Spins and Qubits: Exploring Determinism in Measurements

[giulio_hep]

I'm aware that this is a very basic question, yet I hope to get a non-trivial answer
Let's assume to have an apparatus A (to make measurements) that is oriented in space.
We first point it along the z axis and measure a spin σ_z = 1.
Then we rotate the apparatus through an angle of ½π radians (90 degrees), so that A points along the x axis and we measure the σ_x component of the spin.
Classically, we would expect to get zero but we know that the apparatus will give either σ_x = 1 or σ_x = -1 and we can only foresee that the average of these kind of repeated measurements will be zero.
What I'm unclear about is why we conclude that determinism has broken down, what would make us sure that the result of the second measurement is completely random and there is no more fundamental principle that determines that, even though such a mechanism is unknwon?

 

[Simon Phoenix]

Given that experiment then there isn't any way we could rule out some extra, unknown, mechanism that is making the results appear random - but which, if we only knew the mechanism and its associated variables, would restore determinism. But if there were these kinds of hidden mechanisms then they must be quite peculiar. Considering the case of atoms, if these mechanisms did exist then they can't obey the rules of ordinary thermodynamics because they'd contribute to the specific heat capacity and we would predict a specific heat capacity at variance with experiment (this argument is outlined much better in the first couple of pages of Dirac's classic QM textbook). It's pretty much ruling out hidden variables (at least any hidden variables subject to the laws of thermodynamics).

I'd also appeal to the universality of QM - it applies to everything as far as we can tell. So any hidden mechanism would have to have a similar kind of generality and not just work in an ad-hoc way for one specific system. It would also have to apply to every observable we could conceivably measure.

For me personally the final nail in the coffin was Bell's tour de force when he showed that systems obeying some very general reasonable classical assumptions could not match the quantum predictions. Loosely speaking one of these assumptions is that there exist these hidden properties that exist independently of measurement - the so-called hidden variables (which presumably describe the kind of unknown mechanism you ask about). Another assumption made is that nature is local in that results measured in a lab do not depend on the settings of instruments in some distant lab. Bell's result, his celebrated inequality, is an entirely classical result that could have been derived (and essentially was by Boole - at least mathematically) quite independently of QM.

However, for some reason threads on entanglement seem to get closed down - so probably best to tread lightly :-)

 

[giulio_hep]

Given that experiment then there isn't any way we could rule out some extra, unknown, mechanism that is making the results appear random - but which, if we only knew the mechanism and its associated variables, would restore determinism. But if there were these kinds of hidden mechanisms then they must be quite peculiar. Considering the case of atoms, if these mechanisms did exist then they can't obey the rules of ordinary thermodynamics because they'd contribute to the specific heat capacity and we would predict a specific heat capacity at variance with experiment (this argument is outlined much better in the first couple of pages of Dirac's classic QM textbook). It's pretty much ruling out hidden variables (at least any hidden variables subject to the laws of thermodynamics).

I'd also appeal to the universality of QM - it applies to everything as far as we can tell. So any hidden mechanism would have to have a similar kind of generality and not just work in an ad-hoc way for one specific system. It would also have to apply to every observable we could conceivably measure.

For me personally the final nail in the coffin was Bell's tour de force when he showed that systems obeying some very general reasonable classical assumptions could not match the quantum predictions. Loosely speaking one of these assumptions is that there exist these hidden properties that exist independently of measurement - the so-called hidden variables (which presumably describe the kind of unknown mechanism you ask about). Another assumption made is that nature is local in that results measured in a lab do not depend on the settings of instruments in some distant lab. Bell's result, his celebrated inequality, is an entirely classical result that could have been derived (and essentially was by Boole - at least mathematically) quite independently of QM.

However, for some reason threads on entanglement seem to get closed down - so probably best to tread lightly :-)

Thanks, your answer is quite clear and helpful. I'm just a bit confused about Bell's result (because what I find reasonable is a sentence from the wiki paragraph where in particular Jaynes is mentioned and it is said that "Causes cannot travel faster than light or backward in time, but deduction can.").
Anyway, my question is not about the entanglement but about the measurement of a (single) qubit. In conclusion, if I correctly understand your answer, you are saying that hidden variables are pretty much - but not completely - ruled out.

 

[Simon Phoenix]

I think I'd say local hidden variables haven't 'pretty much' been ruled out they have been ruled out :-)

It's non-local hidden variable theories that cannot be ruled out. The classic example of this is de Broglie/Bohm pilot wave stuff. This is a manifestly non-local theory (the variables are not so 'hidden' either) that, it is claimed, reproduces all the results of QM. That's a claim which doesn't wholly convince me, but that's a matter for another thread.

But the experiment you describe - where you've got a state prep followed by measurement of a single spin direction, is not sensitive enough to allow us to distinguish between QM and some hidden variable version.

I think I'd emphasize the appeal to universality again. You can set up a 'spin-type' measurement for almost any observable, at least in principle. If we have N eigenstates, say, then we can formally consider a projection P_n or P_m with P_n + P_m being the identity. OK we might not get uniform randomness of the measurement results as we do in the spin case you describe, but I think your issue is with randomness anyway. The point here is that if we want there to be some hidden mechanism (or mechanisms) that's orchestrating all this then it's (they're) going to have to work for electrons, photons, molecules, etc - and for every single observable we can imagine - spin, energy, polarization etc

There's also another argument which you might want to look at contained in the introduction of Feynman's famous 'path integral' paper. It's very readable and characteristic of the great man to hit the spot directly.

Essentially the argument is to look at the conditional probabilities of starting at A, moving through some supposed properties B and ending up at C. Classically we construct a chain of conditional probabilities P(C|A) = Σ_B P(C|B) P(B|A). We're saying that something starts at A moves through a possible set of states B and ends up at C - but we don't know which B - and so we sum over all possible paths to obtain the total conditional probability.

QM has an exactly similar rule - but if we don't measure the observable B to find out 'which path' - then this rule applies to amplitudes not probabilities. It's this that's giving us the possibility of interference. The point here is that if we suppose we've started at A and arrived at C and we suppose that we've had to move through some existing definite intermediate states (our unknown mechanism) then we end up with an incorrect formula. Whilst this argument is not watertight - I think it's useful. Of course it was really nailed down by Bell.

 

[giulio_hep]

Thanks so much for your very extensive reply!
I'm fully convinced that my original question is answered:
should they only do experiments with a single qubit, they could not exclude a deterministic explanation (I hope I've not misunderstood).
That's important to me because my post stems from reading a book of Susskind and you seem to agree with me that he started writing a bit too early (immediately after the single qubit/spin experiment) that "determinism has broken down". Your reply does more than this and I can maybe summarize it in 3 parts: local hidden variables, N eigenstates observable, Feynman's path integral.
As far as part 2 and 3 are concerned, I'm tempted to think that if there were an unknown mechanism under the amplitude wave function, that same hidden mechanism would explain all the other phenomena that are described by amplitudes, but I'm probably still missing some important aspects. Regarding part 1, I'm also wondering whether the theorem for local variables is based on the assumption that the result of a measurement is probabilistic and whether the theorem can be applied when one considers that result as deterministic "deduction" (Gerard 't Hooft for example?). Also I don't know if a measurement destroys an entanglement: I suspect yes and that would exclude the most direct proof of randomness, in my humble opinion. Thanks again.