Are The Odds Regarding Particles Definitely Independent Events?

[Metric]

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Doesn’t entanglement entail action at a distance? If so, Metric, I thought this was something that you are arguing against.

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Entanglement is fine and good. Conditional probabilities are fine and good. Non-unitary "wave function collapse" over extended distances is bad and wrong, but I'm willing to pretend such things exist at an "intro to QM" level, since such notions are widely in use.

[flipdeadshot22]

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This seems to me like a philosophical question rather than one that can be proven or shown empirically. A similar situation occurs in statistical mechanics, where the fundamental postulate is that all microstates occur with equal probability. This assumption leads to models that accurately describe real physical systems, but the postulate itself is unproven.

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You should probably attempt to learn a little basic QM before handing off such a question to philosophy. Determining probabilities for quantum mechanical systems to undergo a given set of transitions has a fundamental connection to the commutivity of the operators that define such transitions or "events" as DS put it. Commutivity is basically a mathematical statement that expresses our ability to 'observe independent quantum events' (none of this is standard QM terminology, since i'm trying to keep this simple and avoid making my post 5 pages and filled with Latex coding.)

A good example of this is the non-commutivity of the momentum (p) and position (x) operators, which can mathematically be expressed as [x,p]=ihbar. The fact that the right side of the equation is nonzero means that the two operators do not commute, and that once you have performed a measurement and determined the exact position of the particle, you cannot subsequently measure its momentum with any certainty. So getting back to what david was asking, the odds regarding the measurement of position and after that, the measurement of momentum are not independent.

Contrast this to the measurement of the spin of a particle. Spin is a 3 dimensional property of many quantum systems, and has the property that the measurement of the spin in the x, y or z direction, followed by another measurement in a different direction than the first measurement CAN BE considered independent or, mathematically stated for instance [Sz, Sx] = 0. This shows the commutivity of the spin in the z and x directions (this can be generalized to all directions). This means that if we measure the spin of a particle to be up with 100% certainty, we can follow this up with another measurement to see what the spin is in the x direction and obtain a 0% probability that it exists in this state.

[flipdeadshot22]

Just to correct myself in my above post before metric jumps on me, the commutation relation for the Sz Sx spin operators [Sz, Sx] = 0, is incorrect (these operators actually anti-commute)

Also, in case my post was tl;dr, it applies only to scenarios in which we measure two distinct properties of a system, rather than running multiple trials of measuring a single property (such as that of spin in davids OP.) In the case we are determining whether a system is in a spin up or spin down with a 50/50 chance, then yes this is your basic coinflip with no "hidden variables" or deterministic causes.
The usual form of QM does not say anything about these actual deterministic causes that lie behind the probabilistic quantum phenomena. This fact is often used to claim that QM implies that nature is fundamentally random. Of course, if the usual form of QM is really the ultimate truth, then it is true that nature is fundamentally random. But who says that the usual form of QM really is the ultimate truth? (A serious scientist will never claim that for any current theory.) A priori, one cannot exclude the existence of some hidden variables (not described by the usual form of QM) that provide a deterministic cause for all seemingly random quantum phenomena. I think a good example of this is the Bohm interpretation (check out wiki).

[PairTheBoard]

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A priori, one cannot exclude the existence of some hidden variables (not described by the usual form of QM) that provide a deterministic cause for all seemingly random quantum phenomena.

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I thought that was the whole point to Bell's Theorem. That if you assume such a hidden variable it leads to contradictions that can be observed.

PairTheBoard

[Metric]

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A priori, one cannot exclude the existence of some hidden variables (not described by the usual form of QM) that provide a deterministic cause for all seemingly random quantum phenomena.

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I thought that was the whole point to Bell's Theorem. That if you assume such a hidden variable it leads to contradictions that can be observed.

PairTheBoard

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Bell's theorem basically says that perfectly classical, local, hidden variable theories "aren't good enough" to reproduce experimentally confirmed predictions of QM. That's slightly different from saying there are no hidden variables. There might be hidden variables. But by themselves they cannot be a replacement for QM -- you would have to add some more ingredients (usually to violate locality or something else somewhat disturbing).

[Metric]

Let's do a little example to see things explicitly.

Let's consider a two-qubit system. A single qubit is a system spanned by the states |0> and |1>.

I will call the first qubit "a" and the 2nd qubit "b."

Consider the following state: 1/2 (|0> + |1&gt_a (|0> + |1&gt_b

The probability to measure "a" in state |0> is 1/2, completely independent of whether "b" was measured to be in |0> or |1>, or not measured at all. This is due to the fact that the state is seperatble -- I can express the state as PSI_a PSI_b.

Now consider the following state:
1/root2 (|0>_a |0>_b + |1>_a |1>_b)

Now, the probability to measure "a" in state |0> is still 1/2 if I don't measure "b". BUT, if I do measure "b" to be in state |0>, then there is 100% chance that "a" will also be in |0>. And likewise if "b" was measured to be in state |1>, then "a" will also be discovered to be in state |1> with certainty. Correlations like this are the hallmark of entanglement -- the fact that I can't write this state as PSI_a PSI_b.

The relation to Bell's theorem is that all entangled (pure) states of a system like this one can also be used to violate a Bell inequality, which can't be done classicaly (modulo truly bizarre nonlocal theories).

[gumpzilla]

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I thought that was the whole point to Bell's Theorem. That if you assume such a hidden variable it leads to contradictions that can be observed.

PairTheBoard

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Bell's theorem basically says that perfectly classical, local, hidden variable theories "aren't good enough" to reproduce experimentally confirmed predictions of QM. That's slightly different from saying there are no hidden variables. There might be hidden variables. But by themselves they cannot be a replacement for QM -- you would have to add some more ingredients (usually to violate locality or something else somewhat disturbing).

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Yeah. Something like Bohm/De Broglie's pilot wave trick works just fine as a hidden variable theory, but it's no longer local (meaning it allows for superluminal influences.) At least that's my understanding; I've been meaning to read Bohm's papers on this subject to actually learn what it is he says, but right now I'm just sort of quoting the conventional wisdom.

Regarding conditional probabilities, I've been reading some interesting stuff recently on weak measurement in quantum mechanics. Essentially, the idea is that if you have an ensemble of states that are both pre and post selected (meaning we know the state of the particle at time t_1 and t_2), applying conditional probability to the results of measurements at time t_1 < t < t_2 allows for some really weird, counterintuitive stuff. When I understand it better, I might write up a post on it.

[Metric]

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Yeah. Something like Bohm/De Broglie's pilot wave trick works just fine as a hidden variable theory, but it's no longer local (meaning it allows for superluminal influences.) At least that's my understanding; I've been meaning to read Bohm's papers on this subject to actually learn what it is he says, but right now I'm just sort of quoting the conventional wisdom.

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Yeah -- the Bohm-De Broglie stuff is really weird. It works (at least in the nonrelativistic case -- not sure how far they've gotten with field theory), but it has an extreme feel of jury-rigging about it. And at the end of the day the entire motivation is simply that you don't need to give up the traditional concept of an "always localized" particle -- a concept I'm very happy to live without anyway.

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Regarding conditional probabilities, I've been reading some interesting stuff recently on weak measurement in quantum mechanics. Essentially, the idea is that if you have an ensemble of states that are both pre and post selected (meaning we know the state of the particle at time t_1 and t_2), applying conditional probability to the results of measurements at time t_1 < t < t_2 allows for some really weird, counterintuitive stuff. When I understand it better, I might write up a post on it.

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Are you talking, by chance, about the delayed choice quantum eraser experiment?

http://en.wikipedia.org/wiki/Delayed...quantum_eraser

[gumpzilla]

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Regarding conditional probabilities, I've been reading some interesting stuff recently on weak measurement in quantum mechanics. Essentially, the idea is that if you have an ensemble of states that are both pre and post selected (meaning we know the state of the particle at time t_1 and t_2), applying conditional probability to the results of measurements at time t_1 < t < t_2 allows for some really weird, counterintuitive stuff. When I understand it better, I might write up a post on it.

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Are you talking, by chance, about the delayed choice quantum eraser experiment?

http://en.wikipedia.org/wiki/Delayed...quantum_eraser

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It was not what I had in mind, but it is certainly possible that similar ideas are in play. I have heard of this experiment before but I am not very familiar with it; I'll print out the PRL and add that to my list of reading on this subject.

I don't have my notes on the relevant papers with me, and I'm finding that without them it's hard for me to reconstruct enough of the arguments to really explain them, so I'll try and write more about them in the near future. But, the general idea is that in a situation where you both pre and postselect ensembles (the key thing I'm missing in trying to reconstruct it is why this is necessary), if you employ a von Neumann measuring scheme with a weak enough coupling Hamiltonian, the value that you measure for various observables corresponds to a particular average value for these "generalized states" that consist of a ket evolving forward from t_1 and a bra evolving backwards from t_2.

From this, weird stuff follows. Among other things pointed out:

- The result of the weak measurement of an operator can be greater than the largest eigenvalue of that operator (!)

- To quote one of the section headings from one of these papers, "Two noncommuting observables have definite values in the time period between two measurements." I think this is special behavior based on what your finishing and ending states are; intuitively, the idea is supposed to be that the measurements are "weak" enough that making a measurement of B doesn't perturb the measurement you made of the noncommutative operator A.

- In another one of the papers, a specially constructed state (where a particle is in one of N+1 boxes) is given such that opening any among N of the N+1 boxes will find the particle with certainty. Apparently this is compensated for with some kind of "negative probability." I don't even know what the hell this is supposed to mean yet, and it sounds like complete quackery. But, I was at a talk where some experimental data was shown (by a pretty sharp guy) that apparently could be interpreted in this way. This is what sparked my interest in it.

Like I said, when I understand it more I'll try and write something lengthy up. If you're interested right now, the papers I'm drawing the theoretical statements from are:

Aharonov and Vaidman, PRA 41 11

Aharonov and Vaidman, Journal of Physics A, 24 2315

[PairTheBoard]

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Are you talking, by chance, about the delayed choice quantum eraser experiment?

http://en.wikipedia.org/wiki/Delayed...quantum_eraser

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I have a question about the Quantum Eraser. In this link,

Quantum Eraser Experiment

where they show the diagrams, they say this
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At time T0 when D0 is triggered no interference appears, since the which-way information is contained in the system at that time. At time T1, which in the experiment is some nanoseconds later but could be in principle any time later, when D1/D2/D3/D4 are triggered, we find interference in the correlated subsets of past D0 records undergoing future erasure of the which-way information.
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If the principle is that Erasing the Which-Slit information allows an Interference Pattern as if no Which-Slit was ever gathered, then why couldn't you just direct all the idler photons directly to the BS spliter and erase the information for all the photons. If the idea is as they say, "At time T0 when D0 is triggered no interference appears, since the which-way information is contained in the system at that time", why couldn't you just move the Erasing Apparatus closer so that the Which-Way information is erased by the time the photon reaches D0?

Of course, if the Timing of the Erasure were to affect the Interference pattern at D0 for All photons it would be bad because you could then move the Erasure Apparatus closer and farther away to send faster than light messages.

So I assume it doesn't work that way. Which is funny because it seems like there's no reason why it shouldn't except for the fact that it would then violate faster than light communication.

PairTheBoard