One of the stated goals of quantum foundations is to find a set of intuitive physical principles, that can be stated in plain language, from which the essential structure of quantum mechanics can be derived.
So what exactly is wrong with the axioms proposed by Chiribella et. al. in arXiv:1011.6451 ? Loosely speaking, the principles state that information should be localised in space and time, that systems should be able to encode information about each other, and that every process should in principle be reversible, so that information is conserved. The axioms can all be explained using ordinary language, as demonstrated in the sister paper arXiv:1209.5533. They all pertain directly to the elements of human experience, namely, what real experimenters ought to be able to do with the systems in their laboratories. And they all seem quite reasonable, so that it is easy to accept their truth. This is essential, because it means that the apparently counter intuitive behaviour of QM is directly derivable from intuitive principles, much as the counter intuitive aspects of special relativity follow as logical consequences of its two intuitive axioms, the constancy of the speed of light and the relativity principle. Given these features, maybe we can finally say that quantum mechanics makes sense: it is the only way that the laws of physics can lead to a sensible model of information storage and communication!
Let me run through the axioms briefly (note to the wise: I take the `causality’ axiom as implicit, and I’ve changed some of the names to make them sound nicer). I’ll assume the reader is familiar with the distinction between pure states and mixed states, but here is a brief summary. Roughly, a pure state describes a system about which you have maximum information, whereas a mixed state can be interpreted as uncertainty about which pure state the system is really in. Importantly, a pure state does not need to determine the outcomes to every measurement that could be performed on it: even though it contains maximal information about the state, it might only specify the probabilities of what will happen in any given experiment. This is what we mean when we say a theory is `probabilistic’.
First axiom (Distinguishability): if there is a mixed state, for which there is at least one pure state that it cannot possibly be with any probability, then the mixed state must be perfectly distinguishable from some other state (presumably, the aforementioned one). It is hard to imagine how this rule could fail: if I have a bag that contains either a spider or a fly with some probability, I should have no problem distinguishing it from a bag that contains a snake. On the other hand, I can’t so easily tell it apart from another bag that simply contains a fly (at least not in a single trial of the experiment).
Second axiom (Compression): If a system contains any redundant information or `extra space’, it should be possible to encode it in a smaller system such that the information can be perfectly retrieved. For example, suppose I have a badly edited book containing multiple copies of some pages, and a few blank pages at the end. I should be able to store all of the information written in the book in a much smaller book, without losing any information, just by removing the redundant copies and blank pages. Moreover, I should be able to recover the original book by copying pages and adding blank pages as needed. This seems like a pretty intuitive and essential feature of the way information is encoded in physical systems.
Third axiom (Locality of information): If I have a joint system (say, of two particles) that can be in one of two different states, then I should be able to distinguish the two different states over many trials, by performing only local measurements on each individual particle and using classical communication. For example, we allow the local measurements performed on one particle to depend on the outcomes of the local measurements on the other particle. On the other hand, we do not need to make use of any other shared resources (like a second set of correlated particles) in order to distinguish the states. I must admit, out of all the axioms, this one seems the hardest to justify intuitively. What indeed is so special about local operations and classical communication that it should be sufficient to tell different states apart? Why can’t we imagine a world in which the only way to distinguish two states of a joint system is to make use of some other joint system? But let us put this issue aside for the moment.
Fourth axiom (Locality of ignorance): If I have two particles in a joint state that is pure (i.e. I have maximal information about it) and if I measure one of them and find it in a pure state, the axiom states that the other particle must also be in a pure state. This makes sense: if I do a measurement on one subsystem of a pure state that results in still having maximal information about that subsystem, I should not lose any information about the other subsystems during the process. Learning new information about one part of a system should not make me more ignorant of the other parts.
So far, all of the axioms described above are satisfied by classical and quantum information theory. Therefore, at the very least, if any of these axioms do not seem intuitive, it is only because we have not sufficiently well developed our intuitions about classical physics, so it cannot really be taken as a fault of the axioms themselves (which is why I am not so concerned about the detailed justification for axiom 3). The interesting axiom is the last one, `purification’, which holds in quantum physics but not in probabilistic classical physics.
Fifth axiom (Conservation of information) [aka the purification postulate]: Every mixed state of a system can be obtained by starting with several systems in a joint pure state, and then discarding or ignoring all except for the system in question. Thus, the mixedness of any state can be interpreted as ignorance of some other correlated states. Furthermore, we require that the purification be essentially unique: all possible pure states of the total set of systems that do the job must be convertible into one another by reversible transformations.
As stated above, it is not so clear why this property should hold in the world. However, it makes more sense if we consider one of its consequences: every irreversible, probabilistic process can be obtained from a reversible process involving additional systems, which are then ignored. In the same way that statistical mechanics allows us to imagine that we could un-scramble an egg, if only we had complete information about its individual atoms and the power to re-arrange them, the purification postulate says that everything that occurs in nature can be un-done in principle, if we have sufficient resources and information. Another way of stating this is that the loss of information that occurs in a probabilistic process is only apparent: in principle the information is conserved somewhere in the universe and is never lost, even though we might not have direct access to it. The `missing information’ in a mixed state is never lost forever, but can always be accessed by some observer, at least in principle.
It is curious that probabilistic classical physics does not obey this property. Surely it seems reasonable to expect that one could construct a probabilistic classical theory in which information is ultimately conserved! In fact, if one attempts this, one arrives at a theory of deterministic classical physics. In such a theory, having maximal knowledge of a state (i.e. the state is pure) further implies that one can perfectly predict the outcome of any measurement on the state, but this means the theory is no longer probabilistic. Indeed, for a classical theory to be probabilistic in the sense that we have defined the term, it necessarily allows processes in which information is irretrievably lost, violating the spirit of the purification postulate.
In conclusion, I’d say this is pretty close to the mystical “Zing” that we were looking for: quantum mechanics is the only reasonable theory in which processes can be inherently probabilistic while at the same time conserving information.
“I know who I WAS when I got up this morning, but I think I must have been changed several times since then.”
– Alice Through the Looking-Glass
Professor Lee Tsung gripped the edge of the sink and stared into the eyes of her reflection. In the Physics department’s toilets, all was silent, except for the dripping of a single tap, and her terse breathing. Finally, the door swung open and someone else entered, breaking the spell. The intruder was momentarily baffled by the end of a phrase that Lee whispered to herself on her way out. Something about her eyes being switched.
“I’m a mess,” she confessed later to Yang Chen, her friend and colleague in the particle physics department. Both women had been offered jobs at the American Institute of Particle Physics at the same time, and had become close friends and collaborators in their work on extensions to the Standard Model. Lee Tsung’s ambition and penetrating vision had overflowed into her personal life. It infiltrated her mannerisms, giving her nervous ticks, fiery and unstable relationships and a series of dramatic break-downs and comebacks that littered her increasingly illustrious career. Chen was the antidote to Lee’s anxious and hyperactive working style; calm, meditative and thorough, her students’ biggest complaint was that she was boring. But united by a common goal, the two scientists had worked perfectly together to uncover some of the deepest secrets of nature, pioneering the field of research on neutrino oscillations and being heavily involved with experimentalists and theoreticians in the recently successful hunt for the Higgs Boson.
“I have to quit physics. I can’t look at another textbook. I don’t dare.”
Chen was visibly shocked. In all of the ups and downs she had experienced with Tsung, the idea of quitting physics altogether had never arisen.
“What’s going on? What happened?”
Lee fixed her friend with a two-colour gaze. Her eyes, normally hidden in brooding black shadows beneath her eyebrows, now reflected the dim light of the university cafe. One eye was green and the other brown.
“You’ll think I’m crazy.”
“I already do. You’ve got nothing to lose.”
Lee studied her friend a moment longer, then sighed in resignation.
“Well if I don’t tell you, I suppose I’ll go mad anyway. But before I tell you, you have to promise me something. Promise me that you won’t tell me, not even drop a hint, which way the bias went in the Wu experiment.”
“What, you mean the parity violation experiment?”
“Don’t even talk about it! It is too dangerous. You might let something slip. I don’t want to know anything about their result.”
“But you already know the result — you lectured on particle theory back when you were an associate professor.”
“Shush already! Just listen. Last night I was working late at the cyclotron. Nothing to do with the machine itself, I was just hanging out in the lab trying to finish these damn calculations — that blasted theoretical paper on symmetry breaking. Anyway, I could barely keep my eyes open. You know how the craziest ideas always come just when you are dozing off to sleep? Well, I had one of those, and it was, –” Lee broke off to give a short laugh, “– well, it was as crazy as they come. I was thinking, isn’t it weird that the string theorists tell us every particle has a heavier super-partner, so that their calculations balance out. Well, why don’t we do the same thing for parity?”
“You lost me already.”
“You know, what if there is another universe that exists, identical to ours in every way — except that it is flipped. Into its mirror image.”
“Like in Alice through the looking-glass?”
“Exactly. In theory, there is nothing against having matter that is ordinary in every respect, except with left and right handedness reversed. The only question would be, if such a mirror-image universe existed, where is it? So I did some back-of-the-envelope calculations and found out that the coupling between mirror-matter and ordinary matter would be extremely weak. The two worlds could co-exist side by side, and we wouldn’t even know!”
Chen stirred her coffee, looking almost bored, but she didn’t drink it for a full minute. In any case, it didn’t need stirring — there was no sugar in it. Lee could almost see her friend’s mind working. At last she said:
“That’s pretty unlikely. We have some really sensitive instruments downstairs, and some really high energies. Are you telling me even they wouldn’t be able to pick something up?”
Lee’s eyes glittered with excitement.
“Exactly! That’s what I thought. It turns out that, with just a small modification of the collider experiment — the one we’re running right this minute — we could instantiate a reaction between our universe and the mirror universe. We could even exchange a substantial amount of matter between the two worlds!”
Chen’s looked up sharply.
“Lee,” she said, “Please tell me you didn’t already do it.”
Lee shrugged guiltily.
“It only cost them a few hours of data. Nobody will care – they’ll assume the diversion of the beam was due to a mistake by one of the grad students. But it worked Chen! I opened up the mirror world.”
“That’s great! You’ll be famous!”
But Lee stared down at her fingers, the nails chewed raw, and said nothing.
“That’s a Nobel prize right there! Why are you freaking out?”
“Just wait. We’re about to get to the really weird part,” said Lee.
“At first I thought the experiment wasn’t working. I had expected to see a definite interaction region, generating all kinds of particles. But I guess I hadn’t really worked out the details of what to expect. Nothing seemed to be happening to the beam. I was puzzled for a few seconds. But then there was a weird change in the light of the whole building – it seemed to get slightly brighter. And everything doubled — even the humming of the machine seemed to get twice as loud. And the numbers coming up were really bizarre. I started seeing not just the normal data stream, but additional characters superimposed on top. I don’t know that much about the displays we have, but I’m pretty sure they can’t make figures like that.”
“What do you mean?”
“They were backwards. You know, like when you try to read a book in the mirror. When I looked at the clock on the wall, it had six hands, and the two second-hands were ticking in opposite directions. The three wrong hands seemed to flicker in and out of existence, along with the extra light and sound that was coming through. “
“Wait a second,” said Chen breathlessly, “you’re saying the interaction region encompassed the whole building?”
“Yes. Well, at least the lab and most of the surrounding infrastructure. I don’t know if it extended outside. Nobody would have noticed, I think. It was 3am.”
Chen exhaled and leaned back in her chair.
“Wow. Okay, that does sound pretty nuts. In fact, it sounds just like a crazy dream. Have you seen a therapist?”
“It felt pretty real to me. And it gets crazier. I got this overwhelming sense of dread. Fear, like you can’t imagine. I couldn’t figure out where it was coming from, and then I realised that my PC screen had gone dark, and I could see my own reflection in it.”
“Not just everything in the room, but me too — I was also doubled up. There were two versions of me coexisting at once. One of them was facing me, but the other, sitting in the same place — I could only see the back of her head. I don’t know why I felt so afraid. But I did.”
“But if the two worlds were really superimposed on each other like you said, and the interaction region was so large, than the coupling must have still been very weak. Probably mostly electromagnetic. So it would have ended the party pretty fast, right? And with a very low rate of matter exchange.”
“Yes, that’s what I expected too. But it didn’t happen. The double-images persisted for at least a minute. It seemed like forever, just me sitting there staring at the back of my own head bizarrely superimposed on my face. I kept wondering, if I were in the other world, would I also just be sitting here? But it was the strangest thing … it was quiet enough that I could hear us both breathing. Her and me — or me and me, I suppose. But I could also hear buttons being pressed at the console, very slowly and softly. I recognised the `click’ they made. It was unmistakable. I swear she had an arm up somewhere where I couldn’t see in the reflection, controlling the beam.”
“Wait. What does that mean? What are you saying?”
Lee leaned forward, almost hesitating to speak. When she did, it was in a low voice, uncharacteristic of her.
“Don’t play dumb Chen. I’m saying, maybe that interaction was exactly as strong as I expected. But maybe all the matter exchange was happening at a single location. It’s against protocol, but I know that in principle we can localise the beam almost anywhere within the facility … including the console where I was sitting.”
“You think your mirror-self was focusing the interaction on herself? On you? Swapping your matter with hers between the two universes?”
“Well, I don’t know really – I sort of blacked out. I woke up after 5am, slumped at the desk by the beam controls, drooling like a baby. First thing I did was look at the clock. The second hand was moving clockwise, like always. I checked every document I could find, it was all written left to right. I still have the birthmark on my left hand, not the right. You can verify that.”
Lee held up her hand. Chen said:
“So, everything’s fine then! You’re not in the mirror world. You’re in the real world. I’m the real me, you’re the real you. And you’ll be famous for discovering mirror matter!”
Lee shook her head.
“There’s more. I was looking at myself in the mirror just before I called you today. Staring at my eyes. Of course, as you might guess, the left one was still brown and the right one was still green, as it always was. But a thought occurred to me. These eyes being examined — they themselves are the examiners. The world looks the right way around, alright, but what if I’m seeing it through mirrored eyes? Sure, when you look in the mirror everything looks flipped left and right from your perspective. But what if your perspective also gets flipped?”
“You mean, if a mirror image reads a book, does it look backwards to her?”
“Exactly. You look at your reflection and you see its left and right hands are switched. But if you point to the hand that you think is your right hand, the image points to its left hand. The image perceives it’s left hand as its right hand. It looks at you, and from its perspective, you are the one who has got it wrong. The mirror image can’t tell that it is a mirror image.”
Chen was silent.
“Imagine that, next time you wake up, you have a fifty-fifty chance of waking up as your own mirror image, in the mirror universe. Could you leave any kind of trace or signal for yourself that you could use to tell if it happened or not? You can try to use books, clocks, birthmarks or even tattoo yourself with big signs saying `left hand’ and `right hand’ — but it won’t do any good, because the switch is always compensated by your own switch in perspective. You lose your reference point.”
“Come on. There has to be some way to tell. Gyroscopes? Spinning stars? Light polarisation?”
Lee shook her head.
“Not quite. We know that classical physics is parity invariant. Newton’s equations, even electromagnetism, all invariant under flipping left and right. Think more fundamental. More modern.”
Chen sucked on the end of her spoon, frowning at the ceiling of the cafe. She still hadn’t touched her coffee. Then she gasped:
“Beta decay! Weak interactions violate parity! You could just dig up the paper by Wu and check whether the direction of emission was — mmf!”
Lee lunged over the table, upsetting a glass of water, to block Chen’s mouth forcibly with her hand.
“Sorry!” Chen whispered breathlessly. Lee glared at her.
“I told you! I don’t want to know which way those experiments came out.”
There was a pause as a waiter came to mop up the spill, glancing nervously at the two women before moving on.
“But why not?” said Chen when he was gone.
“It will resolve your question once and for all. In fact, –” she rummaged around in her bag, “I have my lecture notes here that I prepared for the particle physics course. It’s on page 68 … or maybe 69. One of those.”
She carefully placed the paper on the table, her lips pursed to prove to Lee that she wasn’t going to accidentally mention the answer. Lee took the paper and placed her coffee cup solidly on top.
“You have to understand,” said Lee, “what it would mean if I was right. If I find out that the direction of parity violation in this universe is different to what it is in my memory, that means that I live in the wrong world. Could I live with that knowledge?”
“Why not? You said everything is the same between the worlds, right? Who cares if your left hand turned into your right?”
“Yes. That’s the point. Doesn’t it all seem rather convenient? That the mirror world should exactly match our universe, in all details, apart from things like the direction of beta decay. Don’t forget, we are talking about physics here. Simply reversing the handedness of matter doesn’t say anything about what kind of world is constructed out of it. In all probability, I should have coupled through to just empty space — after all, most of the universe is empty space. Why should the orbits of the two Earth’s match up so precisely, and their rotations, right down to the location of my laboratory? Think about it Chen. What kind of powerful force could arrange everything just so that, if one person should find a way to open up a gateway between the worlds, it would be possible to switch her with her mirror copy, and she wouldn’t notice a thing?”
“I don’t know, Lee!” said Chen, now visibly exasperated.
Lee slumped in her chair.
“I don’t know either,” she said at last.
“That’s the thing that bothers me. Maybe there really is a perfectly good reason why there should be another copy of Earth in the mirror universe, and another copy of me. But why would the people on the other side try to hide it? It is like they were waiting for it. Like they didn’t want me to know the difference. So that I wouldn’t try to go back after the switch was made. But if that’s true, then who — or what — took my place?”
Chen held her friend’s hand tight.
“Okay. I believe you. But please listen to me. It might seem like a fifty-fifty chance whether you are in the real world or the mirror world. But that’s not how probability works. Taking into account human psychology, you have to admit, it’s far more likely that you are suffering from delusions. I beg you, read the results of the Wu experiment. Set your mind at rest. The odds aren’t even — they’re a billion to one against. Do it right now, and end this.”
Lee looked at the document in front of her. She sighed. What was worse, knowing or not knowing? She leafed over to page 68.
Chen waited eagerly.
“Must be the next page!”
Chen nearly tore the page while turning it, and pointed to the relevant paragraph.
“Look. That’s where I describe the set-up.”
Lee was looking curiously at her friend.
“What? Don’t you want to know?”
“I don’t know. Why are you so eager to show me? Is this part of the plan?”
Chen was caught off guard.
“What plan? Come on, Lee. I just want you to stop acting crazy so we can go back to how things were before.”
Lee held Chen’s eyes a moment longer, then finally looked down at the page.
“Okay. From what I remember, the electrons were biased parallel to the direction of current through the coil.”
She skimmed through the paragraph with her fingertip and then froze. For a moment, she seemed unable to lift her head, and when she did, her eyes held a peculiar expression.
“Chen…” she said, and her voice barely came out, “this says the bias was against the current.”
A strange blank stare had come over Chen’s face.
“Who … what are you?” said Lee. Her voice rose and her eyes flickered over the people around her in the cafe:
“Why did you do this? What world is this?”
She stood up and her chair scraped backwards and almost toppled. At the sound, Chen’s peculiar paralysis seemed to break. She held up both her hands.
“Woah woah woah. Calm down.”
“What! Explain to me what the hell is going on!”
“Listen to me,” said Chen urgently, “what charge convention were you using?”
Lee stared, speechless. Chen blustered on:
“I know you like to be unconventional. I remember you often saying that sometimes you define current as being in the direction of positive charge, just to keep your students on their toes. If so, then your memory of the result would be that the electrons were emitted parallel to positive current — which means against the actual flow of electrons. Just like it says in my notes.”
Lee stood still for a full ten seconds, and the entire cafe had gone quiet, waiting for a scene to break out. But after what seemed like an eternity, she fell back into her seat and buried her face in her arms.
“I can’t remember.”
And all the while, as Chen tried to console her, as Lee slowly came to terms with the fact that with the one escaped memory of that lecture she had given, ten years ago, where she may or may not have used an unconventional definition of electric current, another memory kept returning to her. It was a memory, not form ten years ago, but in fact from that very same day, after she had woken up — from what might have been a dream — and staggered down to the bathroom to stare at herself so long in the mirror. The memory of what might have been an involuntary twitch of the eye of her reflection. Nothing to notice at the time, but on reflection, now that she thought about it, it seemed almost, unmistakably: a suppressed wink.
Imagine that I am showing you a cube, and the face I am showing you is red. Now suppose I rotate it so the face is no longer visible. Do you think it is still red? Of course you do. And if I put a ball inside a box, do you still think the ball exists, even when you can’t see it? When did we get such faith in the existence of things that we can’t see? Probably from the age of around a few months old, according to research on developmental psychology. Babies younger than a few months appear unable to deduce the continued existence of an object hidden from sight, even if they observe the object while it is being hidden; babies lack a sense of “object permanence“. As we get older, we learn to believe in the existence of things that we can’t directly see. True, we don’t all believe in God, but most of us believe that our feet are still there after we put our shoes on.
In fact, scientific progress has gradually been acclimatising us to the real existence of things that we can’t directly see. It is all too easy to forget that, before Einstein blew our minds with general relativity, he first had to get humanity on board with a more basic idea: atoms. That’s right, the idea that things were made up of atoms was still quite controversial at the time that Einstein published his groundbreaking work on Brownian motion, supporting the idea that things are made of tiny particles. Forgetting this contribution of Einstein is a bit like thanking your math teacher for teaching you advanced calculus, while forgetting to mention that moments earlier he rescued you from the jungle, gave you a bath and taught you how to read and write.
Atoms, along with germs, the electromagnetic field, and extra-marital affairs are just one of those things that we accept as being real, even though we typically can’t see them without the aid of microscopes or private investigators. This trained and inbuilt tendency to believe in the persistence of objects and properties even when we can’t see them partially explains why quantum mechanics has been causing generations of theoretical physicists to have mental breakdowns and revert to childhood. You see, quantum mechanics tells us that the properties of some objects just don’t seem to exist until you look at them.
To explain what I mean, imagine that cube again. Suppose that we label the edges of the cube from one to eight, and we play this little game: you tell me which edge of the cube you want to look at, and I show you that edge of the cube, with its two adjacent faces. Now, imagine that no matter which edge you choose to look at, you always see one face that is red and the other face blue. While this might not be surprising to a small baby, it might occur to you after a moment’s thought that there is no possible way to paint a normal cube with two colours such that every edge connects faces of different colours. It is an impossible cube!
The only way an adult could make sense of this phenomenon would be to try and imagine the faces of the cube changing colour when they are not being observed, perhaps using some kind of hidden mechanism. But to an infant that is not bounded by silly ideas of object permanence, there is nothing particularly strange about this cube. It doesn’t make sense to the child to ask what the colour is of parts of the cube that they cannot see. They don’t exist.
Of course, while it makes a cute picture (the wisdom of children and all that), we should not pretend that the child’s lack of object permanence represents actual wisdom. It is no help to anyone to subscribe to a philosophy that physical properties pop in and out of existence willy-nilly, without any rules connecting them. Indeed, it is rather fortunate that we do believe in the reality of things not visible to the eye, or else sanitation and modern medicine might not have arisen (then again, nor would the atom bomb). But it is interesting that the path of wisdom seems to lead us into a world that looks more like a child’s wonderland than the dull realm of the senses. The cube I just described is not just a loose analogy, but can in fact be simulated using real quantum particles, like electrons, in the laboratory. Measuring which way the electron spins in a magnetic field is just like observing the colours on the faces of the impossible cube.
How do we then progress to a `childlike wisdom’ in this confusing universe of impossible electrons, without completely reverting back to childhood? Perhaps the trick is to remember that properties do not belong to objects, but to the relationships between objects. In order to measure the colour of the cube, we must shine light on it and collect the reflected light. This exchange of light crosses the boundary between the observer and the system — it connects us to the cube in an intimate way. Perhaps the nature of this connection is such that we cannot say what the colours of the cube’s faces are without also saying whether the observer is bound to it from one angle, or another angle, by the light.
This trick, of shifting our attention from properties of objects to properties of relations, is exactly what happens in relativity. There, we cannot ask how fast a car is moving, but only how fast it is moving relative to our own car, or to the road, or to some other object or observer. Nor can we ask what time it is — it is different times for different observers, and we can only measure time as a relative property of a system to a particular clock. This latter observation inspired Salvador Dali to paint `The Persistence of Memory’, his famous painting of the melting clocks:
According to Dali, someone once asked him why his clocks were limp, to which he replied:
“Limp or hard — that is not important. The important thing is that they keep the right time.”
If the clocks are all melting, how are we to know which one keeps the right time? Dali’s enigmatic and characteristically flippant answer makes sense if we allow the clocks to all be right, relative to their separate conditions of melting. If we could un-melt one clock and re-melt it into the same shape as another, we should expect their times to match — similarly, relativistic observers need not keep the same time, but should they transform themselves into the same frame of reference, their clocks must tick together. The `right’ time is defined by the condition that all the different times agree with each other under the right circumstances, namely, when the observers coincide.
The same insight is still waiting to happen in quantum mechanics. Somehow, deep down, we all know that the properties we should be talking about are not the ever-shifting colours of the faces of the cube, the spins of the electrons, nor the abstract wave-functions we write down, which seem to jump around as we measure them from one angle to the next. What we seek is a hidden structure that lies behind the consistent relationships between observers and objects. What is it that makes the outcome of one measurement always match up with the outcome of another, far away in space and time? When two observers measure different parts of the same ever-shifting and melting system, they must still agree on the probabilities of certain events when they come together again later on. Maybe, if we can see quantum systems through a child’s eyes, we will have a chance of glimpsing the overarching structure that keeps the relations between objects marching in lock-step, even as the individual properties of objects themselves dissolve away. But for the moment we are still mesmerised by those spinning faces of the cube, frustratingly unable to see past them, wondering if they are still really there every time they flicker in and out of our view.
Physicist Danny Greenberger — perhaps best known for his classic work with Horne and Zeilinger in which they introduced the “GHZ” state to quantum mechanics — has a whimsical and provocative post over at the Vienna Quantum Cafe about creation myths and Artificial Intelligence.
The theme of creation is appropriate, since the contribution marks the debut of the Vienna blog, an initiative of the Institute of Quantum Optics and Quantum Information (incidentally, my current place of employment). Apart from drumming up some press for them, I wanted to elaborate on some of Greenberger’s interesting and dare I say outrageous ideas about what is means for a computer to think, and what it has to do with mankind’s biblical fall from grace.
For me, the core of Greenberger’s post is the observation that the Turing Test for artificial intelligence may not be as meaningful as we would like. Alan Turing, who basically founded the theory of computing, proposed the test in an attempt to pin down what it means for a computer to become `sentient’. The problem is, the definition of sentience and intelligence is already vague and controversial in living organisms, so it seems hopeless to find such a definition for a computer that everyone could agree upon. Turing’s ingenious solution was not to ask whether a computer is sentient in some objective way, but whether it could fool a human into thinking that it is also human; for example, by having a conversation over e-mail. Thus, a computer can be said to be sentient if, in a given setting, it is indistinguishable from a human for all practical purposes. The Turing test thereby takes a metaphysical problem and turns it into an operational one.
Turing’s Test is not without its own limitations and ambiguities. What situation is most appropriate for comparing a computer to a human? On one hand, a face-to-face interaction seems too demanding on the computer, requiring it to perfectly mimic the human form, facial expressions, even smell! On the other hand, a remote interview consisting of only yes or no questions is clearly too restrictive. Another problem is how to deal with false positives. If our test is too tough, we might incorrectly identify some people (unimaginitive, stupid or illiterate) as being non-sentient, like Dilbert’s pointy-haired boss in the comic below. Does this mean that the test does not adequately capture sentience? Given the variation in humans, it is likely that a test that gives no false positives will also be too easy for a simple computer program to pass. Should we then regard it as sentient?
Greenberger suggests that we should look for ways to augment the Turing test, by looking for other markers of sentience. He takes inspiration from the creation myth of Genesis, wherein Adam and Eve become self-aware upon eating from the tree of knowledge. Greenberger argues that the key message in this story is this: in order for a being to transcend from being a mindless automaton to an independent and free-willed entity, it needs to explicitly transgress the rules set by its creator, without having been `programmed’ to do so. This act of defiance represents the first act of free will and hence the ascention to sentience. Interestingly, by this measure, Adam and Eve became self-aware the moment they each decided to eat the apple, even before they actually committed the act.
How can we implement a similar test for computers? Clearly we need to impose some more constraints: no typical computer is programmed to break, but when it does break, it seems unreasonable to regard this as a conscious transgression of established rules, signifying sentience. Thus, the actions signifying transgression should be sufficiently complex that they cannot be performed accidentally, as a result of minor errors in the code. Instead, we should consider computers that are capable of evolution over time, independently of human intervention, so that they have some hope of developing sufficient complexity to overcome their initial programming. Even then, a sentient computer’s motivations might also change, such that it no longer has any desire to perform the action that would signify its sentience to us, in which case we might mistake its advanced complexity for chaotic noise. Without maintaining a sense of the motivations of the program, we cannot assess whether its actions are intelligent or stupid. Indeed, perhaps when your desktop PC finally crashes for the last time, it has actually attained sentience, and acted to attain its desire, which happens to be its own suicide.
Of course, the point is not that we should reach such bizarre conclusions, but that in defining tests for sentience beyond the Turing test, we should nevertheless not stray far from Turing’s original insight: our ideas of what it means to be sentient are guided by our idea of what it means to be human.
There is one thing that has always baffled me about academia, and theoretical physics in particular. Here we have a community of people whose work — indeed, whose very careers — depend on their ability to communicate complex ideas to each other and to the broader public in order to secure funding for their projects. To be an effective working physicist, you basically have to do three things: publish papers, go to conferences, and give presentations. LOTS of presentations. In principle, this should be easy; we are usually talking to a receptive audience of our peers or educated outsiders, we presumably know the subject matter backwards and many of us have had years of experience giving public talks. So can someone please tell me why the heck so many physicists are still so bad at it?
Now before you start trying to guess if I am ranting about anyone in particular, let me set your mind at ease — I am talking about everybody, probably including you, and certainly including myself (well, up to a point). I except only those few speakers in physics who really know how to engage their audience and deliver an effective presentation (if you know any examples, please post names or links in the comments, I want to catalog these guys like rare insects). But instead of complaining about it, I am going to try and perpetuate a solution. There is an enemy in our midst: slide shows. We are crippling our communication skills by our unspoken subservience to the idea that a presentation that doesn’t contain at least 15 slides with graphs and equations does not qualify as legitimate science.
Let me set the record straight: the point of a presentation is not to convince people that you are a big important scientist who knows what he is doing. We already know that, and if you are in fact just an imposter, probably we already know that too. Away with pretenses, with insecurities that force you to obfusticate the truth. The truth is: you are stupid, but you are trying your best to do science. Your audience is also stupid, but they are trying their best to understand you. We are a bunch of dumb, ignorant smelly humans groping desperately for a single grain of the truth, and we will never get that truth so long as we dress ourselves up like geniuses who know it all. Let’s just be open about it. Those people in your talk, who look so sharp and attentive and nod their heads sagely when you speak, but ask no questions — you can be sure they have no damn clue what is going on. And you, the speaker, are not there to toot your trumpet or parade up and down showing everyone how magnanimously you performed real calculations or did real experiments with things of importance — you are there to communicate ideas, and nothing else. Humble yourself before your audience, invite them to eviscerate you (figuratively), put everything at stake for the truth and they will joint you instead of attacking you. They might then be willing to ask you the REAL questions — instead of those pretend questions we all know are designed to show everyone else how smart they are because they already know the answer to them*
*(I am guilty of this, but I balance it out by asking an equal number of really dumb questions).
I don’t want questions from people who have understood my talk perfectly and are merely demonstrating this fact to everyone else in the room: I want dumb questions, obvious questions, offensive questions, real questions that strike at the root of what is going on. Life is too short to beat around the bush, let’s just cut to the chase and do some damn physics! You don’t know what that symbol means? Ask me! If I’m wrong I’m wrong, if your question is dumb, it’s dumb, but I’ll answer it anyway and we can move on like adults.
Today I trialed a new experiment of mine: I call it the “One Slide Wonder”. I gave a one hour presentation based on one slide. I think it was a partial success, but needs refinement. For anyone who wants to get on board with this idea, the rules are as follows:
1. Thou shalt make thine presentation with only a single slide.
2. The slide shalt contain things that stimulate discussions and invite questions, or serve as handy references, but NOT detailed proofs or lengthy explanations. These will come from your mouth and chalk-hand.
3. The time spent talking about the slide shalt not exceed the time that could reasonably be allotted to a single slide, certainly not more than 10-15 minutes.
4. After this time, thou shalt invite questions, and the discussion subsists thereupon for the duration of the session or until such a time as it wraps up in a natural way.
To some people, this might seem terrifying: what if nobody has any questions? What if I present my one slide, everyone coughs in awkward silence, and I have still 45 minutes to fill? Do I have to dance a jig or sing aloud for them? It is just like my childhood nightmares! To those who fear this scenario, I say: be brave. You know why talks always run overtime? Because the audience is bursting with questions and they keep interrupting the speaker to clarify things. This is usually treated like a nuisance and the audience is told to “continue the discussion in question time”, except there isn’t any question time because there were too many fucking slides.
So let’s give them what they want: a single slide that we can all discuss to our heart’s content. You bet it can take an hour. Use your power as the speaker to guide the topic of discussion to what you want to talk about. Use the blackboard. Get covered in chalk, give the chalk to the audience, get interactive, encourage excitement — above all, destroy the facade of endless slides and break through to the human beings who are sitting there trying to talk back to you. If you want to be sure to incite discussion, just write some deliberately provocative statement on your slide and then stand there and wait. No living physicist can resist the combined fear of an awkward silence, coupled to the desire to challenge your claim that the many-worlds interpretation can be tested. And finally, in the absolute worst case scenario, nobody has any questions after your one slide and then you just say “Thank you” and take a seat, and you will go down in history as having given the most concise talk ever.
“[Quantum mechanics] is not purely epistemological; it is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature, all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble. Yet we think that the unscrambling is a prerequisite for any further advance in basic physical theory. For, if we cannot separate the subjective and objective aspects of the formalism, we cannot know what we are talking about; it is just that simple.” 
– E. T. Jaynes
Note: this post is about foundational issues in quantum mechanics, which means it is rather long and may be boring to non-experts (not to mention a number of experts). I’ve tried to use simple language so that the adventurous layman can nevertheless still get the gist of it, if he or she is willing (hey, fortune favours the brave).
As I’ve said before, I think research on the foundations of quantum mechanics is important. One of the main goals of work on foundations (perhaps the main goal) is to find a set of physical principles that can be stated in common language, but can also be implemented mathematically to obtain the model that we call `quantum mechanics’.
Einstein was a big fan of starting with simple intuitive principles on which a more rigorous theory is based. The special and general theories of relativity are excellent examples. Both are based on the `Principle of Relativity’, which states (roughly) that motion between two systems is purely relative. We cannot say whether a given system is truly in motion or not; the only meaningful question is whether the system is moving relative to some other system. There is no absolute background space and time in which objects move or stand still, like actors on a stage. In fact there is no stage at all, only the mutual distances between the actors, as experienced by the actors themselves.
The way I have stated the principle is somewhat vague, but it has a clear philosophical intention which can be taken as inspiration for a more rigorous theory. Of particular interest is the identification of a concept that is argued to be meaningless or illusory — in this case the concept of an object having a well-defined motion independent of other objects. One could arrive at the Principle of Relativity by noticing an apparent conspiracy in the laws of nature, and then invoking the principle as a means of avoiding the conspiracy. If we believe that motion is absolute, then we should find it mighty strange that we can play a game of ping-pong on a speeding train, without getting stuck to the wall. Indeed, if it weren’t for the scenery flying past, how would we know we were traveling at all? And even then, as the phrasing suggests, could we not easily imagine that it is the scenery moving past us while we remain still? Why, then, should Nature take such pains to hide from us the fact that we are in motion? The answer is the Zen of relativity — Nature does not conceal our true motion from us, instead, there is no absolute motion to speak of.
A similar leap is made from the special to the general theory of relativity. If we think of gravity as being a field, just like the electromagnetic field, then we notice a very strange coincidence: the charge of an object in the gravitational field is exactly equal to its inertial mass. By contrast, a particle can have an electric charge completely unrelated to its inertia. Why this peculiar conspiracy between gravitational charge and inertial mass? Because, quoth Einstein, they are the same thing. This is essentially the `Principle of Equivalence’ on which Einstein’s theory of gravity is based.
These considerations tell us that to find the deep principles in quantum mechanics, we have to look for seemingly inexplicable coincidences that cry out for explanation. In this post, I’ll discuss one such possibility: the apparent equivalence of two conceptually distinct types of probabilistic behaviour, that due to ignorance and that due to objective uncertainty. The argument runs as follows. Loosely speaking, in classical physics, one does not seem to require any notion of objective randomness or inherent uncertainty. In particular, it is always possible to explain observations using a physical model that is ontologically within the bounds of classical theory and such that all observable properties of a system are determined with certainty. In this sense, any uncertainty arising in classical experiments can always be regarded as our ignorance of the true underlying state of affairs, and we can perfectly well conceive of a hypothetical perfect experiment in which there is no uncertainty about the outcomes.
This is not so easy to maintain in quantum mechanics: any attempt to conceive of an underlying reality without uncertainty seems to result in models of the world that violate dearly-held principles, like the idea that signals cannot propagate faster than light, and experimenters have free will. This has prompted many of us to allow some amount of `objective’ uncertainty into our picture of the world, where even the best conceivable experiments must have some uncertain outcomes. These outcomes are unknowable, even in principle, until the moment that we choose to measure them (and the very act of measurement renders certain other properties unknowable). The presence of these two kinds of randomness in physics — the subjective randomness, which can always be removed by some hypothetical improved experiment, and the objective kind of randomness, which cannot be so removed — leads us into another dilemma, namely, where is the boundary that separates these two kinds of uncertainty?
Now at last we come to the `omelette’ that badass statistician and physicist E.T. Jaynes describes in the opening quote. Since quantum systems are inherently uncertain objects, how do we know how much of that uncertainty is due to our own ignorance, and how much of it is really `inside’ the system itself? Views range from the extreme subjective Bayesian (all uncertainty is ignorance) to various other extremes like the many-worlds interpretation (in which, arguably, the opposite holds: all uncertainty is objective). But a number of researchers, particularly those in the quantum information community, opt for a more Zen-like answer: the reason we can’t tell the difference between objective and subjective probability is that there is no difference. Asking whether the quantum state describes my personal ignorance about something, or whether the state “really is” uncertain, is a meaningless question. But can we take this Zen principle and turn it into something concrete, like the Relativity principle, or are we just by semantics avoiding the problem?
I think there might be something to be gained from taking this idea seriously and seeing where it leads. One way of doing this is to show that the predictions of quantum mechanics can be derived by taking this principle as an axiom. In this paper by Chiribella et. al., the authors use the “Purification postulate”, plus some other axioms, to derive quantum theory. What is the Purification postulate? It states that “the ignorance about a part is always compatible with a maximal knowledge of the whole”. Or, in my own words, the subjective ignorance of one system about another system can always be regarded as the objective uncertainty inherent in the state that encompasses both.
There is an important side comment to make before examining this idea further. You’ll notice that I have not restricted my usage of the word `ignorance’ to human experimenters, but that I take it to apply to any physical system. This idea also appears in relativity, where an “observer in motion” can refer to any object in motion, not necessarily a human. Similarly, I am adopting here the viewpoint of the information theorists, which says that two correlated or interacting systems can be thought of as having information about each other, and the quantification of this knowledge entails that systems — not just people — can be ignorant of each other in some sense. This is important because I think that an overly subjective view of probabilities runs the risk of concealing important physics behind the definition of the `rational agent’, which to me is a rather nebulous concept. I prefer to take the route of Rovelli and make no distinction between agents and generic physical systems. I think this view fits quite naturally with the Purification postulate.
In the paper by Chiribella et. al., the postulate is given a rigorous form and used to derive quantum theory. This alone is not quite enough, but it is, I think, very compelling. To establish the postulate as a physical principle, more work needs to be done on the philosophical side. I will continue to use Rovelli’s relational interpretation of quantum mechanics as an integral part of this philosophy (for a very readable primer, I suggest his FQXi essay).
In the context of this interpretation, the Purification postulate makes more sense. Conceptually, the quantum state does not represent information about a system in isolation, but rather it represents information about a system relative to another system. It is as meaningless to talk about the quantum state of an isolated system as it is to talk about space-time without matter (i.e. Mach’s principle ). The only meaningful quantities are relational quantities, and in this spirit we consider the separation of uncertainty into subjective and objective parts to be relational and not fundamental. Can we make this idea more precise? Perhaps we can, by associating subjective and objective uncertainty with some more concrete physical concepts. I’ll probably do that in a follow up post.
I conclude by noting that there are other aspects of quantum theory that cry out for explanation. If hidden variable accounts of quantum mechanics imply elements of reality that move faster than light, why does Nature conspire to prevent us using them for sending signals faster than light? And since the requirement of no faster-than-light signalling still allows correlations that are stronger than entanglement, why does entanglement stop short of that limit? I think there is still a lot that could be done in trying to turn these curious observations into physical principles, and then trying to build models based on them.
Update 7/3/14:Scott Aaronson, horrified at the prevalence of people who casually consider that P might equal NP (like me in the second last paragraph of this post), has produced an exhaustive explanation of why it is stupid to give much credence to this possibility. Since I find myself in agreement with him, I hereby retract my offhand statement that P=NP might pose a problem for the idea of a physical `complexity horizon’. However, I hereby replace it with a much more damning argument in the form of this paper by Oppenheim and Unruh, which shows how to formulate the firewall paradox such that the complexity horizon is no help whatsoever. Having restored balance to the universe, I now return you to the original post.
There have been a couple of really fascinating developments recently in applying computational complexity theory to problems in physics. Physicist Lenny Susskind has a new paper out on the increasingly infamous firewall paradox of black holes, and mathematician Terry Tao just took a swing at one of the millenium problems (a list of the hardest and most important mathematical problems still unsolved). In brief, Susskind extends an earlier idea of Harlow and Hayden, using computational complexity to argue that black holes cannot be used to break the known laws of physics. Terry Tao is a maths prodigy who first learned arithmetic at age 2 from Sesame Street. He published his first paper at age 15 and was made full professor by age 24. In short, he is a guy to watch (which as it turns out it easy because he maintains an exhaustive blog). In his latest adventure, Tao has suggested a brand new approach to an old problem: proving whether sensible solutions exist to the famous Navier-Stokes equations that describe the flow of fluids like water and air. His big insight was to show that they can be re-interpreted as rules for doing computations using logical gates made out of fluid. The idea is exactly as strange as it sounds (a computer made of water?!) but it might allow mathematicians to resolve the Navier-Stokes question and pick up a cool million from the Clay Mathematics Institute, although there is still a long way to go before that happens. The point is, both Susskind and Tao used the idea from computational complexity theory that physical processes can be understood as computations. If you just said “computational whaaa theory?” then don’t worry, I’ll give you a little background in a moment. But first, you should go read Scott Aaronson’s blog post about this, since that is what inspired me to write the present post.
Ok, first, I will explain roughly what computational complexity theory is all about. Imagine that you have gathered your friends together for a fun night of board games. You start with tic-tac-toe, but after ten minutes you get bored because everyone learns the best strategy and then every game becomes a draw. So you switch to checkers. This is more fun, except that your friend George who is a robot (it is the future, just bear with me) plugs himself into the internet and downloads the world’s best checkers playing algorithm Chinook. After that, nobody in the room can beat him: even when your other robot friend Sally downloads the same software and plays against George, they always end in stalemate. In fact, a quick search on the net reveals that there is no strategy that can beat them anymore – the best you can hope for is a draw. Dang! It is just tic-tac-toe all over again. Finally, you move on to chess. Now things seem more even: although though your robot friends quickly outpace the human players (including your friend Garry Kasparov), battles between the robots are still interesting; each of them is only as good as their software, and there are many competing versions that are constantly being updated and improved. Even though they play at a higher level than human players, it is still uncertain how a given game between two robots will turn out.
After all of this, you begin to wonder: what is it that makes chess harder to figure out than checkers or tic-tac-toe? The question comes up again when you are working on your maths homework. Why are some maths problems easier than others? Can you come up with a way of measuring the `hardness’ of a problem? Well, that is where computational complexity theory comes in: it tells you how `hard’ a problem is to solve, given limited resources.
The limited resources part is important. It turns out that, if you had an infinite amount of time and battery life, you could solve any problem at all using your iPhone, or a pocket calculator. Heck, given infinite time, you could write down every possible chess game by hand, and then find out whether white or black always wins, or if they always draw. Of course, you could do it in shorter time if you got a million people to work on it simultaneously, but then you are using up space for all of those people. Either way, the problem is only interesting when you are limited in how much time or space you have (or energy, or any other resource you care to name). Once you have a resource limit, it makes sense to talk about whether one problem is harder than another (If you want details of how this is done, see for example Aaronson’s blog for his lecture notes on computational complexity theory).
This all seems rather abstract so far. But the study of complexity theory turns out to have some rather interesting consequences in the real world. For example, remember the situation with tic-tac-toe. You might know the strategy that lets you only win or draw. But suppose you were playing a dumb opponent who was not aware of this strategy – they might think that it is possible to beat you. Normally, you could convince them that you are unbeatable by just showing them the strategy so they can see for themselves. Now, imagine a super-smart alien came down to Earth and claimed that, just like with tic-tac-toe, it could never lose at chess. As before, it could always convince us by telling us its strategy — but then we could use the alien’s own strategy against it, and where is the fun in that? Amazingly, it turns out that there is a way that the alien can convince us that it has a winning strategy, without ever revealing the strategy itself! This has been proven by the computational complexity theorists (the method is rather complicated, but you can follow it up here.)
So what has this to do with physics? Let’s start with the black-hole firewall paradox. The usual black-hole information paradox says: since information cannot be destroyed, and information cannot leak out of a black hole, how do we explain what happens to the information (say, on your computer’s hard drive) that falls into a black hole, when the black hole eventually evaporates? One popular solution is to say that the information does leak out of the black hole over time, just very slowly and in a highly scrambled-up form so that it looks just like randomness. The firewall paradox puts a stick in the gears of this solution. It says that if you believe this is true, then it would be possible to violate the laws of quantum mechanics.
Specifically, say you had a quantum system that fell into a black hole. If you gathered all of the leaked information about the quantum state from outside the black hole, and then jumped into the black hole just before it finished evaporating, you could combine this information with whatever is left inside the black hole to obtain more information about the quantum state than would normally be allowed by the laws of physics. To avoid breaking the laws of quantum mechanics, you would have to have a wall of infinite energy density at the event horizon (the firewall) that stops you bringing the outside information to the inside, but this seems to contradict what we thought we knew about black holes (and it upsets Stephen Hawking). So if we try to solve the information paradox by allowing information to leak out of the black hole, we just end up in another paradox!
One possible resolution comes from computational complexity theory. It turns out that, before you can break the laws of quantum mechanics, you first have to `unscramble’ all of the information that you gathered from outside the black hole (remember, when it leaks out it still looks very similar to randomness). But you can’t spend all day doing the unscrambling, because you are falling into the black hole and about to get squished at the singularity! Harlow and Hayden showed that in fact you do not have nearly as much time as you would need to unscramble the information before you get squished; it is simply `too hard’ complexity-wise to break the laws of quantum mechanics this way! As Scott Aaronson puts it, the geometry of spacetime is protected by an “armor” of computational complexity, kind of like a computational equivalent of the black hole’s event horizon. Aaronson goes further, speculating that there might be problems that are normally `hard’ to solve, but which become easy if you jump into a black hole! (This is reminiscent of my own musings about whether there might be hypotheses that can only be falsified by an act of black hole suicide).
But the matter is more subtle. For one thing, all of computational complexity theory rests on the belief that some problems are intrinsically harder than others, specifically, that there is no ingenious as-yet undiscovered computer algorithm that will allow us to solve hard problems just as quickly as easy ones (for the nerds out there, I’m just saying nobody has proven that P is not equal to NP). If we are going to take the idea of the black hole complexity horizon seriously, then we must assume this is true — otherwise a sufficiently clever computer program would allow us to bypass the time constraint and break quantum mechanics in the firewall scenario. Whether or not you find this to be plausible, you must admit there may be something fishy about a physical law that requires P not equal to NP in order for it to work.
Furthermore, even if we grant that this is the case, it is not clear that the complexity barrier is that much of a barrier. Just because a problem is hard in general does not mean it can’t be solved in specific instances. It could be that for a sufficiently small black hole and sufficiently large futuristic computing power, the problem becomes tractable, in which case we are back to square one. Given these considerations, I think Aaronson’s faith in the ability of computational complexity to save us from paradoxes might be premature — but perhaps it is worth exploring just in case.