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*a*^{2}, while the chance we will observe steak is*b*^{2}. (Obviously, then, we must have*a*^{2}+*b*^{2}= 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 0.017^{2} = 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.

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Good Lord.

I think that actually made sense.

Careful, Sean, you might get your membership in the Kool Kwantum Kids Klub revoked 😉

beautiful post, sean. and that puppy clinches it.:)

‘But there’s a catch. The outcomes of quantum processes can never be predicted precisely, but only in terms of probabilities. So quantum computation doesn’t invariably give the right answer.’ – http://www.nature.com/news/2006/060220/full/060220-10.html

A very useful, reliable step forward for computing….

*Golf clap*

Thanks for the applause, is it because I’ve been trying to get interest in a more reliable future computer scheme which is based on award-winning wafer scale technology? See http://www.ivorcatt.com/3ew.htm ? (Linked from my page)

So we may now contemplate a simple desk-top calculator?, that is sitting alone waiting for an office user to interact.

The calculator is performing all kinds of calculations in anticipation of its next function!

Next time you switch on a “casio” calculator, just because a “zero” pops onto screen, do not take it for granted, internally the casio may have performed mathematical miricles in order to arrive at the default “zero”!

I think I almost got that. I’ve bookmarked this, and promise to try again when I haven’t had three glasses of wine.

Afterthought: Or, maybe I should try again after another two.

OK,

So how do we get this to extremely large numbers into their prime factors. It seems like a lot of work for a single bit of information. The “promise” of quantum computing is that very large problems become accessible. It seems like this approach would lead to more “virtual calculations” to get the the answer than traditional methods.

What am I missing?

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Thanks Sean. I took their paper home to read over the weekend but didn’t make much progress on it. Your precis was very helpful.

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Can a salad be rotated into a

cookedsteak? Imagine how easy this could make food preparation: make something simple, then rotate its wavefunction to something more elaborate. But I suspect a conservation law is violated here. I guess we won’t know for sure until that grant application for the Quantum Food Sciences Institute comes through….Elliot, I didn’t really tackle the “quantum computation” part of the experiment, only the “quantum interrogation” part. That would be another post. But the basic philosophy is the same — you take advantage of superposition, which means that states are not simply described by bits but by

functionsof bits, to sneakily do many calculations at once.Hmm… I have bookmarked this entry for my own reference when we get to the quantum theory unit of the course. If I can do it properly, not only will my students better appreciate the weird wonderfulness of quantum theory, but they will also be hungry. The cafeteria should offer me a percentage of the take that day…

Okay… very fine explanation!

But why doesn’t this then allow spacelike communication with entangled states? Imagine a “channel utility” midway between the two communicating sites that sends out paired entangled states to each of them. And site A does a “receipt” from site B by executing this Quantum Zeno thing and finding with high probability whether the state is pure or not without observing it. If it is we can conclude (in a perfect universe) that the collapse was due to an observation made at site B. B of course transmits a bit just by observing the state. Because A didn’t collapse the state itself, the usual popular arguments against spacelike communication through entanglement appear to fail.

BTW the channel described above is only used for communicating B to A. Communications A to B use a different channel.

There is a fourth thing about quantum mechanics to be explained – how the quantum chef can not only prepare states of superposition of steak and salad, but also, having once prepared them, rotate them, so that e.g. salad -> -steak and steak -> salad.

This quote is misleading. The point is that you can make the probability of getting the wrong answer as low as you want. You can make it 1^(-15) if you like, or 1^(-100) if that’s how much accuracy you need. There have been probabilistic algorithms in computer science since long before quantum computing was thought of; quantum computers just let us put additional algorithms into the general class of probabilistic algorithms with arbitrarily low probability of giving the wrong answer.

For the lay people in the crowd, “simplified” while the “complexity” comes later in entanglement issues.

Murray Gellman called this process, “Plectics.” So under the heading of “interrogation,” I am wondering.

Can one really make the precision of measurement arbitrarily small via the quantum Zeno effect? Assuming it takes a smaller (or larger) amount of energy to rotate the state vector as the increment decreases, don’t you either approach doing nothing to the salad/steak or pumping an infinite amount of energy into your apparatus to make the rotation? Even the former seems forbidden by uncertainty, as you can’t apply zero energy to the apparatus.

The main problem here is that it’s well known that puppies will get excited about pretty much anything, be it steak or salad. They are permanently in waveform-collapsing mode.

I like puppies.

I’m sure there’s a housetraining analogy that could be constructed involving the superposition of a puppy in a crate in the state of having to pee or not. I suppose one might use a quantum Zeno effect to never have to take the puppy out on a cold night: If you can observe it contantly, maybe it will never have to go?

While a state of existance may be known in a macrosense, it still applies at a microsense as well?

G -> H -> … -> SU(3) x SU(2) x U(1) -> SU(3) x U(1)How you see then in a global perspective, having everything “inclusive” sets the stage for the standard model to be expressed? This became, complex, yet it arose from Planck Time?