range entanglement, or use interference to solve problems faster.) For more see GIQTM.

    Compared to quantum copy-protected software, one major difference here is that we don’t need to satisfy any strong security definition (as of course we know that the brain doesn’t)—we just want something to be unpredictable by outsiders in a Knightian sense.

  20. fred Says:

    Scott #19

    But isn’t it true that *any* quantum system is finite. So, to clone a system one could just generate systematically all the possible quantum systems with the same number of states (a gigantic amount), and one of those will be exactly like the original system?
    Sure, there’s a lot of “waste” produced, but if I do this with your brain, I know that one of the systems will be an exact copy of your brain (i.e. same inputs will give same outputs, etc).

  21. BLANDCorporatio Says:

    So the only thing that makes one precious and worthy of moral considerations are glitches. Rather repulsive, imo.

  22. Scott Says:

    Fred #19: This is where it becomes relevant that you don’t have exponential resources in our universe, but have “merely” ~10122 qubits available…

  23. Ian Says:

    Hey Scott, just curious how long I may have to hold my breath to find out if you were going to take Gil’s bait in comment 5 🙂

  24. Scott Says:

    BLAND #20: Why do you choose to call the things that make someone individual “glitches,” rather than (say) “quirks”?

    Ian #22: Nope, not taking that bait. 🙂

  25. BLANDCorporatio Says:

    irt. Scott #23:

    Because it accurately captures what your proposal relies on.

    GITQM posits freebits carried by photons from the Big Bang. So someone living deep enough in a lead planet for a long enough time suddenly isn’t an interesting person anymore because they’ve been disconnected from the Universe’s prime source of static?

    Or the more recent ~requirement that the conscious system be “buffeted around by chaotically-amplified microscopic [quantum] fluctuations”. OTOH it seems plausible the brain is somewhat resilient to such fluctuations (I’m inside EM fields galore right now). But accepting that such fluctuations are what ultimately make one morally worthy, surely having more of them would make us more quirky and interesting and what not, eh? But we have names for such states– “intoxicated” being one– and they are usually not associated with good judgement. As in, if I were high on freebits or whatever, I doubt I would be in a state that should be recognized as competent, and my decisions therein binding, as opposed to the relatively sober state I am in right now.

  26. JimV Says:

    Thanks for another interesting post.

    I’e noticed both in playing and in programming video games, that a small element of unpredictability is crucial to making the game interesting. (How about video pool, you may ask. Yes, I wrote a video pool game and found that it needed a small amount of random friction.) So perhaps universes without the No Cloning theorem are so uninteresting that high-order civilizations never form. (Another semi-anthropic argument.)

  27. adamt Says:


    Since I’m reading Section 2 of Penrose’s new book, “Fashion, Faith and Fantasy in the New Physics of the Universe” right now I wonder how this picture of personal identity would be changed if objective reduction along the lines of Penrose proposal turns out to be true. In this picture a quantum system composing “Scott” over the course of your life would be subject to finite many *real* objective reductions shattering the linear quantum evolution of your personal identity. Doesn’t this mean that it is at least possible that some some multiple reduction of all observables of “Scott” might occur at some time t0 and the state that describes this “Scott” could be described classically and cloned?

    I guess this gets into the interaction between the no cloning theorem and a picture of the universe where the quantum formalism is just an approximation up to some energy/time threshold. What sorts of wrenches get thrown into play then?

  28. adamt Says:

    BLANDCorporatio #25,

    I think you are conflating your own sense of personal identity as a moral person or conscious with Scott’s mere assertion that “you” in some very fundamental universal sense might be unique. All the mystery lies in the “you” unfortunately so this big integer takes the broad view of the entire history of the physical system that nominally describes this “you.”

  29. James Cross Says:

    Scott #17

    I see now. When I was reading history, I was thinking past but the integer represents the entire life past and future.

    So this would imply a deterministic universe where everything from Big Bang to whatever unfolds mechanically. I can’t see having identity in this would amount to any more than being a cog in giant machine. You may be a one of a kind cog but still just a cog with everything in life predetermined. It reminds me somewhat of bad trips on Salvia divinorium.

    See section on Machinescapes

    Or read this account:

    “I resonate with much of what you’ve described, especially that sensation of being squished and pressed by some flipping (or rolling, as I’ve always described it) force, accompanied by a feeling that I’d always been there, things had been this way all along, and that the rest of my life had been the hallucination, one that is only just broken out of.”

  30. JimV Says:

    Shouldn’t the Big Integer also include or be entangled with all the effects of your life? So that all the people who read and enjoy your book and blog are in it, and all the people you teach and all the people taught by them, etc., and all those who use your theorems or extend them?

    Note also (as of course you know) your number is unpredictable until you and all your effects in all the branches of the MWI have been totally forgotten. If finite, it still could be rather impressively large; and maybe the one with biggest Big Number wins.

  31. BLANDCorporatio Says:

    irt. adamt #28:

    Indeed I am, because I understood that as motivation for the whole GITQM thing in the first place. Scott is careful enough to frame GITQM and related postings as speculative answers, but they are answers to this problem (among others): if there’s a thousand of identical yous running around, what are YOU worth?

    Well, my guess would be that if a person is worth anything, it’s not dependent on how many of them there are available. And it’s certainly distasteful to me to think the worth lies in a glitch. (I am aware of the powerlessness of aesthetic arguments when confronted with reality, but GITQM is essentially an aesthetic argument as well, so there.)

  32. BLANDCorporatio Says:

    irt. JimV #26:

    There’s a difference between unpredictability in practice, and its “in principle” variant. Unpredictability in practice is easy, for sufficiently lazy/weak predictors. Classical computers do it all the time with pseudo-randomness.

    And I could see myself being a classical computer thinking it is a person, influenced by what I perceive to be the behavior of other things I believe to be persons when I do my decisions. I see no reason why my decisions should carry more weight or deserve more moral respect if they were also influenced by freebits from space who had nothing else to do with any parties involved, or with which nucleus spontaneously popped that day or some other such arbitrary quantum event.

  33. Michael P Says:

    How would No Cloning, Personal Identity, and all that work in a Many-World interpretation described by Everett and David Deutsch?

  34. Gil Kalai Says:

    Hi Scott, everybody,

    One closely related discussion to the issue of predictability was in my 2012 debate with Aram Harrow (Aram’s third post.) . Aram proposed to identify a noiseless (predictability requires noiselessness ) “imaginary quantum computer” by starting with a noisy one and adding to it all the sources of noise. After a long exchange of comments we came to the conclusion that the only way to do it is to consider the entire universe or something drastic like that. Unlike my previous comment this is not a consequence of the point of view of “no quantum fault tolerance” as  we came to agreement about it.

    That predictability requires noiselessness is discussed in my paper with Guy Kindler (The third point in page 4, and the second bullet in B.1.) It is about noisy BosonSampling but it applies more generally.

    Finally in the expanded version of my recent paper on the quantum computer puzzle, several related things are discussed in Section 7.

  35. Scott Says:

    Michael #33: They work the same as in a non-many-worlds view. Since none of my speculations here (unlike Penrose’s) involve the slightest tampering with the rules of orthodox QM, this entire discussion is strangely unaffected by what you believe or don’t believe about the reality of the other branches. (But see GIQTM if you want a longer answer.)

  36. joe Says:

    Greame Smith has laid down the gauntlet:

    ” People continue to refuse to talk about CTCs consistently. It’s just too much fun to say deep-sounding things!

  37. Charles Stromeyer Jr. Says:

    Might the ability to suprise the universe with a new integer be related to the idea that ‘causal order’ is intrinsically linked to the capacity to make a free choice (i.e., only correlated with events in the future of the lightcone) which is explained more precisely in this 1 page paper by Colbeck & Renner:

  38. Michael P Says:

    Scott #35: I thought Everett view includes cloning of personal identities into a variety of worlds, so to speak.

  39. Scott Says:

    joe #36: Gauntlet accepted, with a side of fries and a large beverage. I will put the mathematical consistency of the CTC Turing machine model we define up against the mathematical consistency of anything whatsoever—and certainly up against the mathematical consistency of quantum field theory, the centerpiece of my friend (or former friend? 😉 ) Graeme’s field of physics. Mohammad, Giulio, and I rigorously define and characterize some interesting complexity classes, full stop. It’s true that the behavior of the computational model becomes rather strange when you consider superpositions or mixtures of inputs in the non-CTC region, and insist that such things be handled within the model itself. So, that’s why we don’t consider that! 🙂

    In any case, the bad behavior of CTCs on mixtures of inputs is by now well-known, and is a necessary consequence of wanting to discuss a world with CTCs in the first place, and can only be avoided by essentially defining CTCs out of existence and replacing them with static quantum advice states (the “solution,” if you can call it that, that Graeme et al. proposed), and was already explicitly addressed in our paper, and is orthogonal to the issue (computability theory of finding fixed points) that we wanted to talk about.

    But all that could be swept aside, if not for Graeme’s snark about “deep-sounding things.” I defy Graeme or anyone else to point to any needless pontification in our paper. The paper is pretty much just pure closed timelike beef: rigorous statements and proofs of results (some of them pretty surprising and nontrivial) characterizing a clear and well-defined set of complexity classes.

  40. Sniffnoy Says:

    Oh, huh, it didn’t even occur to me that, the polynomial-time version having been solved earlier, the computability version hadn’t been done! That’s really neat!

  41. BLANDCorporatio Says:

    irt. Charles Stromeyer Jr. #37:

    That’s a cute definition of freedom.

    My problem with such definitions though is that they imply that the actually important decisions a person makes shouldn’t be free (consider: would you like votes for the PotUS to be uninfluenced by past knowledge of the candidates?).

    On the one hand, one can say so what, obviously things that matter cannot be free. But on the other hand, when freedom [of will] is associated so strongly, in many a conversation, with what makes a person so special and valuable, it seems dangerous to equate the term freedom with something that only trivialities could enjoy.

  42. fred Says:

    Scott #22

    “This is where it becomes relevant that you don’t have exponential resources in our universe, but have “merely” ~10122 qubits available…”

    One problem with this type of discussion is the constant switch between the “theoretical” and the “practical”, whenever convenient.

    The non-cloneable principle doesn’t prevent Apple, Msft, Google from “duplicating” complex lumps of matter so that they all do the same non trivial logical tasks in the exact same manner.

    But then you bring up:

    “A system wouldn’t have to behave as a quantum computer in any interesting sense for it to be buffeted around by chaotically-amplified microscopic fluctuations that are ultimately traceable to quantum-mechanical degrees of freedom.”

    But if what makes you a unique beautiful snow flakes is QM noise, why even bring up the non-clonable principle then?
    The notion of cloning is really non-sensical at that level of detail once you realize that copying any finite region of space/matter gives another region that’s *always* going to be fundamentally different from the original one because they both have to occupy a different position in the universe – one will be slightly closer to the cloning apparatus, one will be slightly closer to Mount Everest, one will be slightly further from the Sirius, etc. They will be bathed by different streams of neutrino, gravitons, photons, etc.
    So any “practical” cloning of a system that’s sensitive to subatomic noise would require to not only clone the subsystem but also its entire environment, forcing you to create a perfect symmetry of the entire universe along a plane (like a mirror).

  43. Charles Stromeyer Jr. Says:

    Regarding comment 41 from Bland Corporation:

    Here is a rigorous mathematical proof of the general Colbeck-Renner Theorem from a different author:

    I think the use of “freedom of choice” is reasonable within this context, but I might be wrong?

  44. Sniffnoy Says:

    Charles: If you’re linking to arXiv, would you mind linking to the abstract rather than directly to the PDF? From the abstract, one can easily click through to the PDF; not so the reverse. And the abstract allows one to do things like see different versions of the paper, search for other things by the same authors, etc. It’s just generally more useful. Thank you!

  45. Daniel Bilar Says:

    Hi Prof. Aaronson

    I thought of your cloning theory musings post when I read this item on Marletto (Oxford) work building on “non-probabilistic superinformation”
    searchign perhaps for a successor to quantum theory.

    TlDR: deterministic theory exhibits unpredictability as a consequence of the impossibility of cloning certain states.

    “There are two things: one, my work shows that unpredictability can arise under deterministic theories, and that it is a direct consequence of the impossibility of cloning certain sets of states,” Marletto told “That unpredictability can arise under deterministic theories may seem a little surprising at first. But the point is that ‘unpredictability’ just means that it is impossible to build a predictor—a machine that would reliably predict the outcome of a single measurement of given observable on a system prepared in a given state. This impossibility is just like that of the no-cloning theorem, and does not require any probabilistic structure. Probabilities, instead, come into play only when considering patterns occurring in repeated experiments.

    “Two, this work updates and generalizes the decision-theory approach to the Born rule in quantum theory, which was proposed to reconcile deterministic unitary quantum theory without the Born rule with the appearance of stochasticity in quantum experiments. In particular, it shows that most of the assumptions of that approach are not, as previously thought, subjective decision-theoretic axioms, but follow from physical properties of superinformation theories. It also establishes under what conditions superinformation theories support that argument, thereby defining a class of theoretical possibilities in which the successor of quantum theory might be sought.”

    Read more at:

  46. BLANDCorporatio Says:

    irt. Charles Stromeyer Jr. #43:

    I’m not claiming the one-page paper you linked to was “wrong” (if there are problems with it and the paper it is meant to buttress, I’m not sophisticated enough to discern).

    My problem with this is one of terminology, mixed with a paranoia about a bit of “motte and bailey” going on.

    The one-page paper, and the new one you link to, seem to be fairly rigorous mathematical arguments about well-defined situations. The papers themselves may be correct.

    My problem is what I suspect to be a Gray tendency to be charmed by Mathematics’ apparent ability to tackle deep meaningful questions about the human condition … as long as we ignore all the complications that make those questions interesting and restrict ourselves to toy models that our analysis can make mince meat out of.

    As in, I’m fearful of an extrapolation from a jargon redefinition of “free” in the context of quantum experiments, to “free” in the context of such things as free will and so on.

    And the reason I find this trivializing is that when people fight for freedom, they don’t fight for your right to choose the mocha instead of your usual chai latte because a neutrino pinged your brain or whatever. The notion of freedom that was interesting/relevant enough to fight about concerns choices such as who should our leader be, whether a pregnancy should be continued, or who to marry.

    And while that last one isn’t usually associated with rational conscious calculation, I think it has more in common with a deliberation process informed by experience than it does with a quantum coin toss. I mean, like, I love you with all my heart, honey, but it’s not you, it’s that an alpha decay happened in my brain– how absurd is that.

    I thank you for the second link however. I will read it soon, because of a layman’s interest in things physics. It’s just that the interest turns skeptical when such things get quoted near discussions about consciousness or free will, or any other favorite topic of Grays everywhere.

  47. JimV Says:

    BC @32: Thanks for the reply, although it came across as a non sequitur. There were no morals involved in my speculation, and the difference between pseudo-random action (as in some, but not necessarily all computer games) and true, non-cyclic randomness is a matter of scale. E.g., it is possible that a universe in which only pseudo-random events were possible would not support the full range of complexity in biological evolution that this universe does. However, in a computer game that takes over only a few days, pseudo-randomness may be sufficient.

    Note: I am using the term pseudo-randomness as I thought you were using it, but am not sure it is the correct technical term.

  48. BLANDCorporatio Says:

    irt. JimV #47:

    Hey now. There’s a lot of stuff on the table here, such as predictability, personal identity, freedom (and quantum money, and security and so on) … I just related what you said (games with randomness are more interesting) to what I care about (the value of unpredictability for a person’s moral standing), because I thought you posted the first to support the second paranthesis.

    It’s my bad if I saw too much though.

    About whether it’s possible that a universe without a source of true randomness may be more limited in biological complexity, yes it’s possible. OTOH chaos doesn’t need non-determinism, and can mix things up pretty well, which makes the contrary of your statement, “it is possible that a universe in which only pseudo-random events were possible would support the full range of complexity in biological evolution that our universe does”, also plausible. After all, what would be the grounds to choose between the two?

  49. Anonymous Mind Says:

    Maybe the Higgs boson is a soul particle — seat of your identity. A Higgs boson has the superpower of giving mass and could travel though the brain giving and getting mass (memories). Maybe it even travels out of the brain and comes back — out of body.

    If a fundamental particle can be conscious with free will, I think its parent, the finite universe, is also conscious with free will.

    Rebecca Newberger Goldstein and other philosophers were joking about the Higgs boson being a soul in a youtube video.

    If the soul is small like a Higgs Boson it could be given a new home in much more durable ever improving robot body.

  50. Ben Standeven Says:

    @ Charles Stromeyer Jr. #43, BLANDCorporatio #46:

    Note that Scott’s freebits are actually not free in the sense of Colbeck and Renner, since their observed state is correlated with their state at any earlier time.

  51. Concerned 3rd Party Says:

    Hi Prof.Aaronson et al.,

    First time commenter here…reading through GIQTM at the moment…

    “Look—I don’t know if any of you are like me, and have ever gotten depressed by reflecting that all of your life experiences, all your joys and sorrows and loves and losses, every itch and flick of your finger, could in principle be encoded by a huge but finite string of bits, and therefore by a single positive integer. (Really? No one else gets depressed about that?) It’s kind of like: given that this integer has existed since before there was a universe, and will continue to exist after the universe has degenerated into a thin gruel of radiation, what’s the point of even going through the motions? You know?”

    I am not entirely sure this picture, if it were true, would lead to depression. It actually sounds beautiful to me…
    Is it due to a loss of the sense of there being some value to personal identity and subjective experience while we are around? Maybe such evaluation is within the Big Integer too, and knowing all the ingredients of the finest Belgian chocolate would not diminish the experience of tasting it?

    As does your poetic assertion regarding the eternal(ly recurring) marriage of transcience and uniqueness…
    And I do not know enough to be make a judgment of truth here…
    Apologies if that was entirely unhelpful…

  52. mjgeddes Says:

    Hello again Scott,

    Since what I call ‘my big breakthrough’ on the consciousness puzzle a few months back, my views seem to have been rapidly crystallizing in a flurry of insights as I move from the initial vague intuitions towards something more precise and formal.

    I feel I’m really starting to close in definite answers – my intellectual wagons are circling around the concepts of ‘personal identity’ and ‘time’ and the circles are getting smaller as I tighten the noose…

    You see, I really think the puzzles of ‘consciousness’, ‘time’ , ‘personal identity’ and ‘free will’ are all strongly related, in the sense that if you fully understood one of them, you’d grasp the secret to all of them. I get the impression you think the same thing, although you’ve never explicitly said so.

    Some of my views of the ‘triple-aspect’ theory of reality have crystallized, so I have to say a few things about that just to update what i said in the earlier thread.

    Earlier I was still considering some weak forms of dualism. My update is that I now fully reject all forms of dualism as wrong-headed. I had said that there were 3 fundamental properties of reality
    (information, matter (fields), consciousness).

    I now think there is *one* fundamental element of reality, which I call the ‘Info-Cognition Field’. This manifests itself as 3 *apparent* properties (information, fields and consciousness), but there’s really only 1 thing there (neutral monism). It’s analogous to the way that ‘electromagnetism’ manifests as ‘electricity’ and ‘magnetism’.

    I had earlier made the claim that consciousness was ‘the arrow of time’. I think this was right on the money. The secret is ‘time’!

    I now think that ‘qualia’ is a basic fundamental unit of ‘time’ or ‘causality’. It’s a unit of ‘causal continuity’ and this also defines an ‘identity’.

    You see, every object in the universe has an ‘identity’, which we can *define* as the degree of localization in space and time. In other words, the ‘identity’ of the object is only defined to the extent that we can ‘pin down’ the location of the object in the multiverse.

    So that then is ‘consciousness’ (or ‘qualia’). It’s the fundamental unit of ‘identity’ or ‘causal continuity’, present to some degree in all things.

    More complex identities (or consciousness) are built up from simpler ones, exactly analogous to the way that more complex forms of matter are built from simpler ones.

    The ‘combination problem’ for qualia is resolved by seeing that you ‘paste together’ smaller units of ‘causal continuity’ into larger ones…. a unit of ‘causal continuity’ is simply a thermodynamic ‘propensity’ for events to happen in one way rather than another way.

    Decision theory can be interpreted as a theory of how to ‘paste together’ different preferences or ‘qualia’ into a coherent whole. So ‘decision theory’ is really a high-level theory of ‘qualia’! Similarly, at an each higher level of abstraction, values or ethics would emerge as ‘qualia’ get pasted together into ever more complex minds. In this picture, axiology (theory of values) emerges as a higher-level form of decision theory.

    What is the relationship between information, matter and consciousness?

    Each ‘contains’ the other 2, just as I thought. Each property can be *approximately* defined as a combination of the other 2. They’re not separate, but they’re not identical either. Reductionism is an *approximation* , not an absolute principle. So the 3 properties (information, matter and consciousness) are still ontologically distinct, even though they can be approximately defined in terms of each other.

    How does all this relate to your ideas as regards GIQTM? I think my ideas match up well with yours!

    If qualia (consciousness) is indeed the basic unit of ‘time’ or ‘causal continuity’, and is a ‘thermodynamic propensity’, then indeed, consciousness directly participates in the thermodynamic arrow of time, just as you claim!

    Furthermore, if consciousness is also the basic unit of ‘identity’, then again, the puzzle of free will would be directly linked to that of ‘identity’.

    On the basis of this, I regard GIQTM as a real candidate for a possible solution to these puzzles!

    However, although I like the general ideas in GIQTM, I’m still skeptical of the specific QM mechanism proposed (no-cloning).

    Something still isn’t quite right. In particular, I still really can’t see why ‘identity’ should be unique…the problem is that according to MWI of QM, we are being ‘forked off’ all the time into multiple copies, so doesn’t this suggest that low-level quantum states aren’t really that important for personal identity?

    But Scott, I certainly think you are on the right track overall.

  53. jonas Says:

    Sniffnoy: the first page of the PDF has a link back to the abstract page on ArXiv. Most PDFs on ArXiv have such a link.

  54. Psycicle Says:

    I’m not sure whether or not you’ve seen this. In short, no-cloning can be broken in exponential time when the quantum state is interacting with the environment.

    I’m not sure how the contents of this article affect or don’t affect your speculations regarding no-cloning and personal identity.

  55. Jay Says:

    Psyclicle #54,

    This is indeed very interesting, and I wish Scott would discuss it for its own. However, I’m not convinced it causes any problem for the freebit picture.

    More specifically, what Craig Gidney showed is that there exists some combinations of (quantum computations+partial measurements) which allow guessing the quantum state post-measurement even without knowing the initial state. However, he also noted that this is not true for all computations.

    In terms of the freebit picture, if you have long-lived freebits that contribute only in some specific situations (say, when more freebits have just arrived or when there’s some important decision to make or both), then this completly escapes Craig Gidney’s demonstration.

  56. Jon K. Says:


    Thanks for sharing the Constructor Theory link, Daniel Bilar. I think it will have more appeal to me than most others around here. Questioning Bell’s theorem has cause some kerfuffles in the past. 😉

  57. Marnie Dunsmore Says:

  58. Marnie Dunsmore Says:

  59. Neel Krishnaswami Says:

    What if unclonability is our universe’s way around the paradoxes of self-reference, like the unsolvability of the halting problem and Gödel’s Incompleteness Theorem?

    In a certain sense, this is a well-known result (in the mathematical sense of “well-known” meaning “there are people who know it”). (I say “certain sense”, because of course this is an informal question, and there are multiple ways to formalize it.)

    The paradoxes of self-reference are all instances of the Lawvere fixed point theorem, which states that in a cartesian closed category, if there is a point-surjective morphism A -> (X^A), then every endomorphism X -> X has a fixed point. “Clonability” (which logicians call “contraction”) amounts to the condition that the category is cartesian. If you drop that condition, then his fixed-point theorem doesn’t go through.

    In fact, you can prove that for a set theory over linear logic (ie, monoidal closed categories), the naive comprehension axiom is sound. See (e.g.) Kazushige Terui’s Light Affine Set Theory: A Naive Set Theory of Polynomial Time.

    Basically, it takes all three of self-reference, contraction, and closure to get the fixed point theorem to go through. Drop any one of them and the proof fails. For the proof theorist, it is certainly most elegant to drop contraction, so it is nice the universe agrees.

  60. Alex B. Aible Says:

    What exactly is the relationship between the no-cloning theorem and the supposed empty set?

  61. Darrell Burgan Says:

    Doesn’t the no-cloning theorem actually suggest that each of us cannot possibly be reduced to a unique integers? Finite integers can certainly be copied. If no-cloning is true, then there is some bit of us that cannot be replicated, no matter how hard we try.

  62. Giorgio Castriota Scanderbeg Says:

    So, copying yourself is like xeroxing a toilet-paper? It definitively does the job!

    Hi, Scott I just finished a ten full days readings of all your posts and I am in the position to say that your is the best blog in the blogosphere. Great.

    (Yes I know it: 10 days to produce an unwitty Steven Wright like jokes? But i’m very proud to have understood just 1/100000 of the blog’s contents and it is what I deserve.)

    PS: next copy of myself, I swear, I’ll not chase blonde girls in my high school, but unknown x on the blackboard.

  63. Job Says:

    Isn’t the no-cloning theorem really the no-knowing-that-you-cloned theorem?

    You might be able to inadvertently, with low probability, be able to clone a quantum state, but you wouldn’t be able to prove it?

    At best, it’s the universe’s way to ensure the plausibility of personal identity and free will.

    But you know it’s all just a game.

  64. BLANDCorporatio Says:

    irt. Darrell Burgan #61:

    “Doesn’t the no-cloning theorem actually suggest that each of us cannot possibly be reduced to a unique integers?”

    Not necessarily. The No-cloning theorem merely states that you cannot copy an unknown quantum state. It is conceivable that a quantum state could be represented by some integer, but since you don’t know what it is, all’s according to theorem.

    (If you do know the quantum state/integer, then you can simply prepare copies of it. This is not controversial and can be done all the time for quantum states.)

  65. BLANDCorporatio Says:

    irt. Job #63:

    I suppose you can look at it like this: you can prepare any quantum state you wish, so maybe the one you prepare happens to equal the state of some system whose state you don’t know.

    As to whether it’s the universe’s way to ensure [the plausibility of] free will and personal identity, meh. People just don’t want to abandon animistic instincts, apparently.

  66. Jr Says:

    What is so evil about copy-protection? I mean, it is a pain in the #!$£ that you can not transfer your programs when you change computers but how are you going to enable software developers to make money without it?

  67. John Sidles Says:

    As the comments on this fine Shtetl Optimized topic wind down, perhaps it would not be amiss to cite a few further references (all of which can be found on-line as free-as-in-freedom preprints).

    In brief: (1) William Wootters’ and Wojciech Zurek’s “The no-cloning theorem” (2009) emphasize the interlocking roles of causality, relativity, gauge-invariance, and entanglement; (2) Aaron Fenyes’ “Limitations on cloning in classical mechanics” carefully extends definitions of cloning, and (3) Nicholas Teh’s “On classical cloning and no-cloning” (2012), which is the longest of the articles, emphasizes the contrasting roles of symplectic versus metric isomorphism in the context of Kahlerian/Hamiltonian dynamical flows.

    These authors provide abundant references and plenty of concrete suggestions for further research; moreover their suggestions have the great merit of being reasonably student-accessible, and finally these articles, in aggregate, beautifully illustrate the synergistic roles of physicality, naturality, and universality in quantum research.

    Specifically in regard to the feasibility of demonstrating quantum superiority, versus the infeasibility of said demonstration, it is reasonable (as it seems to me) to foresee that considerable advances will be required in our understanding of quantum cloning — including advances along the specific lines that Wooters, Zurek, Fenyes, and Teh survey — as a precondition for there to be any very substantial likelihood of settling this question experimentally and/or theoretically and/or mathematically, anytime soon.

    That’s one reason (among many) why it’s very good news (as it seems to me), for quantum researchers of all levels and all subdisciplines, that this marvelously stimulating no-cloning literature is out there.

  68. Scott Says:

    Job #63: Yes, of course, you can clone with some small success probability (which you can prove decreases exponentially with the number of qubits in the state you’re trying to clone). For a reasonably large state (say, >1000 qubits), the success probability is so tiny as to be irrelevant in practice.

  69. Scott Says:

    Darrell #61: No, if each of us is described by a finite number of qubits, then there is a positive integer that captures whatever there is to know about you to any desired precision. But that integer might not be fully knowable by anyone else, even given arbitrarily advanced future technology, since the measurements needed to recover it might not be compatible with quantum mechanics.

  70. Scott Says:

    Alex #60: Sorry, I don’t understand the question. Who claimed there was a relationship between the No-Cloning Theorem and the empty set, and in what context?

    (The main relation that I can think of offhand: according to the No-Cloning Theorem, the set of quantum operations that succeed in cloning an unknown state, is equal to the empty set. 🙂 )

  71. jonas Says:

    Scott Re #69: what do you mean by “anyone else”? Else compared to who or what?

    Jr Re #66: there were at least three ways you can want to pay software developers even if software didn’t have legal or technical copy protection.

    1. You pay software developers to develop software that is so useful for you that you are willing to pay for it even if other people can use the same software. There will never come a time when there is already perfect software for everything you want to do.

    2. You pay software developers to develop software, and run the software on your own computers as services such that other people physically don’t have access to the program, only some of its input and output.

    3. Or you pay developers to provide understanding or support for existing software when you have the program and can in theory decode it, but don’t have the time or ability to do it.

    In all of these cases, “you” can mean a company, not an individual, and “develop software” can mean modifying existing software.

  72. v Says:

    Excuse me if I’m overly blunt here, but… I’m surprised you’re so late to this kind of philosophy (regarding the last part of your post, about quantum no-cloning and human minds etc)

    To me it seems quite clear that there are random and uncontrollable/unpredictable fluctuations in our brains on a much bigger scale than the quantum scale. And I think their effect (and the effects on all smaller levels, including quantum effects) is more or less zeroed out (be it by some error-correcting mechanisms or more likely just by the very nature of randomness to average itself out).
    And even if you could convince me that through some freak chaos-theory butterfly effect a certain bump in the brownian motion of two atoms or a certain collapse of the spin of an electron caused me to change my decision on a question, I would still think that is the exception, and not the rule of how my brain works. I want to think my brain is a consistent, logical, and so completely deterministic and predictable system instead of a set or quantum dice or any other “magic” that we will be forever unable to understand.

  73. Peter J Says:

    Scott, you wrote:

    “I also think that the Penrose-Lucas argument, based on Gödel’s Theorem, for why the brain has to work that way is fundamentally flawed.”

    Could you please elaborate a little bit on this? What is your main reason for dismissing the Penrose-Lucas argument? Many people have raised objections to the Lucas-Penrose argument (e.g. Bringsjord & Xiao 2000; LaForte, Hayes and Ford 1998). However, different critics usually attack different points in the argument allowing that the points objected to by other critics are in fact correct. Thus, the opponents are often contradicting each other and it appears to me that the Gödelian argument has not been refuted. What is your opinion about that?

  74. Scott Says:

    Peter #73: I certainly don’t agree with everything that’s been said against the Penrose-Lucas argument, but I also think that most of the usual objections to it are complementary to each other, emphasizing different aspects of the same fatal flaw.

    The simplest way to see the problem, is the one Alan Turing already pointed out in 1950: namely, there seems to be no reason whatsoever, neither from first principles nor from the actual, hit-or-miss history of set theory, to imagine that human being have a knowably sound procedure for deciding the consistency of an arbitrary formal system. So an AI that passes the Turing Test wouldn’t need such a procedure either. But then there’s no Gödelian obstruction to such an AI being written.

    Or, to put the same point another way: there seems to be no mathematical obstruction to an AI behaving indistinguishably from Erdöos or Grothendieck or Gödel himself or any other human mathematician who ever existed. The only obstruction is to an AI behaving like a hypothetical omniscient and infallible mathematician.

    Or, to put the same point a third way: if we insisted on faulting an AI for not being able to assent to things like,

    “The AI cannot assent to the truth of this mathematical sentence,”

    then to be consistent, we should also build a wiring diagram for a given human mathematician’s brain, and then use that wiring diagram to build a mathematical sentence corresponding to

    “The human mathematician cannot assent to the truth of this mathematical sentence.”

    In no case, without saying anything more about the nature of the human brain, etc., have we said anything to differentiate humans from machines, as long as we’re scrupulous about applying the same rules to both.