You might wonder, is the inability to uniquely determine measurement outcomes just a lack of complete knowledge about the system? Or put differently, could you gain some additional information beyond the quantum state that will help you determine outcomes exactly? The answer is "no", quantum phenomena are fundamentally probabilistic in nature and one way to show this is using an argument first made by John Bell in the 1960s.
Bell's theorem, as the result is called, says that there is no hidden mechanism that decides the outcome of a quantum measurement. The way the argument works is that if you assume that there is some unknown factor that manufactures the statistical results of a measurement, then you can establish a limit on the amount of correlation among certain properties you can measure, which you can state as a Bell inequality. You can then show that there are always some quantum states that can violate the inequality. In this post, we attempt to describe a simple example of Bell's theorem at work.
We will have to be a little bit more mathematical than usual in some parts but nothing too advanced is need: some arithmetic, absolute values, inequalities, a little trigonometry (really, just knowing what sine and cosine are), simple algebra such as if x = a + b and y = a - b then a = (x + y)/2, and taking "products" of states, that is |a) x |b) = |a)|b) = |a,b).
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| A typical experiment for testing a Bell inequality. A source emits many entangled pairs of electrons to detectors held by Alice on the left and Bob on the right. Each detector has 2 outcomes +1 or -1 for measuring the spin. There is a switch that lets Alice choose one of two possible directions for the spin measurement, labeled a and a'. Bob has a similar switch with settings b and b'. |
Consider the spin of an electron as a qubit, with two states |u) for up and |d) for down that can be distinguished perfectly. Qubits can be in superposition and following what we've done previously, we can have states like
|+) = 1/sqrt(2) |u) + 1/sqrt(2) |d)
|-) = 1/sqrt(2) |u) - 1/sqrt(2) |d).
Bell inequalities are often mentioned in the context of an experiment performed with entangled qubits, an example of which is shown above. In the setup, we have a source of pair of electrons in an entangled state
|Bell) = 1/sqrt(2) |u,u) + 1/sqrt(2) |d,d).
There are two detectors for measuring the spin of the electrons. We have Alice operating the left detector and Bob handling the right one. Each detector has two possible measurement settings, which we label a or a' on Alice's side, and b or b' on Bob's side. For now, it does not matter what the settings refer to exactly, only that Alice and Bob have 2 options for measuring the spin.
Whichever setting they use, they get one of two possible outcomes, which we will call +1 or -1. Again it does not matter which outcome you call +1 or -1, but to be consistent, we will usually call the nearest state when you move clockwise from |u) the +1 outcome. A short sample of the results of this experiment is also provided above.
Using Alice's and Bob's measurement results for electrons belonging to the same entangled pair, we can calculate the correlation
C = [ (number of +1 pairs) - (number of -1 pairs) ] / (total number of detected pairs).
We have a minus for the -1 pairs because they refer to outcomes that are anti-correlated. Observe that if Alice and Bob always both measure "|u) or |d)", then all detected pairs will be +1 pairs and C = +1. similarly if Alice and Both always both measures "|u) or |d)" but Bob has his detector inverted (so that |u) is the "+1" outcome for Alice while it is the -1 outcome for Bob), then all detected pairs will be -1 pairs and C = -1 in that case.
Alice and Bob will choose randomly between the measurement settings available to them so the only thing we can tell for certain is that C is a number between -1 and 1. For example, suppose that Alice measures "|u) or |d)" while Bob measures "|+) or |-)". If we write |u) and |d) in terms of |+) and |-), we find that
|u,u) = 1/sqrt(2) |u,+) + 1/sqrt(2) |u,-),
|d,d) = 1/sqrt(2) |d,+) - 1/sqrt(2) |d,-).
then
|Bell) = 1/2 |u,+) + 1/2 |d,+) + 1/2 |u,-) - 1/2 |d,-).
This means that there is 1/4 probability that you get any one of the four possible outcomes when Alice and Bob's measurements are taken together. This translates to getting, on average, the same number of +1 pairs as -1 pairs, which means C = 0 for this example.
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| Circular slice of the sphere of qubit states that contains the states |u) and |+). We measure the spin along the direction indicated by a pair of points |x) and |x') on this circle, where x is the angle between |x) and |u). |
Generally, we let Alice or Bob choose a measurement setting that is a certain angle away from |u) in that circle on the set of qubit states that contains |u) and |+) (automatically |d) and |-), too, since these are oppositely directed). The setting can be identified with an orientation in this circle, described by pair of opposite points, which we label |x) and |x'), and are given by
|x) = cos(x/2) |u) + sin(x/2) |d),
|x') = -sin(x/2) |u) + cos(x/2) |d).
Note that when x = 0°, we get the pair |u), |d) and when x = 90°, we get the pair |+) , |-).
We now have everything we need to "derive" a Bell inequality. The simplest version requires two settings for each detector, as we have in the experiment displayed earlier. Let A(a) be Alice's result (+1 or -1) on the left detector when it is set at a, and B(b) be Bob's result in the right detector when it is set at b, and same goes for a' and b'. Let C(a,b) be the correlation of measurement outcomes when Alice's detector setting is a and Bob's setting is b, and so on.
Suppose there was some hidden variable H that determines the outcome for each entangled pair of electrons based on the setting of the detector. For simplicity, we will assume that H takes only 2 values, 0 or 1. (We can make the argument for an H that takes many more values but the calculation is simpler when we only have two.) If H determines the correlation, then
C(a,b) = p(0) A(a|0) B(b|0) + p(1) A(a|1) B(b|1)
where p(0) is the fraction of measured pairs with H = 0, p(1) is the same with H = 1, A(a|0) is Alice's result given that the detector setting is a and H = 0, and the same goes for the other possibilities. Since we assume H must be 0 or 1, we also have p(0) + p(1) = 1.
Consider C(a,b) - C(a,b'), which is equal to
p(0) A(a|0) B(b|0) + p(1) A(a|1) B(b|1) - p(0) A(a|0) B(b'|0) - p(1) A(a|1) B(b'|1).
Now we add the following terms to it:
p(0) A(a|0) B(b|0) A(a'|0) B(b'|0) + p(1) A(a|1) B(b|1) A(a'|1) B(b'|1)
- p(0) A(a|0) B(b'|0) A(a'|0) B(b|0) - p(1) A(a|1) B(b'|1) A(a'|1) B(b|1).
That might look like a mess but if you look carefully, all that was is a fancy way of adding zero, because we are adding then subtracting the same stuff. If we group similar terms, we find that
C(a,b) - C(a,b') = p(0) A(a|0)B(b|0) [ 1 + A(a'|0)B(b'|0) ]
+ p(1) A(a|1)B(b|1) [ 1 + A(a'|1)B(b'|1) ]
- p(0) A(a|0)B(b'|0) [ 1 + A(a'|0)B(b|0) ]
- p(1) A(a|1)B(b'|1) [ 1 + A(a'|1)B(b|1) ].
Because the product A(a|H)B(b|H) is +1 or -1, its absolute value is +1, whatever the value of H is and even if we had setting a' or b'. Also, we know that p(0) and p(1) are bigger than or equal to zero.
An important result concerning absolute values is the so-called triangle inequality, which says that if you had any triangle with sides of lengths x, y, and z, then it must be true that
|x - y| ≤ z ≤ x + y.
You can check with examples to convince yourself that it's true. Like for a right triangle with sides of lengths 3, 4, and 5, you have |3-4| = 1 < 5 < 3+4 = 7.
If we take the absolute value of C(a,b) - C(a,b'), we can use the triangle inequality and |A(a|H)B(b|H)| = 1, etc. to get
|C(a,b) - C(a,b')| ≤ p(0) [ 1 + A(a'|0)B(b'|0) ] + p(1) [ 1 + A(a'|1)B(b'|1) ]
+ p(0) [ 1 + A(a'|0)B(b|0) ] + p(1) [ 1 + A(a'|1)B(b|1) ].
From the way we define correlations, and since p(0) + p(1) = 1, we can write this as
|C(a,b) - C(a,b')| ≤ 2 + C(a'b') + C(a',b).
Now this is what we get if we had added the messy stuff that was equal to zero. But we could have subtracted it instead, in which case we would end up with
|C(a,b) - C(a,b')| ≤ 2 - C(a'b') - C(a',b).
Because it should not matter whether we add or subtract zero, the inequality must be true when the terms on the right side achieve the smaller value, which we can write as
|C(a,b) - C(a,b')| ≤ 2 - |C(a'b') + C(a',b)|.
Finally, writing the inequality in its usual form,
|C(a,b) - C(a,b')| + |C(a'b') + C(a',b)| ≤ 2.
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| Measurement settings for Alice and Bob that achieve the maximum amount of violation of the Bell inequality we derived. |
So the question now is, do the correlations obtained from an actual experiment with electrons violate the inequality? The answer is yes, but as we have seen earlier, it is necessary to choose the detector settings for Alice and Bob to actually observe the violation. The simplest choice of settings is shown above, and ideally it would produce the maximum violation possible.
The calculation needed to show the violation is straightforward but a bit tedious so we will describe it in detail in the following.
Recall how the states |x) and |x') are related to |u) and |d). From that, we can write
|u) = cos(x/2) |x) - sin(x/2) |x')
|d) = sin(x/2) |x) + cos(x/2) |x').
To compute the correlations for the setting chosen in the picture, we need to write the state |Bell) in terms of the measurements that Alice and Bob perform.
When the settings are a and b:
|u,u) = cos(x/2) |u,x) - sin(x/2) |u,x')
|d,d) = sin(x/2) |d,x) + cos(x/2) |d,x').
If we label |u) and |x) by +1 and |d) and |x') by -1, the correlation C(a,b) can be computed by multiplying each value of a measurement with the probabilities of states that give that value. So we get
C(a,b) = (+1) ( p[|u,x)] + p[|d,x')] ) + (-1) ( p[|u,x')] + p[|d,x)] )
= cos^2(x/2) - sin^2(x/2)
= cos(x).
For setting b, we have x = 45° so C(a,b) = 1/sqrt(2). For setting b', we have x = 135° so C(a,b') = -1/sqrt(2).
To calculate the correlations C(a',b) and C(a',b'), recall that
|u,u) + |d,d) = |+,+) + |-,-)
so for Bob, we can write |+) and |-) in terms of |x) and |x'):
|+, +) = [ cos(x/2) + sin(x/2) ]/sqrt(2) ] |+, x) + [ cos(x/2) - sin(x/2) ]/sqrt(2) |+, x'),
|-, -) = [ cos(x/2) - sin(x/2) ]/sqrt(2) ] |-, x) - [ cos(x/2) + sin(x/2) ]/sqrt(2) |-, x').
Using cos(45°) = sin(45°) = 1/sqrt(2) and the trigonometric identity sin(x + y) = sin(x) cos(y) + cos(x) sin(y), we find that
C(a',b) = (+1) ( p[|+, x)] + p[|-, x')] ) + (-1) ( p[|+, x')] + p[|-, x)] )
= sin^2(45° + x/2) - sin^2(45° - x/2).
Using your handy calculator, you can check that C(a'b) = 1/sqrt(2) for x = 45° and C(a',b') = 1/sqrt(2) for x = 135°.
Putting all the correlations together to check the inequality, we get
|C(a,b) - C(a,b')| + |C(a'b') + C(a',b)| = 4/sqrt(2) = 2 sqrt(2) > 2.
The main assumption used in the Bell inequality is that any hidden mechanism that produces correlated results can only affect its immediate vicinity, since any sort of influence it makes is limited by the speed of light, according to special relativity. Since we should not be so eager to give up on Einstein's work, because entangled states can produce violations of a Bell inequality, it means no such local mechanism exists. The behavior of quantum objects represents a truly probabilistic phenomenon.
As my old advisor Berge Englert would say: "This is a brutal fact of life. In a very profound sense, quantum mechanics is about learning to live with it. "
Reference:
B.-G. Englert, Lectures on Quantum Mechanics: Basic Matters, pp. 4-8 (World Scientific, Singapore, 2006).
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