Sorry for the radio silence around here of late. I don’t know about anyone else, but I’ve been traveling like a mad person. The good news is that I just got back from UC Davis, where I had the chance to meet John Conway for the first time in person.
The bad news is: no time for blogging. But I recently received an email pointing out that some links have died in an old post, which I proceeded to update. And that gave me the idea of stooping to a classic blogospheric move in times of sparse content: reposting old stuff! So here is the post in question, from several years ago. If people don’t complain too loudly, maybe we’ll dig up some more ancient blogging and bring it back to the surface.
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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.
For me (author of “Quantum Reality”) the best one-sentence summary of the essence of quantum mechanics was formulated by UC Berkeley physicist Henry Stapp: “Things that could have happened but did not, influence the things that do.”
Had to meet John Conway? You don’t like him much?
Yeah, that was some sloppy typing. Fixed.
Great post, Sean. Thanx.
“Things that could have happened but did not, influence the things that do.”
Like saying, “I love you.”
It makes me think of that quote by Goethe:
“…the moment one definitely commits oneself, then Providence moves too. All sorts of things occur to help one that would never otherwise have occurred. A whole stream of events issues from the decision, raising in one’s favor all manner of unforeseen incidents and meetings and material assistance, which no man could have dreamed would have come his way.”
I liked the “Back from the Future” article in Discover Mag concerning reverse causality as well as the other articles relating to relativity and quantum mechanics. (April 2010) Very thought provoking. I’d never read the magazine before, but plan to subscribe.
More of this sort of thing please, and less of the other stuff.
Julia wrote, “I’d never read the magazine before, but plan to subscribe.”
Ah, well, that should produce some results right there, if I understand correctly…
I assume this post is all about Elitzur’s “bomb testing” setup. I actually heard about this setup from Elitzur himself! He gave a talk in Tel Aviv University. I was astounded by this, even though I had already learned the maths of quantum mechanics. Wonderful stuff.
Quantum Mechanics: the dreams stuff is made of.
So you could say that Life got in the way? Ahahah. (Oh, not that John Conway. Sorry.)
Good (re)post though. I vaguely remember it, but this time around it makes sense. So thanks for the improvement that must have been discreetely snuck in there. 🙂
I did not read this the first time, but this and Schroedinger’s cat have never sat well with me. In my first quantum mechanics course, the first lecture, it was presented that QM is a statistical model and probabilities of outcomes will always assume a sufficient sample of events/particles to be meaningful.
Perhaps that instructor was wrong. But, if there is merit in that, how do you justify taking a statistical model and then apply it to a single event and develop all these profound and wonderfully counter-intuitive ideas around it.
Should you not analyze a billion boxes with a billion sleeping or not sleeping puppies in it and then get the statistical answer? I hate to be so anal retentive but I have never heard a good explanation of this and it seems to be part of many of the arguments about the philosophy of QM like deBroglie-Bohm vs. Pauli, etc.
@SteveB
I could be wrong, but it seems that if you accept QM is just a statistical model that only makes sense for a sufficiently large sample size, you are implicitly accepting a “hidden variables” interpretation of QM. There’s nothing wrong with that, but it brings a whole new counter-intuitive idea: non-locality, or the “spooky action at a distance”.
This “rotate the wave function” thing sounds really terrific. Ninety degrees is exactly what I had in mind. When do we move on to lesson 2?
Hi Sean,
What happens if the sleeping puppy/superbomb was itself a quantum system? For example instead of the sleeping puppy being inside or not we were trying to determine if there was a sleeping puppy or a dead puppy, and without being able to look inside the puppy state is both dead and sleeping. By performing the “non-destructive” measurement you’re still collapsing the system into one of both possibilities.. what’s interesting is to do so without actually disturbing it. Is that correct?
To take the original Elitzur setup it would be like having the bomb be a superposition of “good” and “bad”, with the former blowing up when a photon triggers it. So using the polarization of light you can measure that the bomb is “good”, and hence necessarily collapse the quantum bomb into that state, without a photon or anything physical actually interacting with it!
This explanation seems to make a lot of sense, except for the “Start with some food in the (salad) state.” How do you know that you’re starting with food in that state unless you observe it? And once you observe it, won’t it be stuck in (salad) forever?
@Chris
I think the idea is that by “rotating” the state by 45 degrees or 1 degree, the system is put back into a superposition. Of course, this operation only works on quantum diets.
Quantum mechanics is the most misleading thing ever invented, ever.
In experiment, “experimenter” begins with an intension to measure, wheather in the box exists a puppy – “a large box lies before us, and we would like to know whether there is a sleeping puppy inside”. Than experimenter knows very well, that he can measure the puppy existence only with a steak. Right after that, “experimenter” explains amazing things about his measuring device – the steak, that the steak is not a 100% steak, but also a salad. This is first contradiction with assumption, that his measuring device is steak, the device that can make the effect manifest. That means, that the experimenter does not know, wheather his device is steak or salad, and therefore he do not know, if he is measuring the effect or not. The effect is ! puppys weak up because of steak !. “Experimenter” than declares, that this uncertainity of his measurment device, the eigenstates of his device, are amazing. But “experimenter” probably does not know, that this is totally unscientific – to measure distinct effect with uncertain device – the device, which unit is uncertain. Third and finally, “experimenter” clearly declares, that he is measuring wheather the food is steak or salad, using the puppy in the box, and “experimenter” does not know, wheather the puppy is there or not. Absolutely nonsensical. The miracle of stupidity. The result of such measurement is called collapse of wavefunction. In other words, “experimenter” is measuring the charge in region using the device, he do not know whather it is charge-meter or length-meter. After the measurement, he will discover the state of his meter – charge or meter. That means, if the meter is charge, and the charge is in area, the discovery will be charge. If the meter is lenght, the discovery is nonsesical, because to measure charge with length-meter is stupidity. That means, the collapse of wavefunction can be valid or nonsense, depending on the method of measurement. If this is all what we need to know about quantum mechanics, than goodby quantum mechanics.
To find some puppies without waking them up is contradiction with assumption, that “nice juicy steak, the aromas will awaken the puppy”. That means, you cannot find puppy without waking it, or you cannot measure the charge without device beeing influenced by the charge. That is the sense of measurement – to record the influence of the cause of influence, to verify existence of influence. That means, that you must use the device that can record the influence – the steak and ears, not the salad and ears. The rotation of steak-salad device is sensitivity of device – salad = 0% sensitivity, steak = 100% sensitivity. If 45% sensitivity is not enough to detect the puppy, or the charge, the device is not capable to perform the measurement, and the device must be improved. If you use 100% steak (charge) device, than if the influence is zero, that means, the puppy (charge) does not exists in the box, and it absolutely does not mean, that the 100% steak (charge) device has turned into 100% salad (meter) device. If the measurement is no influence, than the result of measurement is not distinct, and the result is – no puppy (charge) or insufficient sensitivity. If we are shure that the sensitivity is sufficient, than the result is no puppy (charge). If we are not shure that the sensitivity is sufficient, then we must look in the box, to get the valid result – that means, we must improve the sensitivity. If we can look in the box without using stake-salad device, that means we have other device with sufficient sensitivity – eyes – that the result of looking in the box with eyes is, that we will know, wheather the sensitivity of the steak-salad device was sufficient or insufficient – nothing else. That means, we do not know about our stake-salad device, how much it is stake and how much it is salad, we just know, that the steak-salad device sesitivity was sufficient or insufficient. It is insufficient, if the result of stake-salad device is in contradiction with eyes device – more sensitive device.
If we know that there is puppy inside the box, and the 97% stake device is not enough sensitive to measure the content of the box, to wake up the puppy, that is absolutely not “sure sign there is a puppy inside”, but absolutely shure sign, that our 97% stake device is not sensitive enough, to detect the puppy. Quantum Zeno effect is just misleading contradictive unscientific argumentation, and the author and supporters of this argumentation have little idea the trouble they are causing. Now we simply replace “there is a puppy in the box” with “the argument (expression) is true”, and you will get, that you can know wheather the argument (expression) is true or false without thinking about the argument (expression).
Quantum mechanics is the most misleading thing ever invented, ever.
similar to chris’s point –
“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).”
the puppy’s already collapsed the wavefunction, even if we’re not aware of that. So how can we convert a collapsed wavefunction back into a superposition?
Scientific thinking is evaluation of noncontradiction of argument or expression and its conformity with reality. The concept of detection is fundamentaly based on the quantitative record of real influence. Without influence there is no way to make quantitative record. Influence means existence. Existence means influence, whether it is direct or indirect. If something exists without influence, than there is no way to detect the properties of the entity, and than there is no way to constitute corresponding scientific formalism. If we cannot detect existing entities, that does not mean an existence without influence, but insufficient detection capabilities. If we cannot detect existing entities, we cannot know about their existence. Using methods of logical derivation, mathematical derivation, analogy, probability, thought experiments, intuition – conscious methods – we can assume existence of nondetected entities. However, the fundamental requirement of trueness is noncontradiction and fundamental requirement of reality is detection. Because the truenes depends on evolutionary state of scientific knowledge, detection is most important requirement of scientific knowledge. Methods of detection must be deterministic and causal, therefore the level of determinism and causal understanding, determines the level of the methods of detection. Direct detection is fundamental basis of any conscious understanding. Direct detection may be insufficient or incorect. Indirect detection may be misleading. Arguments in the article above are contradictious, logicaly inconsistent, and nonsystematic. It is important to ask, what is intension of this argumentation ? Is intesion of “experimenter” to detect noninfluencing, nonmanifesting entity ? Is intension of “experimenter” to postulate reality on contradictious arguments ?
@Clive: Think of a photon, you can collapse it into a horizontal polarization then pass it through a polarization rotor and put it back in a superposition of vertical and horizontal.
This is a nice introduction to the actual experiment that might make things clearer: http://physics.illinois.edu/people/kwiat/interaction-free-measurements.asp
(by the way it shouldn’t be a surprise that a collapsed wavefunction doesn’t stay in a specific eigenstate forever, generally it can time evolve back into a superposition on its own anyway if no measurement is made, and by making measurements you can put it in specific states whenever you like)
“Every observable can be represented by a Hermitian operator, the eigenvalues of which are the various possible values that would be obtained on measurement. Immediately after a measurement the state of the system is the corresponding eigenstate associated with that eigenvalue.” Existence is observable. Existence operator has two eigenvalues – true and false (1 or 0). If existence measurement can destroy entity being measured, than the eigenstate of entity 0 is associated with the result of measurement – eigenvalue 1. In such case the postulate above is contradictious, because nondestroying measurements exist. The postulate above can be true, only if any existence measurement is nondestroying. If measurement of existence of the photon destroys the photon, the postulate above is not true. If the entity being measured does not exist, than the measurement is nondestroying, because it is not possible to destroy something that does not exist. If existence measurement can create an entity, than the existence operator is not distinct, because – or the eigenstate is 1 and associated eigenvalue is 0, or the eigenstate is 1 and associated eigenvalue is 1. Therefore the existence operator depends on the method of measurement. If quantum mechanics is funadamentally based on the postulate above, than quantum mechanics is fundamentally dependent on the nondestroying measurements, and destroing existence measurement of the photon cannot be described by quantum mechanics, or is in contradiction with quantum mechanics fundamentals.
Thanks softvision! That’s what I was gonna say.
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Re: Post #14 (@ Tevong)
I wonder the same thing. The puppy itself is quantum, right? That is, can’t we say that the puppy is in a superposition of existence/nonexistence (say, with equal probability of being in either state)? When we first shove the plate of food into the box, does the plate collapse the puppy/non-puppy wavefunction into a definite state of existence or nonexistence?