The Teleporter's Dilemma

One of the annoying things about trying to explain quantum mechanics to a general audience is that the weirdness of the theory forces you to use incredibly convoluted examples. Pop-science books about quantum physics are full of schemes that the producers of the Saw movies would reject as implausibly complicated.

I wish I was posting to say that I had found a way around this, but I haven't. So here's another entry in the thriller-movie school of quantum analogies.

Imagine that you and a friend are out hiking, and find yourselves kidnapped by a sinister conspiracy of some sort. You're taken to a remote island, and shown an apparatus consisting of a dial on the floor and a remote control with a single button. You press the button, and the needle on the dial turns in a clockwise direction. There's a mark on the rim of the dial at one position, but no other distinguishing features.

Your captors explain that there is another dial/ remote system elsewhere on the island, identical in every respect except it doesn't have a mark on it. You are told that your friend will be taken to that apparatus, and given the remote. Your task is to get the needles on both dials pointing in exactly the direction indicated by the mark on the first dial. If you succeed, you'll be set free and given a billion dollars. If you fail, a nuclear weapon will be detonated in Los Angeles. (If we're going to do convoluted thriller plots, here, why not play for high stakes?).

The catch: after you are separated, you will only be allowed to communicate one single number to your friend. You can talk for a few minutes before you're split up, but after that, you will only be allowed to send one number, a time in seconds, to your friend, to tell him (or her) how to get the needle pointing in the correct direction.

What do you do?

One thing to try would be to measure the amount of time that the mechanism runs before the needle reaches the mark, and send that number to your friend. This would work, provided that the two needles start out pointing in the same direction. They might, but they might not, and are you willing to risk LA on that?

What you need to do this is some sort of common reference point, so you know that the two dials are pointing in the same direction. Since you were kidnapped while on a hiking trip, you're each carrying a compass, and that gives you the extra information you need. You can agree in advance that your one number will be the time in seconds after the needle passes due north. Then, because you share a reference direction, you know that you're working with a common starting point, and the time that you send is sufficient information to get the needle pointing in the right direction. LA is saved, and you both get rich.

i-93c2cd62cf30fc2c972215e9d473b34d-sm_fig8-1.jpgThis tortured scenario is an illustration of the idea behind quantum teleportation. In teleportation experiments, the goal is to faithfully send a copy of a particular quantum state. This is typically done with polarized photons, and a polarized photon can be thought of as some combination of horizontal and vertical polarizations, which you can represent as an arrow pointing in some arbitrary direction, as shown in the cartoon at right. Transmitting a photon state from one place to another means getting a photon whose arrow points in a particular direction at the destination.

If the photon were a single classical object, you could do this by measuring two numbers, the amount of vertical polarization and the amount of horizontal polarization going into the superposition. Quantum mechanics doesn't let you do this, though-- if you try to measure the state of a single photon, you will either find horizontal or vertical polarization, and there is no way to extract the relative proportions of the two. In Copenhagen Interpretation language, the act of measuring the photon causes it to collapse into one of the two possible states, destroying the information about the other state. Given a large enough number of identically prepared photons, you could reconstruct the state by measuring the relative probabilities of horizontal and vertical, but if you've only got one, you're stuck.

You might hope that there would be some mechanism to allow you to make an exact copy of the single photon you have, and send that along, but that also turns out to be impossible. The formal proof of this "no-cloning theorem" (physicists have a real gift for names...) needs more math than I'm going to go into here, but you don't go too far wrong by blaming it on the usual quantum limits on information. Quantum physics (uncertainty and the like) says that there's no way to measure the exact state of a single photon, but if you can make perfect copies of the single photon, you can get around that limitation-- just make a bunch of copies, and measure the probabilities of horizontal and vertical, and then you know the state of the original.

You can transmit a perfect copy of a single photon, though, by using entanglement. You and the person you want to send the state to can share a pair of "entangled" photons, which are two photons in a state such that their polarizations are undefined but correlated-- if you measure one to be vertical, then the other will always be horizontal. If you measure one to be at an angle 45 degrees clockwise from vertical, the other will be 45 degrees counter-clockwise from vertical. These are the states used to test Bell's theorem and demonstrate that local realism cannot be correct.

If you each have one photon from an entangled pair, that provides a common reference point for your measurements, exactly like the compass in the thriller scenario above. What you do is to make a joint measurement of the photon whose state you're trying to send, and your part of the entangled pair. You ask, essentially, "Do these two photons have the same polarization, or opposite polarizations?" When you do that, you know that the other part of the entangled pair is instantaneously put into a state that is related to the original photon state in some particular way. Exactly what state that is depends on the outcome of your measurement (it turns out that there are four possible outcomes to the joint measurement, for reasons that don't really matter), but if you send your friend the result of your measurement, he (or she) knows exactly what to do to put it into the state of the initial photon you were trying to send, without ever knowing what that state is.

So, LA is saved, and you both get rich. Wait...

You might reasonably ask, at this point, whether this runs afoul of the "no-cloning" theorem mentioned above. It doesn't, because the act of making the joint measurement destroys the initial state. The photon you started with is now in some entangled state with the photon you had from the entangled pair-- their states are correlated in a particular way, but the precise state of each is not defined.

Schematically, the process looks like this (the dog pictures, showing my parents' late, lamented Labrador RD and Kate's parents' Boston Terrier Truman, indicate that this is a figure from the book-in-progress):


There are three photons. Initially, Photon 1 is in a definite state, represented by an arrow pointing in a particular direction, while Photon 2 and Photon 3 are entangled-- they don't have a definite state, but when measured, their states will be correlated. After the entangling measurement and the sharing of the measurement result, Photon 3 is in the same definite state that Photon 1 started in, while Photon 1 and Photon 2 are left in a different entangled state, with undefined but correlated polarizations.

And that's how quantum teleportation is like the plot of an implausible thriller movie.

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Make that a sachel nuke detonated at 1111 Constitution Ave. NW, Washington, DC and it's win-win. Doing LA is entirely satisfactory if you fricassee the Belmont Learning Center (the world's first $billion high school for being built atop a toxic waste dump, amongst other novel cost-saving measures).

Good explanation. Was able to step through the calculation given your description and found that if the measurement came up with equal polarizations, the 3rd was 90 degrees off (basically - the coefficients of the state switched places) of the original state, and if the measurement showed opposite polarizations, the 3rd was in the original state. Wee teleportation.

Tortured, yes, but also inconsistent.

Your task is to get the needles on both dials pointing in exactly the same direction. ... This would work, provided that the two needles start out pointing in the same direction.

That's an odd assumption to make, seeing as that is your goal. No?

What you need to do this is some sort of common reference point.

And if you have one, why not just point the needles that way? Say due north?

You can agree in advance that your one number will be the time in seconds after the needle passes due north.

What, is the needle moving the whole time? You said it moves when the button is pressed. Does it keep moving forever after that or something? And even so, how does a mark on your dial do any good, since the other one has no such mark. The mark could be anywhere, N, S, E, W, whatever. Telling "hey, it is 3 seconds before it reaches some direction, but I can't tell you which direction" doesn't seem very useful for anything.

That's an odd assumption to make, seeing as that is your goal. No?

The goal is to get them both pointed in the direction indicated by the mark on the first dial. I'll edit the post to make that clearer.

It's a contrived scenario, but it has to be to capture the essential elements of the teleportation problem.

I thought that implausibility was de rigeur for thriller movies?!

Hey, is it true that the creationists are going to rip-off your blog and start their own called "Unprincipled Certainty"?!

If this turns into another damn Saw movie...

The main thing your example solution did was short circuit my reading of the rest of your article. Why on earth are you squigging over time, when partner can just whip out his compass and observe the direction of the mark? How are you synchronizing the time without infinite precision synchronized compass watches? Why do we know that the B field in the remote location is pointing in the right direction for this to work? Can't we compromise - give the hiker the money and nuke LA?

Okay, I had a chance to read this offline now and I think there are several flaws in your plot. This is true with any thriller, so don't feel bad.

Oh, you didn't feel bad. Never mind. Moving right along now....

If both the captives are in the same room at the start, before the one is taken to the other room which is a duplicate of the first, he/she ought to be able to simply move the needle to the same position they saw in the first room, especially if they have a compass at hand to measure the direction!

You can't "agree in advance that your one number will be the time in seconds after the needle passes due north" because the second needle may be at a position where it need not pass due north in order to align!

You could employ a more complex system whereby, say, it would be the number of seconds after it's passed due north once already.

But if they both have compasses, all they need communicate is the number of seconds as a compass direction. Recall that compass directions are divided not only into degrees, but also into minutes and seconds. Of course, they'd need an electronic compass (if there is such a thing) for that kind of accuracy.

You'll note that not one of my "solutions" involves quantum physics, but you never specified that they're quantum physicists or even knowledgeable about such things. You only specified that they're hikers! Ah-HA!

Okay, so now tell me to take a hike....