Classical analogue of quantum contextuality

[Demystifier]

QM is often thought to be strange due to contextuality - the fact that measurement outcomes depend on measurement itself. Today a nice paper
http://lanl.arxiv.org/abs/1703.07550
appeared that presents a classical analogue of quantum contextuality. I hope it can help to make contextuality look more intuitive and less strange.

 

[LeandroMdO]

I thought this insight was known ever since Kochen and Specker gave their spherical model of a single spin.

 

[Demystifier]

I thought this insight was known ever since Kochen and Specker gave their spherical model of a single spin.

To a certain extent It was known, but it seems that it was not sufficiently widely known.

 

[stevendaryl]

I think that there is nothing too mysterious about contextuality. The paper's example of clapping your hands on a coin tumbling through space is a good example. Contextuality to me means that your interaction with some system produces a result that is not a pre-existing property of the system. However, in most such cases from classical mechanics, people don't think of such an interaction as a "measurement". If I play a game of chess, the outcome will be win/lose/draw, but I wouldn't think of the chess game as a measurement of the win/lose/draw property of the chessboard. So to me, the weird thing about contextuality in quantum mechanics is the uneasy tension between contextuality and the claim that you've actually measured something. In Bohmian mechanics, there is a sense in which you haven't actually measured anything when you've measured spin.

I guess the nice thing about the coin example is that there actually is a property that you can measure--the coin really does have an orientation, and if you clap your hands just right, you can measure it. But if you clap your hands in the wrong way, you get what appears to be a measurement of the coin orientation, although it's an artifact of your clapping, rather than a pre-existing property of the coin. So, now that I think about it, it is a pretty good analogy for quantum measurements: If a particle is prepared to be spin-up, you'll measure spin-up, so you're measuring a pre-existing quantity. If the particle is prepared to be a superposition of spin-up and spin-down, you'll get what looks like a spin measurement that doesn't correspond to a pre-existing quantity.

 

[ShayanJ]

I think that there is nothing too mysterious about contextuality. The paper's example of clapping your hands on a coin tumbling through space is a good example. Contextuality to me means that your interaction with some system produces a result that is not a pre-existing property of the system. However, in most such cases from classical mechanics, people don't think of such an interaction as a "measurement". If I play a game of chess, the outcome will be win/lose/draw, but I wouldn't think of the chess game as a measurement of the win/lose/draw property of the chessboard. So to me, the weird thing about contextuality in quantum mechanics is the uneasy tension between contextuality and the claim that you've actually measured something. In Bohmian mechanics, there is a sense in which you haven't actually measured anything when you've measured spin.

I guess the nice thing about the coin example is that there actually is a property that you can measure--the coin really does have an orientation, and if you clap your hands just right, you can measure it. But if you clap your hands in the wrong way, you get what appears to be a measurement of the coin orientation, although it's an artifact of your clapping, rather than a pre-existing property of the coin. So, now that I think about it, it is a pretty good analogy for quantum measurements: If a particle is prepared to be spin-up, you'll measure spin-up, so you're measuring a pre-existing quantity. If the particle is prepared to be a superposition of spin-up and spin-down, you'll get what looks like a spin measurement that doesn't correspond to a pre-existing quantity.

This picture works for a spin-1/2 particle whose spin part of the state is a pure state. But it doesn't work for e.g. either of the particles that are in a Bell state. Their spins are simply not aligned in any particular direction!

 

[martinbn]

As they point out if the angles for repeated measurements are not zero or ninety degrees the coin toss experiment is nothing like the QM case so the mystery actually remains. Also if you don't read the BM parts there is nothing in the paper.

 

[stevendaryl]

This picture works for a spin-1/2 particle whose spin part of the state is a pure state. But it doesn't work for e.g. either of the particles that are in a Bell state. Their spins are simply not aligned in any particular direction!

I don't see a huge difference for contextuality between measuring the spin in the z-direction for a particle that is in a superposition of spin-up and spin-down and measuring the spin in the z-direction of one particle of entangled pair. In both cases, there is no pre-existing value for the z-component of spin. You're certainly right that in the case of entangled particles, there is no pre-existing value for the spin in any other direction, either. The point is that the same measurement process (send a particle through a Stern-Gerlach device) sometimes reveals a pre-existing value, and sometimes doesn't.

 

[ueit]

I think that this paper does a poor job in providing a classical analog to quantum contextuality. In QM the measured value may depend on changes in the measurement settings far away from the place of measurement. The coin toss example does not deal with that.

Of course, there is nothing mysterious in the fact that the configuration of distant objects may alter a measurement result, but this influence can only be understood classically using the concept of a field. In a double-slit experiment the particle "knows" if a distant slit is open or closed because the field produced by the barrier at the particle's location contains that information. So, a different measurement setting could give you a different result not merely because the particle/instrument interaction changes but because the particle itself has a different evolution prior to the time of the measurement.

The paper is also wrong in claiming that there is no preexisting value. Of course it is. One can determine that value by recording the coin toss with a camera and analyzing the position of the coin at the desired time. The hand-clapping "measurement" is not required.

Andrei

 

[ShayanJ]

I don't see a huge difference for contextuality between measuring the spin in the z-direction for a particle that is in a superposition of spin-up and spin-down and measuring the spin in the z-direction of one particle of entangled pair. In both cases, there is no pre-existing value for the z-component of spin. You're certainly right that in the case of entangled particles, there is no pre-existing value for the spin in any other direction, either. The point is that the same measurement process (send a particle through a Stern-Gerlach device) sometimes reveals a pre-existing value, and sometimes doesn't.

My point is that, if it was only for the pure states, we could say that there does exist a measurement that results in a pre-existing value and the fact that we're getting a contextual result is because we're doing a wrong measurement. But this logic breaks down for a particle in an entangled pair, a right measurement simply doesn't exist. This makes it just a simple analogy that breaks too soon to be of any use.

 

[stevendaryl]

My point is that, if it was only for the pure states, we could say that there does exist a measurement that results in a pre-existing value and the fact that we're getting a contextual result is because we're doing a wrong measurement. But this logic breaks down for a particle in an entangled pair, a right measurement simply doesn't exist. This makes it just a simple analogy that breaks too soon to be of any use.

I guess I agree. Which to me means that it's much too glib to explain away EPR by saying "quantum measurements are contextual". That's a true statement, but it doesn't help to explain what's weird about EPR.

On the other hand, for Bohmian mechanics, spin measurements are always contextual (there is no property of having a spin component in the z-direction).

 

[stevendaryl]

My point is that, if it was only for the pure states, we could say that there does exist a measurement that results in a pre-existing value and the fact that we're getting a contextual result is because we're doing a wrong measurement. But this logic breaks down for a particle in an entangled pair, a right measurement simply doesn't exist. This makes it just a simple analogy that breaks too soon to be of any use.

Another comment about your comment: If we're trying to explain why Alice's measurement of spin in the z-direction is correlated with Bob's measurement of spin in the z-direction, why is it relevant whether or not there is some other direction (maybe the x-direction) for which the particle has a definite value for spin?

 

[ShayanJ]

Another comment about your comment: If we're trying to explain why Alice's measurement of spin in the z-direction is correlated with Bob's measurement of spin in the z-direction, why is it relevant whether or not there is some other direction (maybe the x-direction) for which the particle has a definite value for spin?

I don't quite understand the question because if there was a direction for which the particles had a definite value for spin, they wouldn't be entangled!

 

[Demystifier]

Also if you don't read the BM parts there is nothing in the paper.

What do you mean by "nothing"? Nothing interesting? Nothing nontrivial? Nothing new? Nothing mathematical?

 

[stevendaryl]

I don't quite understand the question because if there was a direction for which the particles had a definite value for spin, they wouldn't be entangled!

What I'm saying is that the contextuality of measurements of the z-component of spin seems to be the same, regardless of whether there is a definite value for spin in another direction. Having a definite value in the x-direction doesn't explain contextuality of measurements in the z-direction, and lacking a definite value in the x-direction doesn't explain contextuality of measurements in the z-direction.

 

[martinbn]

What do you mean by "nothing"? Nothing interesting? Nothing nontrivial? Nothing new? Nothing mathematical?

Yes to all of these. (or am I supposed to say no, how do I answer negative questions clearly? what I mean is none of these is in the paper outside the sections about BM)

 

[stevendaryl]

What I'm saying is that the contextuality of measurements of the z-component of spin seems to be the same, regardless of whether there is a definite value for spin in another direction. Having a definite value in the x-direction doesn't explain contextuality of measurements in the z-direction, and lacking a definite value in the x-direction doesn't explain contextuality of measurements in the z-direction.

The weird thing about quantum mechanics is that a measurement that is contextual is completely indistinguishable from a measurement that actually reveals a pre-existing value. To use the example of clapping a coin between your hands: If the coin is oriented parallel to your hands, then the clapping is gentle. If the coin is oriented perpendicular to your hands, then the clapping is violent--the edge of the coin digs into your hands. The latter case is an irreversible change; it causes heat.

In the quantum case, you can figure out from nonlocal information whether Bob's measurement revealed a pre-existing value or whether it was contextual. But there is no difference between those cases that is visible to Bob. At least not for a single "run" of the experiment.

 

[ShayanJ]

What I'm saying is that the contextuality of measurements of the z-component of spin seems to be the same, regardless of whether there is a definite value for spin in another direction. Having a definite value in the x-direction doesn't explain contextuality of measurements in the z-direction, and lacking a definite value in the x-direction doesn't explain contextuality of measurements in the z-direction.

What does "contextuality in the z-direction" mean? How can we determine whether the results of our measurements are contextual or not if we only measure the spins in only one direction?
Sorry but its easy for me to get confused in such discussions because I'm still in the process of learning this part of physics.

 

[stevendaryl]

What does "contextuality in the z-direction" mean?

It means that a measurement of spin component in the z-direction results in a value that is not a pre-existing value, but is produced by the act of measurement.

How can we determine whether the results of our measurements are contextual or not if we only measure the spins in only one direction?

The theory tells you whether it's contextual or not. You can't tell by measurements alone (that is, there is no difference for Bob whether his measurement reveals a pre-existing spin value or not).

 

[zonde]

My point is that, if it was only for the pure states, we could say that there does exist a measurement that results in a pre-existing value and the fact that we're getting a contextual result is because we're doing a wrong measurement. But this logic breaks down for a particle in an entangled pair, a right measurement simply doesn't exist. This makes it just a simple analogy that breaks too soon to be of any use.

You can't claim that pre-existing does not exist because they can't be the same for all particles in ensemble. The most that you can say is even if pre-existing values exist the model is still incomplete and can't explain entangled particles.

 

[ShayanJ]

It means that a measurement of spin component in the z-direction results in a value that is not a pre-existing value, but is produced by the act of measurement.

Yeah, I know. But contextuality, as far as I know, is proved by having quantities related to different directions and proving that they couldn't be pre-existing altogether. How does that make sense for only one direction?

 

[ShayanJ]

You can't claim that pre-existing does not exist because they can't be the same for all particles in ensemble. The most that you can say is even if pre-existing values exist the model is still incomplete and can't explain entangled particles.

I was talking about standard QM. Each of the particles in an entangled pair are in an unpolarized improper mixed state which means all directions have the same probability.

 

[zonde]

I'm was talking about standard QM.

Me too.

 

[stevendaryl]

Yeah, I know. But contextuality, as far as I know, is proved by having quantities related to different directions and proving that they couldn't be pre-existing altogether. How does that make sense for only one direction?

Yes, the experimental proof of contextuality in the z-direction depends on considering other directions, but the fact of contextuality doesn't depend on this. If you prepare an electron to be spin-up in the x-direction, then the spin in the z-direction is completely undetermined, which makes a measurement of spin in the z-direction contextual, even though in this case, the electron is not part of an entangled pair.

I think there is a sense in which the EPR experiment is not that different from the case of measuring the z-component of spin for a particle prepared to be spin-up in the x-direction. In both cases, you're preparing a system in a particular pure state, and then you're measuring an observable for which that state is not an eigenstate:

• Case A: Prepare a single electron in the state $\frac{1}{\sqrt{2}} (|u\rangle - |d\rangle)$. Measure $\sigma_z$. The original state was not an eigenstate, so you get a nondeterministic outcome.

• Case B: Prepare a pair of particles in the state $\frac{1}{\sqrt{2}} (|u\rangle |d\rangle - |d\rangle |u\rangle)$. Measure $(\sigma_{\hat{a}}, \sigma_{\hat{b}})$ (that is, do the composite measurement of measuring spin in the $\hat{a}$ direction for the first particle and $\hat{b}$ direction for the second particle). The original state was not an eigenstate, so you get a nondeterministic outcome.

So there isn't a big difference, as far as I can see. Except that in Case A, you might believe that there is a local hidden-variable governing your seemingly nondeterministic outcome, while in Case B, you can prove that there is not.

 

[Nugatory]

The paper is also wrong in claiming that there is no preexisting value. Of course it is. One can determine that value by recording the coin toss with a camera and analyzing the position of the coin at the desired time. The hand-clapping "measurement" is not required.

Any analogy based on a classical system will be subject to that criticism (and will ultimately fail as anything more than an analogy) because we have a satisfactory local and non-contextual hidden-variable theory for classical systems. However, the paper isn't quite exactly claiming that there is no preexisting value. It is claiming that we can analyze the system as if there is no preexisting value (which is a reasonable treatment of naked-eye observation of a spinning coin) and we will end up with a useful analogy... And as analogies go, this one is probably about as good it gets.