Kind of a technical question, but typing it out might provide some inspiration, or failing that, somebody might have a good suggestion in the comments.
Here's the issue: I'm starting on a chapter about quantum teleportation for the book, and one of the key steps in the teleportation scheme is an entangling measurement of two of the particles. If you're teleporting a photon polarization state, the easy way to do it is to make a joint measurement of the polarization of the photon whose polarization you want to "teleport" and one photon from the entangled pair you're using for the teleportation, and measure them onto the "Bell basis," of four states that look like this:
State 1: VV + HH
State 2: VV - HH
State 3: HV + VH
State 4: HV - VH
Now, that makes perfect sense to a physicist, but my goal is to explain this to 1) non-physicists, 2) without equations, and 3) with minimal hand-waving. And my question is, what's the best way to describe this.
It's easy enough to describe the basic essence of this measurement-- basically, you're asking whether the two photons have the same polarization or opposite polarizations. That's straightforward enough, and can easily be explained to a layman. The thing that's hard to explain is why there are four answers, rather than two.
Common sense would say that "Do you have the same polarization?" is a yes-no question, with only two answers. But there are four possible wavefunctions in the Bell basis, in two pairs that differ only by a phase factor. As someone who's been doing this for a while, I'm familiar enough with this sort of thing that I no longer wonder what that phase factor means.
Unfortunately, familiarity doesn't necessarily equal understanding, certainly not at the level of being able to explain it to someone who doesn't already have some idea of what's going on. And I'm really not sure how to say anything about this.
The real meaning of the sign in VV +/- HH is something like the relative phase between the HH and VV parts of the wavefunction. I'm not sure how to explain that, though-- in particular, I'm not sure how to convince anybody that that makes VV+HH and VV-HH distinct states.
If you're talking about a single particle, the usual way of vidualizing this sort of thing physically is via the "Bloch sphere," where you express the state as a vector pointing to a point on the surface of a sphere. In this picture, the two different signs correspond to points on opposite sides of the sphere. I'd rather not get into that, though. And anyway, I don't think I've seen a two-particle version of the Bloch sphere, presumably because it would need to be in four or more dimenstions.
If you think about it in terms of the polarization of a single particle, and describe it as a vector, (V+H) would be a polarization of 45 degrees up and to the right, and (V-H) would be a polarization of 45 degree up and to the left. But, of course, HV+VH is not the same thing as (V+H)(V+H), so that doesn't quite work. That might be the best chance at explaining it, though, and as lies-to-children go, it might not be that bad.
I'd be happy to have a better explanation, though, so if anybody knows of one, leave it in the comments. I'll be looking through the various quantum books that I have to see if I can find anything...
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Hmm... Isn't this the key point? the "reality" of all the quantum stuff: there is this layer of reality that we cannot directly observe, all we can see is the "square" of this layer as it were. There are two square roots always for the layer we can see, and we can see the effects of it because they "really" are different from each other, and have different effects later, even if we can't see below the "squared" floor.
Well reading that seems babbling, but whatever, it ran through my head.
I'm not sure how to explain that, though-- in particular, I'm not sure how to convince anybody that that makes VV+HH and VV-HH distinct states.
Have you explained why H+V and H-V are distinct states? i.e. have you allready introduce something like a Hadamard gate?
Another suggestion is that the Bell basis measurement can be thought of as a controlled not, followed by a Hadamard on the controlled qubit, followed by a measurement in the HH, HV,VH,VV basis. This emphasizes that there are four outcomes. Using the fact that the controlled not checks whether the bits are the same or different (the parity) and then noting that the Hadamard is used to move into superposition might be a path towards explaining the phase. But all of this is predicated on a prior discussion of the Hadamard gate.
Have you explained why H+V and H-V are distinct states? i.e. have you allready introduce something like a Hadamard gate?
No, and I was hoping not to get into that level of detail.
I've been trying to deal mostly with probability distributions rather than wavefunctions, as I think that's a little more comprehensible. I think the geometrical explanation of the difference between H+V and H-V (namely, that they're orthogonal states in a basis rotated by 45 degrees, only not in those words) would probably suffice for that.
I'm not planning to go through all the algebra to show that the four outcomes are simple rotations of the original state, either, so there's going to be some hand-waving already. One more might not be fatal, but if there's a better way, I'd like to find it.
I like the geometrical approach and making the analogy with a single particle, as you mention.
With one particle, you have H and V as one basis, but the thing about QM is that you can use a different basis (any two vectors on the unit circle in the H-V plane). So H+V and V-H are just 45 degree rotations of H and V.
Likewise with two particles you have a four dimensional space: HH, HV, VH, VV. The four Bell states are just rotated basis vectors, as in the one-particle case. I wouldn't worry about HV+VH not being the same thing as (H+V)(H+V); that's irrelevant. The Bell states are rotations of the natural basis vectors for the space.
Maybe I'm missing something. There are 4 states as 2*2=4
Show them HH, HV, VH, VV. Then that these other combinations are more useful.
Just like for problems on an plane inclined at 45 degrees, the natural coordingates are x+y and x-y.
The hard part is with spins, why do |+->-|-+> and |+->+|-+> have different magnitudes for the total spin - 0 and 1
That one's tough, 2*2=3+1 but still.......
The four states are distinct because there are measurements that separate them. Maybe it is more intuitive to talk about those measurements than about any elements of the formalism (e.g. I view the Bloch sphere as no more than a mathematical apparatus). Just a thought...
Use geometry and shapes, best way to describe it is to Visualize it in your mind using a visual metaphor, and THEN describe back from the metaphor.
Einstein said "If I can't visualize it, I can't understand it"
Always visualize!
He was onto something with that always making it geometrical and visual, use placeholder shapes, etc. The best metaphor is that of a circuit, as if there was an invisible wire and you were transmitting the state.
Use a 3D program like Truespace (they have a free version at their site), A trial version of Maya or 3DS max if they are available, or ask someone from the art department to animate it for you.
Here is an excellent site over here, maybe he can write a java applet for you? See below.
http://www.phy.ntnu.edu.tw/ntnujava/index.php
I'd say that quantum phase generally represents an interference resource. We can measure the quantum state of a two q-bit system, which will tell us, inter alia, about the relative phases of the two q-bits, then we can use that relative phase as a resource in interactions with other quantum systems to measure properties of those other systems. Once we have characterized a quantum state by measurement, we can stop measuring it and use it. I think you have to get into the use that the quantum phase can be put to, even if you don't talk about the mathematical structure. It's surely important that the difference between (H+V)(H+V) and (HH+VV) can be used, for example, and how it can be used should be describable in terms of the geometry of experimental apparatus, instead of in terms of the relatively more abstract Bloch sphere.
I guess it might be useful to tie your discussion of a two q-bit state to whatever discussion you have of interference for a simple scalar wave function, and how that can be used for simpler applications than teleportation.
I think Perry (#5) is on the right track. HH, VV, HV, and VH could be thought of as somehow linearly independent. Call them x1, x2, x3, x4. Supposing you knew the value of x1 and x2, you could re-encode them into different equations: the values of x1+x2 and x1-x2. Similarly for x3 and x4.
Then, you can hand-wave away why you can't also specify x1-x3 and x1+x3: they aren't independent in the first place (that is, at least one of the labels match). Now, that's an not entirely accurate hand-wave. But that's the point of a hand-wave anyway, isn't it?