Quantum mechanics, as we all know, is weird. It’s weird enough in its own right, but when some determined experimenters do tricks that really bring out the weirdness in all its glory, and the results are conveyed to us by well-intentioned but occasionally murky vulgarizations in the popular press, it can seem even weirder than usual.
Last week was a classic example: the computer that could figure out the answer without actually doing a calculation! (See Uncertain Principles, Crooked Timber, 3 Quarks Daily.) The articles refer to an experiment performed by Onur Hosten and collaborators in Paul Kwiat‘s group at Urbana-Champaign, involving an ingenious series of quantum-mechanical miracles. On the surface, these results seem nearly impossible to make sense of. (Indeed, Brad DeLong has nearly given up hope.) How can you get an answer without doing a calculation? Half of the problem is that imprecise language makes the experiment seem even more fantastical than it really is — the other half is that it really is quite astonishing.
Let me make a stab at explaining, perhaps not the entire exercise in quantum computation, but at least the most surprising part of the whole story — how you can detect something without actually looking at it. The substance of everything that I will say is simply a translation of the nice explanation of quantum interrogation at Kwiat’s page, with the exception that I will forgo the typically violent metaphors of blowing up bombs and killing cats in favor of a discussion of cute little puppies.
So here is our problem: a large box lies before us, and we would like to know whether there is a sleeping puppy inside. Except that, sensitive souls that we are, it’s really important that we don’t wake up the puppy. Furthermore, due to circumstances too complicated to get into right now, we only have one technique at our disposal: the ability to pass an item of food into a small flap in the box. If the food is something uninteresting to puppies, like a salad, we will get no reaction — the puppy will just keep slumbering peacefully, oblivious to the food. But if the food is something delicious (from the canine point of view), like a nice juicy steak, the aromas will awaken the puppy, which will begin to bark like mad.
It would seem that we are stuck. If we stick a salad into the box, we don’t learn anything, as from the outside we can’t tell the difference between a sleeping puppy and no puppy at all. If we stick a steak into the box, we will definitely learn whether there is a puppy in there, but only because it will wake up and start barking if it’s there, and that would break our over-sensitive hearts. Puppies need their sleep, after all.
Fortunately, we are not only very considerate, we are also excellent experimental physicists with a keen grasp of quantum mechanics. Quantum mechanics, according to the conventional interpretations that are good enough for our purposes here, says three crucial and amazing things.
- First, objects can exist in “superpositions” of the characteristics we can measure about them. For example, if we have an item of food, according to old-fashioned classical mechanics it could perhaps be “salad” or “steak.” But according to quantum mechanics, the true state of the food could be a combination, known as a wavefunction, which takes the form (food) = a(salad) + b(steak), where a and b are some numerical coefficients. That is not to say (as you might get the impression) that we are not sure whether the food is salad or steak; rather, it really is a simultaneous superposition of both possibilities.
- The second amazing thing is that we can never observe the food to be in such a superposition; whenever we (or sleeping puppies) observe the food, we always find that it appears to be either salad or steak. (Eigenstates of the food operator, for you experts.) The numerical coefficients a and b tell us the probability of measuring either alternative; the chance we will observe salad is a2, while the chance we will observe steak is b2. (Obviously, then, we must have a2 + b2 = 1, since the total probability must add up to one [at least, in a world in which the only kinds of food are salad and steak, which we are assuming for simplicity].)
- Third and finally, the act of observing the food changes its state once and for all, to be purely whatever we have observed it to be. If we look and it’s salad, the state of the food item is henceforth (food) = (salad), while if we saw that it was steak we would have (food) = (steak). That’s the “collapse of the wavefunction.”
You can read all that again, it’s okay. It contains everything important you need to know about quantum mechanics; the rest is just some equations to make it look like science.
Now let’s put it to work to find some puppies without waking them up. Imagine we have our morsel of food, and that we are able to manipulate its wavefunction; that is, we can do various operations on the state described by (food) = a(salad) + b(steak). In particular, imagine that we can rotate that wavefunction, without actually observing it. In using this language, we are thinking of the state of the food as a vector in a two-dimensional space, whose axes are labeled (salad) and (steak). The components of the vector are just (a, b). And then “rotate” just means what it sounds like: rotate that vector in its two-dimensional space. A rotation by ninety degrees, for example, turns (salad) into (steak), and (steak) into -(salad); that minus sign is really there, but doesn’t affect the probabilities, since they are given by the square of the coefficients. This operation of rotating the food vector without observing it is perfectly legitimate, since, if we didn’t know the state beforehand, we still don’t know it afterwards.
So what happens? Start with some food in the (salad) state. Stick it into the box; whether there is a puppy inside or not, no barking ensues, as puppies wouldn’t be interested in salad anyway. Now rotate the state by ninety degrees, converting it into the (steak) state. We stick it into the box again; the puppy, unfortunately, observes the steak (by smelling it, most likely) and starts barking. Okay, that didn’t do us much good.
But now imagine starting with the food in the (salad) state, and rotating it by 45 degrees instead of ninety degrees. We are then in an equal superposition, (food) = a(salad) + a(steak), with a given by one over the square root of two (about 0.71). If we were to observe it (which we won’t), there would be a 50% chance (i.e., [one over the square root of two]2) that we would see salad, and a 50% chance that we would see steak. Now stick it into the box — what happens? If there is no puppy in there, nothing happens. If there is a puppy, we have a 50% chance that the puppy thinks it’s salad and stays asleep, and a 50% chance that the puppy thinks it’s steak and starts barking. Either way, the puppy has observed the food, and collapsed the wavefunction into either purely (salad) or purely (steak). So, if we don’t hear any barking, either there’s no puppy and the state is still in a 45-degree superposition, or there is a puppy in there and the food is in the pure (salad) state.
Let’s assume that we didn’t hear any barking. Next, carefully, without observing the food ourselves, take it out of the box and rotate the state by another 45 degrees. If there were no puppy in the box, all that we’ve done is two consecutive rotations by 45 degrees, which is simply a single rotation by 90 degrees; we’ve turned a pure (salad) state into a pure (steak) state. But if there is a puppy in there, and we didn’t hear it bark, the state that emerged from the box was not a superposition, but a pure (salad) state. Our rotation therefore turns it back into the state (food) = 0.71(salad) + 0.71(steak). And now we observe it ourselves. If there were no puppy in the box, after all that manipulation we have a pure (steak) state, and we observe the food to be steak with probability one. But if there is a puppy inside, even in the case that we didn’t hear it bark, our final observation has a (0.71)2 = 0.5 chance of finding that the food is salad! So, if we happen to go through all that work and measure the food to be salad at the end of our procedure, we can be sure there is a puppy inside the box, even though we didn’t disturb it! The existence of the puppy affected the state, even though we didn’t (in this branch of the wavefunction, where the puppy didn’t start barking) actually interact with the puppy at all. That’s “non-destructive quantum measurement,” and it’s the truly amazing part of this whole story.
But it gets better. Note that, if there were a puppy in the box in the above story, there was a 50% chance that it would start barking, despite our wishes not to disturb it. Is there any way to detect the puppy, without worrying that we might wake it up? You know there is. Start with the food again in the (salad) state. Now rotate it by just one degree, rather than by 45 degrees. That leaves the food in a state (food) = 0.999(salad) + 0.017(steak). [Because cos(1 degree) = 0.999 and sin(1 degree) = 0.017, if you must know.] Stick the food into the box. The chance that the puppy smells steak and starts barking is 0.0172 = 0.0003, a tiny number indeed. Now pull the food out, and rotate the state by another 1 degree without observing it. Stick back into the box, and repeat 90 times. If there is no puppy in there, we’ve just done a rotation by 90 degrees, and the food ends up in the purely (steak) state. If there is a puppy in there, we must accept that there is some chance of waking it up — but it’s only 90*0.0003, which is less than three percent! Meanwhile, if there is a puppy in there and it doesn’t bark, when we observe the final state there is a better than 97% chance that we will measure it to be (salad) — a sure sign there is a puppy inside! Thus, we have about a 95% chance of knowing for sure that there is a puppy in there, without waking it up. It’s obvious enough that this procedure can, in principle, be improved as much as we like, by rotating the state by arbitrarily tiny intervals and sticking the food into the box a correspondingly large number of times. This is the “quantum Zeno effect,” named after a Greek philosopher who had little idea the trouble he was causing.
So, through the miracle of quantum mechanics, we can detect whether there is a puppy in the box, even though we never disturb its state. Of course there is always some probability that we do wake it up, but by being careful we can make that probability as small as we like. We’ve taken profound advantage of the most mysterious features of quantum mechanics — superposition and collapse of the wavefunction. In a real sense, quantum mechanics allows us to arrange a system in which the existence of some feature — in our case, the puppy in the box — affects the evolution of the wavefunction, even if we don’t directly access (or disturb) that feature.
Now we simply replace “there is a puppy in the box” with “the result of the desired calculation is x.” In other words, we arrange an experiment so that the final quantum state will look a certain way if the calculation has a certain answer, even if we don’t technically “do” the calculation. That’s all there is to it, really — if I may blithely pass over the heroic efforts of some extremely talented experimenters.
Quantum mechanics is the coolest thing ever invented, ever.
Update: Be sure not to miss Paul Kwiat’s clarification of some of these issues.
agm:
Thanks.
Ok — and please forgive me as I am certainly no trained physicist — but it seems to me what this really is, for lack of a better description, is a RAM bus for a quantum computer. I could perform (x) iterations of the same operation on a dataset, in effect “save” the current waveform state on the register out to a separate set of data (“RAM”), then use the result register to perform another set of answer-space reductions with another algorithm, collapse the system at the end, and determine the intermedia values after the fact.
Again, please let me know if I am completely off base with my understanding of this. 😛
I have one question. Granted that this is a thought experiment, how is it that the non-detection of the salad by the puppy counts as a wave-collapsing ‘observation’ for the purposes of quantum mechanics? After all, the puppy can just as easily not detect the salad when the (food) is outside of the box, implying that the quantum waveform would collapse before you could put the (food) into the box, which doesn’t make sense.
Pingback: hausblog » Blog Archive » Quantum interrogation | Cosmic Variance
Wouldn’t it be easier just to use a cat. I thought it was traditional to use a cat for physics analogies!
🙂
I’m not a physiicst… so bear with me. An issue that is puzzling me. The argument is based on the pre-supposition that you can start the food (salad) state and can then rotate the wavefunction to any intermediate state. Aren’t you collapsing the wavefunction when you start with food(salad)?
To my mind, in a non-determinate state, food = a(salad) + b(teak), where a = b. Also, how do you rotate the wave-function partially. Any observation will collapse it by rotatiing by 45 degrees. How do you bias the wavefunction without collapsing it?
Pingback: Coyote Mercury » Blog Archive » The Education of Greyhound Phoebe, Chapter the Third
How can you put a puppy in a box, without anybody knowing it’s in there? for this to work, would’nt you need a perfect horizon, with no information leaking out, no matter how small or in which direction through time. Even if you built a machine to put the puppy in the box, you would still know what the machine was for and how can you build a machine without someone knowing what it’s for. If Alice falls down a rabbit hole, she ether leaves a teddy bear at the entrance or does not leave a teddy bear( A teddy bear is a small amount of information about what went down the rabbit hole, which is left for all to see, there is a chance you already know the answer anyhow.) If there is no teddy bear, then the possibility of alice being down the rabbit hole is Irrelavant, it just a rabbit hole..
Scott #52: Under the colorful assumptions of this story, the sleeping puppy is measuring the food-value of the thing we stick in the box. In particular, even if it’s salad, the puppy has measured it to be salad. A little poetic license there, perhaps.
Dibyendu #55: We are definitely assuming that the food starts in a pure-salad state. That’s okay, we can look at the wavefunction and manipulate it in the course of preparing our apparatus. What we can’t do is measure the food-value until the very end. But, as mentioned in the post, rotating the wavefunction doesn’t count as a “measurement,” since we don’t know the value of the wavefunction after we’re finished. It could be anything, depending on where it started.
Pingback: Yin Zhangqi’s Blog » å¥½çš„æ–‡ç« éƒ½èƒ½è®©äººçœ¼å‰ä¸€äº®
Enjoyed the post. Thanks.
Pingback: Paul Kwiat on quantum computation | Cosmic Variance
Pingback: rakhesh shows … » Quantum interrogation
Ulrich’s objection does not seem to be merely semantical, but rather cuts to the heart of one problem with the explanation. If the food is not “really” in both states at once, but merely has a probability of being salad or steak, that would seem to affect our ability to rotate the eigenvector. Just because the food is in an unknown state (but with known probabilities) does not, by itself, mean that we can freely change it to any other unknown state (with known probabilities). (And worse yet, as Dibyendu pointed out, the experiment requires the food to be in a non-superposition to begin with: it has to already be salad. If we adjust our apparatus as Sean suggested, that itself is a measurement, even if we don’t actually directly measure the individual salad particles that we conduct the actual experiment with: if the state is known by any means whatsoever, the state is known. This would seem to further impair our ability to rotate its state.)
That leads to a fundamental problem with quantum computing: if one assumes the ability to alter the state of quantum particles, no matter what state they’re in to begin with, then logically one could turn (say) neutrons into protons (by changing one down quark into an up quark, thus changing the neutron’s down+down+up into a proton’s down+up+up), and thus turn (say) lead into gold (albeit neutron-deficient gold, which would need more neutrons quickly to avoid fissioning…or maybe that’d be the point, and one could use this to turn ordinary matter into quickly decaying fuel for a nuclear reactor: say, turn nitrogen-14 w/7 neutrons – harvested from the air – into silicon-14 w/0 neutrons, and watch it fly apart in a cloud of energized protons – easy to capture for power production). But this possibility – a direct logical consequence of this unrestricted state alteration – appears to fly in the face of known physics, therefore this state alteration appears impossible (at least, without restrictions that would invalidate the puppy experiment analogy as it is currently written).
There’s also a problem with the fact that the puppy (if present) is performing an observation, although this may be more a problem with the analogy than with the core concept. In reality, the “puppy” would usually not be somethng ordinarily capable of performing a measurement in the same sense as we do, so the “puppy” couldn’t collapse the wavefunction (and thus presumably stop any further eigenvector rotation) on its very first measurement.
OK, how does one rotate the state of unobserved food? How would you put it in the box?
Does it become “observed” if a robot puts it in the box? Does moving it to the flap rotate anything?
How would you confirm that you have rotated the state, and can that be done without observing the food?
Update: I read the article, and…sure enough, although they did do some interesting experiments with one “rotation” (actually, the “food” is polarized photons, and what’s being rotated is apparently the polarization – from horizontal to vertical and back – not actually the probability in and of itself, although it affects other things in the system) and measurement, the Zeno chain is still just a thought experiment.
This reminds of the puzzle where you come to a fork in the road, and there are two guys one who always lies, and one who always tells the true. They both know where the roads lead, and you want to know which is the right path, say to Town. Trick is you can only ask ONE question to one of the persons. What question could you ask? one answer is: “If I were to ask you, if this is the corrent path to town, would you say yes?” You get the right answer is all cases.
I mostly get it, but I think he might be wrong by stating that the probability could be improved as much as you like…might the accuracy depend on the cost of the calculation? For example, if trying to factor a very large prime, you might just be trading one set of problems for another…if trying to improve the probability of your result turns out to be as computational expensive as a more straightforward (brute force) approach.
>>OK, how does one rotate the state of unobserved food? How would you put it in the box?
>>Does it become “observed” if a robot puts it in the box? Does moving it to the flap rotate anything?
>>How would you confirm that you have rotated the state, and can that be done without observing the food?
It’s just a metaphor. Maybe this will help you understand the basic concepts:
http://video.google.com/videoplay?docid=-4237751840526284618&q=quantum
Pingback: Beware of the Dogma » Quantum Interrogation (no, it’s not something they do at Gitmo)
Pingback: Ex Cathedra » Blog Archive » Quantum Puppies
Pingback: mutter » Blog Archive » Quantum interrogation and puppies
Pingback: Watermark
48. Rob G on Feb 28th, 2006 at 2:27 pm
Ulrich, Your objections seem to be largely philosophical and/or semantic in nature. Do you have any objections to the conclusions?
Which conclusions exactly?
I agree, physics is partly maths and partly philosophy/semantics. You need the latter to figure out how the maths relates to the world. You don’t need it if you want to earn your living doing physics. The result: lousy semantics, pseudoproblems, gratuitous solutions.
I also have a comment on Adrian’s comment (63.) but need some time to formulate it.
63. Adrian on Feb 28th, 2006 at 11:24 pm
Just because the food is in an unknown state (but with known probabilities) does not, by itself, mean that we can freely change it to any other unknown state (with known probabilities).
Hello Adrian,
An unknown state with known probabilities is a contradiction in terms. A quantum state isn’t a classical state (a collection of possessed properties). It’s an algorithm that is used to calculate the probabilities of the possible outcomes of measurements. If we know the state, then we know the probabilities, and if we don’t know it, then we don’t know them. (Those who believe that quantum states are “states of knowledge” would consider even the notion of an unknown quantum state a contradiction in terms.)
65. Adrian on Feb 28th, 2006 at 11:53 pm
actually, the “food” is polarized photons, and what’s being rotated is apparently the polarization – from horizontal to vertical and back – not actually the probability in and of itself…
Obviously one can’t rotate probabilities! What one knows how to rotate in this particular contex is an algorithm, inasmuch as this can be thought of as a one dimensional mathematical construct (a so-called “ray”) that can be rotated in a more than one-dimensional mathematical construct (a so-called Hilbert space). But this algorithm is what determines the probabilities of the possible outcomes of all possible measurements.
the Zeno chain is still just a thought experiment.
One would think it was exciting news if what was once a thought experiment came out different than predicted. Strangely, it is exciting news every time what was once a thought experiment comes out exactly as predicted. (By the way, various more or less equivalent experiments have actually been done.)
Sean, thanks for presentin QM in an understandable, albeit oversimplied manner for us old duffers and carpenters. If more physicists still spoke English then I suppose more people would pick up on it. I understand now, why you were not tenured at the UofC, since students showing up for your class would naturally stay for the rest, which defeats the economic goals of the University, and of course other Profs who would rather spend time picking their noses at the alter of ego.