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The wave function determines the probabilities of an outcome, and the wave function is parameterized "shaped" depending on the measurement set.

i.e. two different measurement set would lead to two different wave function description. Let's call A measurement set=>wave function A, and B measurement set=>wave function B.

Having two entangled particles X and Y, there is something I can't understand,

Supose we make successive measurement with A experimental set to successive generated entangled particles X : X, X',X'', X''' ...

So measurements should match with the expected probabilities outcomes predicted by the wave function for A.

Then the entangled counterpart measured with B experimental set to successive generated entangled particles Y : Y, Y',Y'', Y''' ...

So measurements should match with the expected probabilities outcomes predicted by the wave function for B.

How can the entangled particles satisfy both wave equation and at the same time, satisfy the entanglement condition?

If it can't satisfy both, so decoherence of the entangled counterpart could be detected by a deviation of the outcomes and the expected probabilities. If it can satisfy both, then there is no effect in being entangled.

Where is the problem?

Thanks in advance for any help about understanding this

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Hi HDE, You phrase your Question and your response to John Schanck in such a way that it looks a little as if you might be having trouble with the classical concept of correlation, as much as with the more subtle quantum mechanical concept of entanglement. As a diagnostic, I'd like you either to tell us about the difference between classical correlations and quantum entanglement or to tell us about classical correlations without worrying about quantum mechanics at all. –  Peter Morgan May 27 '11 at 1:58
    
@Peter Morgan, I would like to know about those subtle differences you mention, an answer covering the topic would be great, thanks –  HDE May 27 '11 at 12:21

2 Answers 2

up vote 5 down vote accepted

Suppose the entangled particles are generated in a singlet state, $\frac{1}{\sqrt{2}} (\vert 01 \rangle - \vert 10 \rangle)$. If measurement set A is the standard basis, $\{ \vert 0 \rangle, \vert 1 \rangle \}$, then we would expect the result to be "zero" with probability $\tfrac{1}{2}$ and "one" with probability $\tfrac{1}{2}$.

If I've read your question correctly, it seems that your confusion comes from believing that the measurement results of entangled particles must always be identical. Barring experimental error, this is only the case when the measurement bases, A and B, are identical. If A and B were both the standard basis, then there would be perfect correlation between the A measurements and the B measurements.

If on the other hand, B is a basis like $\{ \vert + \rangle, \vert - \rangle \}$ where $\vert + \rangle = \frac{1}{\sqrt{2}}(\vert 0 \rangle + \vert 1 \rangle)$, and $\vert - \rangle = \frac{1}{\sqrt{2}}(\vert 0 \rangle - \vert 1 \rangle)$, then we only expect a 50% correlation between the results of A measurements on X and B measurements on Y.

This correlation is still quite good, and is the foundation of Bell-type inequalities, but it is not so good that either party learns anything about the other's choice of basis (or in your terminology: sees a deviation in expected probability based on the decoherence of their counterpart). Both parties always expect to see the "zero" and "one" states with 50% probability - the entanglement only affects the correlation between the remote parties results were they to compare them.

EDIT: @HDE, With respect to the results of measurements in a fixed basis, non-entangled particles are much like weighted coin flips. So the experiment in which the parties have non-entangled particles is not much different from if we let two parties choose arbitrarily weighted coins and had them flip those coins to generate their results. Now, the probability that both parties get the same result is a function of which coin they each chose, if they each chose a coin that's weighted 90% towards heads, then they'll get the same result around 82% of the time. 81% of the time they'll both get heads, and 1% of the time they'll both get tails.

Let's look at entanglement, here the probability that both parties get the same result is given by $\cos^2({\theta})$ where $\theta$ is the "angle" between their bases. In my example with singlets above the angle was $\pi/4$ and so there was a 50% correlation between the parties' results. With an angle of $\pi/8$ we see an 85% correlation. This is the key difference, the correlation is governed by the similarity of the bases, not the frequency of particular outcomes. An outcome that occurs 10% of the time will yield a coincidence with probability $\cos^2({\theta})$, which will generally not be 10%. To emphasize this, let's think about the weighted coin flips. If Alice gets a tails on a 90% heads coin, then even in an optimistic case, where Bob chooses the same coin, she only expects Bob to get a tails 10% of the time. With entangled particles, when Alice sees a tails, even if it only occurs one in a million times, she expects Bob to see a tails with certainty assuming he chose the same basis, or with probability .85 if he chose one $\pi/8$ radians off, and so on.

Hope that helps

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I suspect that you know the answer that I am looking for, but I still can't understand you, I don't think that results of entangled particles must always be identical, but I really don't know what difference to expect about two entangled particles experiment compared with a pair of non-entangled particles one, that's the core of the question, if it were possible, and you could add more explanation or examples without the use of bra-ket notation, It will be great, anyway thanks for your answer –  HDE May 26 '11 at 2:45
    
Although I don't know the source or definition of $\theta$ or "the angle between their bases" meaning, the weighted coin example enlighten it, thanks +1 –  HDE May 30 '11 at 14:29
1  
The $\cos^2({\theta})$ result can be derived from the fidelity of pure states (en.wikipedia.org/wiki/Fidelity_of_quantum_states#Pure_states) and from the Chaucy-Schwarz inequality (see unapologetic.wordpress.com/2009/04/17/inner-products-and-angles to learn about the geometric interpretation of Chaucy-Schwarz) applied to corresponding elements of each parties' basis. The fidelity of pure states is the absolute value of their inner product - the result is a probability amplitude, so it must be squared to obtain a true probability. –  John Schanck Jun 2 '11 at 15:29

Suppose we put two people in a room and ask them to toss coins. We expect them to get half heads and half tails. We also expect that when we look at both their coins together they will get opposites, head-and-tail, as often as they get the same, head-and-head or tail-and-tail. If the two coins come up randomly, but they're always either both heads or both tails, we never see head-and-tail, that's a correlation, and we might look for an explanation of that, even though we wouldn't have to look for any explanation of the way each coin came up. The Reichenbach common cause principle expresses precisely this requirement.

At this point I'll point out (coarsely) the difference between an empiricist and a realist (and noting that there are instrumentalists, naive or structural realists, etc.). An empiricist looks at the correlations and, finding that those correlations are always there, figures out a way to use them, without much caring what explanation or causes there might be for the correlations. Some people are temperamentally quite happy not to care how or why, they're happy to use it. A realist cares about how and why it happens. The proportions of Physicists who think in one way or the other, and who think in both ways depending on the situation they face in their research, goes in cycles. Of course an empiricist is still curious, it's just that their curiosity runs in different pathways. It's important that there are both ways of thinking and that there's an interplay between them.

Suppose now that we do a different experiment. A psychologist gives two people two buttons each, and asks them to press either one or the other, and then looks to see both what the patterns of the individual button presses are and what correlations there are between them. The psychologist carries out lots of tests, for a whole day at a time, and pays the subjects more if they maintain an average of 100 button presses per minute. That's 50000 events per day, say, in contrast to that many events per second, or far more still, in a Physics experiment. Nonetheless, lots of data. In each test, different feedback is given about what the other person has pressed. In one test, we give audible feedback immediately after the button is pressed, in another half a second after the button is pressed, in another we give visual feedback, perhaps of different colors and perhaps in different positions. We try lots of different alternatives, and discover that there are different correlations in each case. People are not very good at being random, but after an hour of something as boring as this someone who is taking the project seriously would get into a vibe with the other person. Finally, we could also introduce a third person whose job it is to try to get the two other people to be synchronized. They can control the feedback lights however they like. Imagine any configuration and different kinds of feedback you like and speculate on what the results would be.

The physics does just this. There are two detectors and a central source. If there's no source, the detectors "trigger" every now and then, which is called the "dark rate". When we introduce different central sources, we see different statistics for both the rate at which we see events and for the correlations between the events. We also see different rates and correlations if we introduce different detectors. Without knowing why there are different rates and correlations, we can see that there are different rates and correlations. We can use those different rates and correlations.

Where quantum theory differs from classical probability is that we describe the results we get for a single type of source for all different types of detectors, and, separately, the results for a single type of detector for all different types of sources, whereas for classical probability we describe the results we get for each single type of source and detector.

There are so many ramifications that I'm uncertain where to stop. For example, a "detector" can have a compound structure. It can be two detectors, your A and B, either close together or far apart, perhaps set to trigger as a pair only when they trigger at the same time. The electronics in real experiments can be byzantine, although it's the job of experimentalists to be able to tell other experimentalists what their electronics does so they can do the same thing, ideally in a different way.

Notoriously, there are certain kinds of explanations for the statistics we observe in Bell-inequality violating experiments that are not possible. They can't be explained by the idea that the central source sends what might be called classical particles to the two detectors. People have proposed other kinds of causal models, but Physicists have for the most part not found those models useful.

Again, this is embarrassingly incomplete, but the nature of your Question and of your response to John Schanck's more-or-less passable account in terms of the nitty gritty of quantum mechanics suggested to me that this type of narrative might be more useful for you.

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thanks for the contribution –  HDE May 30 '11 at 14:30

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