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.
An “onion skin” as a calorimeter, has determined particle identifications from the collisions and resulting interaction? Glast?
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‘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.’ – Ben
It’s impractical, to give it accuracy you lose the size advantages of quantum-scale computing. There are good electronics ideas which are out there that get no funding. If you look back to the beginning of nuclear power (and I’m a fan of nuclear power), similar mysticism and novelty were used to sell it to the public. Today it is called propaganda. Don’t believe what you’re told! (Media question, c 1945: ‘What about radioactive waste?’ Physicist’s reply: ‘Hospitals desperately need it to treat cancer, it’s a real life-saver, and it can also be used to preserve food by killing bacteria…’)
Dear Sean,
Very nice exposition. Only one question–what is a measurement? Can only experimenters and puppies make them?
🙂
-Sam
Hey Sean, great post on counterfactual probabilities.
How about putting some “extra” factuals into the equations nad experimental setup?
How about the experimental setup as stated by you above, but make the Salad [COLD] and the Steak [HOT]!
Whilst enjoying the philosophical implications of your post, I challenge you to make the changes, and then give us the results? 😉
It seems that the introduction of thermal values to the experiment, have a far more profound counterfactual physical miracle!
In “order” to clarify the 2nd law implications:
http://en.wikipedia.org/wiki/Perpetual_motion
A simple setup of outcome could be apllied to :
In an otherwise completely empty Newtonian universe, a single particle could travel for ever at constant velocity with no violation of the laws of physics — though of course no energy could be extracted from it without slowing it down. For example, [an electron can spin around a nucleus in an atom of matter indefinitely unless it or the atom is disrupted in some way.]?
The authors have a good probability of creating a condensate, wherby Electrons can be forced to remain in constant motion around an atom indefinitely?
I think I need to learn that trick of rotating salads by 90 degrees to turn them into steaks… (Mmmm… steak.)
For what it’s worth: In one of his papers (a research paper no less), Michael Berry mentioned that it is silly to think of apples and oranges as being in different states of quantum fruitness.
Dateline Midnight March 31st 2010..First quantum computer goes on sale,..Time 01:01..computer purchased by Paul Valletta with a Cheque that is postdated with the counterfactual probabilistic (date-time) algorithm.
The postdated cheque defeats all bank clearing systems that try to process it through their inferior clearing systems, the closer they get to clearing the cheque, the farther the post-dated cheque alters it date!
13:00 hrs April 1st, Quantum computer closes all its windows and doors, shuts down and refuses to compute in any way unless it is switched off..continues to listen to an abstract track by Paul Simon, that has the catchy line..You Can Call Me Hal?…
Then there are my dogs, which would eat salad or steak just as readily, rendering the entire experiment moot….
So, someone please correct me if I am wrong here, but the “trick” is using a separate serial qubit operation to inspect the result state of a “standard” register in a quantum computer?
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Is it a given, that if a Quantum Computer can give answers to a sum without going through the calculation process, equals, the parallel to a Quantum Computer giving an answer to a question that has not been proposed?
You do not need a Quantum Computer for this surely, Microsoft have cornered this market with XP!
I think cooper hits the nail on the head. This works because you get the puppy to do an observation instead of you, but someone is still doing an observation. Actually, that seems to be the point: if there’s a puppy, the wavefunction gets remixed whether or not the puppy makes a ruckus, whereas the wavefunction is the same if there is no puppy. Pretty slick.
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Sean: “Quantum mechanics, according to the conventional interpretations that are good enough for our purposes here, says three crucial and amazing things.”
Not good enough, since these interpretations are at the root of much ‘quantum confusion’.
Sean: “First, objects can exist in “superpositions” of the characteristics we can measure about them.”
QM assigns probabilities to possible measurement outcomes on the basis of actual measurement outcomes. That’s all there is to it – unfortunately, because this leaves so much open to debate and speculation.
Sean: “According to quantum mechanics, the true state of the food could be a combination … 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.”
All that QM tells us is that if we make the appropriate measurement, we will find salad with probability A and steak with probability B. (A and B are the squares of the respective absolute values of a and b.) This is definitely not the same as saying that the food “really is a simultaneous superposition of both possibilities”.
Sean: “The second amazing thing is that we can never observe the food to be in such a superposition; whenever we observe the food, we always find that it appears to be either salad or steak.”
There is nothing amazing about it if you keep in mind what I just said.
Sean: “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.”
As said (but it cannot be repeated to often), the quantum formalism is an algorithm for assigning probabilities to possible measurement outcomes on the basis of actual measurement outcomes. This algorithm (be it in the form of a density operator, a so-called state vector, or a so-called wave function) tells us nothing whatever about the objective, actual, physical (or whatever) state of the food between measurements.
Sean: “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’.”
Before the measurement the probabilities A and B are both greater than 0 (and thus less than 1). After the measurement either A=1 and B=0 or A=0 and B=1. Why should that be amazing, given that QM assigns probability on the basis of measurement outcomes? A new measurement outcome means new probabilities.
Sean: “You can read all that again, it’s okay. It contains everything important you need to know about quantum mechanics.”
There really is sooo much more. Take (for instance) a look at ThisQuantumWorld.com.
What I always liked about the way the bomb problem was set up was that it was more of an actual experiment that you could do, if you were good enough — isolate the bombs from any light, make a detector that’s responsive to a single photon, that kind of thing. And that led to experiments that were actually done, like the one at LANL. In your example, it’s not clear how you’d set up interference between the salad and steak states, so it seems like it’s just some kind of analogy.
Not to complain too much, I’m all for non-violent, puppy-based experiments. But I don’t think it makes it clear that the effect is real and tested (to some degree), not some physicist’s dream.
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.
What’s the ontology that supports this?
The question of whether the waves are something “real” or a function to describe and predict phenomena in a convenient way is a matter of taste. I personally like to regard a probability wave, even in 3N – dimensional
space, as a real thing, certainly as more than a tool for mathematical calculations … Quite generally, how could we rely on probability predictions if by this notion we do not refer to something real and
objective? [Max Born, Dover publ., 1964, “Natural Philosophy of Cause and Chance”, p. 107.]
This is the general ontological (epiontic, according to Zurek) problem. But there are specific ontologies (Bohm-de Broglie pilot waves, advanced-retarded actions, time symmetrical two-state formalism, etc.).
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Gyan asks: What’s the ontology that supports this?
We ain’t got no ontology. We don’t need no ontology. I don’t have to show you any stinking ontology!
This whole thread is brilliant, but I particularly liked the ontology post!
T
Ulrich,
Your objections seem to be largely philosophical and/or semantic in nature. Do you have any objections to the conclusions?
The more serious question remains, if the puppy is consumed without having been cooked, how can you be sure it was skinned?
Controlling temperature might be very important?