* * * * * * * * * * * * * *
* * * * * * * * * * * * * *
Contents Are Settings Communicated in Results? The Full Results The Assumptions Hardy’s Choices Counterfactuals Textbook Quantum Theory The Javascript Program Bell’s Inequalities What Does Alice See?
Quantum theory tells us that there can be correlations between the results of measurements of spatially-separated systems which are difficult to reconcile with special relativity theory. The purpose of this page is to use a javascript program to demonstrate and allow the exploration of this kind of correlation in the context of measurements of the spins of two spin one-half particles.
If you make a successful measurement of the spin of such a particle in any given direction, then you can only get one of two possible results. Either your measurement will tell you that the spin is in the direction of measurement or it will tell you that the spin is in the opposite direction.
Suppose that a system is set up at a central point which, at pre-arranged times, produces pairs of particles with each pair in a given pair-state spin wavefunction. One particle of each pair travels towards Alice while the other particle moves towards Bob. Alice chooses a measurement direction θ_{A} in a particular plane and records r_{A} = + if she measures the spin to be in that direction or r_{A} = − if she measures the spin in the opposite direction. Alice then sends her result to Bob. Meanwhile, Bob has independently chosen a measurement direction θ_{B}, recorded his own measurement r_{B} and sent his result to Alice. After a succession of such pairs, each produced in the same pair-state wavefunction, the two experimenters each have, and can share, a list of results of the form (θ_{A}, r_{A}; θ_{B}, r_{B}).
Our fundamental interest here is in how information flows in this situation, and in whether the predicted and observed consequences of quantum theory are compatible with our intuitions about this flow, as might, for example, be expressed in the following animation:
The first information flow here is in the movement of the particles from the central point to Alice and to Bob. The seond flow is in the records of the results moving between Alice and Bob and into their common listings. Notice that the particle which moves to Alice’s laboratory can certainly carry information about the initial quantum state of the particle pair, but that it could also carry information about the initial state of the particle pair which goes beyond the description given by the quantum state. For example, it could carry a label identifying the exact time at which the particle pair separated. One aim of our investigation is to understand whether we can imagine any such labels which could explain the observed quantum correlations. It seems intuitively obvious that there is no way that the particle travelling to Bob’s lab can carry information about how Alice intends to measure her particle when it reaches her. After all, as is indicated by the dial movement in the animation, she could decide on her setting after the particles have separated. Indeed, it is possible to imagine that both particles and records are moving at the speed of light, in which case, according to special relativity theory, there would be no conceivable way in which information could be carried from the space-time point of Alice’s decision to the space-time point of Bob’s measurement. In other words, there is no way in which a choice of measurement direction by Alice can affect the result of the measurement which has been chosen by Bob. It also needs to be stressed that complete records do not exist until they have been collected together.
It is sufficient for the degree of generality we wish to investigate here, to specify the spin wavefunction for the pair using just one parameter p, with ½ ≤ p ≤ 1.
Experiment indicates that, given the parameter p and the directions θ_{A} and θ_{B}, the successive results (r_{A}, r_{B}) are independent and identically distributed. This means that it is sufficient in the investigation of the probabilities of the values (r_{A}, r_{B}) to combine the results of any sequence of n measurements and merely to record the results either in the form (p, θ_{A}, θ_{B}, n, n_{++}, n_{+−}, n_{−+}, n_{−−}) or in the form (p, θ_{A}, θ_{B}, n, f_{++}, f_{+−}, f_{−+}, f_{−−}).
Here n_{++} is the number of times, out of n, that a pair is measured and A and B both record +, n_{+−} the number of times that A records + and B −, n_{−+} the number of times that A records − and B +, and n_{−−} the number of times that A and B both record −.
The f's are the frequencies of the various results, so that
f_{++} = n_{++}/n, f_{+−} = n_{+−}/n, f_{−+} = n_{−+}/n, and f_{−−} = n_{−−}/n.
Define n^{A}_{+} = n_{++} + n_{+−}, n^{A}_{−} = n_{−+} + n_{−−}, n^{B}_{+} = n_{++} + n_{−+}, n^{B}_{−} = n_{+−} + n_{−−} and
f^{A}_{+} = n^{A}_{+}/n, f^{A}_{−} = n^{A}_{−}/n, f^{B}_{+} = n^{B}_{+}/n, f^{B}_{−} = n^{B}_{−}/n.
Then n^{A}_{+} is the number of times out of n that A records +, n^{A}_{−} is the number of times that A records −, n^{B}_{+} is the number of times that B records +, and n^{B}_{−} is the number of times that B records −. f^{A}_{+}, f^{A}_{−}, f^{B}_{+}, and f^{B}_{−} are the corresponding frequencies.
Note that f_{++} + f_{+−} + f_{−+} + f_{−−} = f^{A}_{+} + f^{A}_{−} = f^{B}_{+} + f^{B}_{−} = 1.
It is somewhat simpler to use frequencies as, for large n, these should approximate the corresponding quantum probabilities. We shall use the notation pr_{± ±}, pr^{A}_{±}, and pr^{B}_{±} for these various limit-of-frequency probabilities. The javascript program driving this webpage is a genuine simulation. If it is working correctly then, after a run of 10,000 measurement-simulations, the frequencies produced are very likely to be within 0.015 of the theoretical quantum probabilities.
Note that, where necessary, the n's, f's, and pr's are to be considered as functions of p, θ_{A}, and θ_{B}.
We have seen that by the time an experiment is concluded, Alice knows Bob’s measurement direction and his result. She can also be assumed to know p. The first questions to be investigated, however, are whether it would be possible for Alice to learn the value of p, just by looking at the results of her measurements, and whether it is possible for her to learn the value of θ_{B}, just by looking at the results of her measurements.
We can investigate these questions by making various choices of θ_{A} and looking at how the frequency f^{A}_{+} at which Alice sees the plus result depends on p or on θ_{B}. This can be done by playing with the following sliders. Of course, the same questions arise for Bob and can be investigated in the same way.
Set measurement direction for Alice using this slider: θ_{A} = 0
Set p using this slider:
p = 0.5
Set measurement direction for Bob using this slider: θ_{B} = 0
Alice
number of measurements with the current settings: 0
Bob
number of measurements with the current settings: 0
Set θ_{A} to 0. Then, regardless of Bob’s choice of θ_{B}, f^{A}_{+} will eventually approach p. This means that Alice can learn the value of p just by looking at her results. Hence anyone controlling the p-slider can use it to send messages to Alice. These messages travel no faster than the speed of the measured particles. This is entirely consistent with special relativity theory.
If θ_{A} = 180 then, regardless of Bob’s choice of θ_{B}, f^{A}_{+} will eventually approach 1−p. On the other hand, if θ_{A} = 90 then f^{A}_{+} will approach ½ independent of p. In this case, the variance of f^{A}_{+} is maximal and so the approximation to the long-term value is likely to be least accurate. The same is true when p = ½, when, independent of the choices of θ_{A} and θ_{B}, both f^{A}_{+} and f^{B}_{+} approach ½.
Now fix p and choose any value for θ_{A}. The value approached by f^{A}_{+} in a long runs of tests will be stubbornly independent of θ_{B}. Thus it appears that moving his slider does not enable Bob to signal to Alice through her results. This is just as well as, as explained in the initial animation, it would enable him to send signals to Alice faster than would be allowed by the speed of light. This would violate special relativity.
The most obvious correlations between Alice and Bob’s results arise when Alice and Bob arrange always to choose the same directions. Setting the individual sliders to make this happen is tedious, so use the following radio buttons to cause them to move together or independently.
choose directions independently
make equal choices
When Alice and Bob make equal choices, they can discover that the pair-state wavefunction of the particles is symmetric. When p = ½, they will see this by finding that whatever result Alice records for some direction, Bob will necessarily record the same result. In other words, there is perfect correlation between the results in such a case. For any p, there is also perfect correlation if θ_{A} = θ_{B} = 0 or if θ_{A} = θ_{B} = 180. In all these cases we have pr_{+ −} = pr_{− +} = 0. Otherwise θ_{A} = θ_{B} merely implies that pr_{+ −} = pr_{− +}, or equivalently that pr^{A}_{+} = pr^{B}_{+}. Now the correlation is probabilistic rather than absolute, so that although in the long term Alice and Bob’s frequencies will agree, they won’t necessarily record the same result every individual experiment. This effect is strongest for values close to p = 1 and θ_{A} = θ_{B} = 90, 270.
At the end of each experimental run, Alice and Bob can construct a table recording the frequencies with which the various possible joint results have been observed. Such a table is presented below. Alice and Bob’s recent results for the current settings are given at the top of the table. “Never” indicates that the frequency with which a particular result has been seen is zero, and “always” that that frequency is one. The radio buttons above for making independent or equal choices of directions will also affect the sliders here. The perfect correlations mentioned above, when pr_{+ −} = pr_{− +} = 0, will produce two “nevers” in the corresponding rows of the table.
Set measurement direction for Alice using this slider: θ_{A} = 0
Set p using this slider:
p = 0.5
Set measurement direction for Bob using this slider: θ_{B} = 0
Alice
Bob
Alice | |
Bob |
p | θ_{A} | θ_{B} | n | f^{A}_{+} | f^{B}_{+} | f_{++} | f_{+−} | f_{−+} | f_{−−} |
---|
We would like to be able to explain these results in a way which is compatible with the two fundamental assumptions introduced above:
Assumption One: Given the parameter p and the directions θ_{A} and θ_{B}, the successive results (r_{A}, r_{B}) are independent and identically distributed.
Assumption Two: There is no way that a choice of measurement direction by Alice can affect the result of the measurement which has been chosen by Bob.
There are certainly two ways of explaining the results if we ignore assumption two. One is by textbook quantum theory and the other is by the javascript program which is driving this web page.
Suppose we could explain the results in a way compatible with both assumptions. Then our explanation might be taken to amount to:
Assumption Three: The results of any actual experiment are decided as soon as the particles separate.
Hidden in the word “actual” in this assumption are issues about what happens if there are probabilistic processes going on in each individual particle during their separation. As long as, in accordance with assumption two, these processes are not affected by the chosen measurement on the other particle, we can deal with these issues by imagining that, in each actual case, random events with the necessary probability distributions will occur at the moment of separation (“the dice are thrown”) after which the outcome of these process has been determined. This places restrictions on the probability distributions pr_{± ±} for the observed results. These are discussed briefly in the section here on Bell’s inequalities and at much greater length elsewhere, including in Bell 1987, in Maudlin 2002, and in Redhead 1987. However, at least for the moment, these details can be left to one side.
One way of satisfying assumptions one to three is to assume that the particles in each pair carry instruction listings consisting of a function r_{A}(θ) for the A-particle and r_{B}(θ) for the B-particle. These functions will have values in the set {+, −}, and the idea is that if Alice chooses the measurement direction θ_{A} and Bob the direction θ_{B}, she will obtain result r_{A}(θ_{A}) while he will obtain r_{B}(θ_{B}). r_{A} and r_{B} are functions of the possible angles which Alice and Bob can choose for their measurement directions. In general, any angle in [0, 360] will be possible, but more restricted sets will also be of interest. The instructions with the A-particle and the instructions with the B-particle may be correlated; in other words, when an experimenter prepares an actual pair of particles, that preparation will be a “common cause” of the instructions given to the particles and so the instructions might include the correlation noted above, that if p = ½ then, for all θ, r_{A}(θ) = r_{B}(θ). However, no A-instruction can be dependent on any choice which Bob could make after the particles have separated. It is not assumed that every A-particle or every B-particle has the same instructions, but rather that there is some probability distribution on a set of possible instruction pairs for the particle pairs.
Example One: If we assume that θ_{A} and θ_{B} are kept fixed for all time, then the functions r_{A} and r_{B} collapse down, for each particle, to single values (either + or −) and we can explain the experimental results for these fixed angles, by assuming that these value-pairs are distributed at the time of separation with the probabilities pr_{± ±} which are the asymptotic limits of the corresponding f_{± ±}. The javascript program does something rather similar, but ignoring assumption two and taking the actual values of θ_{A} and θ_{B} into account in the generation of each pair of outcomes.
Example Two: If we set p = 1, then it turns out that, for all θ_{A} and θ_{B}, Alice’s results are independent of Bob’s results. In other words, pr_{++} = pr^{A}_{+} pr^{B}_{+}. This is equivalent to each of the three equations pr_{+−} = pr^{A}_{+} pr^{B}_{−}, pr_{+−} = pr^{A}_{+} pr^{B}_{−}, and pr_{−−} = pr^{A}_{−} pr^{B}_{−}. If we make explicit the dependence on the parameters (θ_{A}, θ_{B}), and note that the outcome of our initial investigation tells us that pr^{A}_{+} is independent of θ_{B} and that pr^{B}_{+} is independent of θ_{A}, we would write the equation pr_{++} = pr^{A}_{+} pr^{B}_{+} in the form pr_{++}(θ_{A}, θ_{B}) = pr^{A}_{+}(θ_{A}) pr^{B}_{+}(θ_{B}). Then, for p = 1, we can explain the results for every θ_{A} and θ_{B} by assuming that r_{A} is a family of random variables indexed by θ_{A} such that Prob(r_{A}(θ_{A}) = +) = pr^{A}_{+}(θ_{A}) while Prob(r_{B}(θ_{B}) = +) = pr^{B}_{+}(θ_{B}).
If there was some such explanation uniform across all the possible values of (p, θ_{A}, θ_{B}) then we would have a local model of this part of quantum theory as a probabilistic theory compatible with special relativity theory. Unfortunately, as we shall see in the next investigation, this is not possible.
Lucien Hardy (1998, 1993, 1992) has provided a particularly clear illustration of the problem of correlations in quantum mechanics. For each p between ½ and 1, Alice and Bob are each given the choice of just two possible (p-dependent) measurement directions which we shall denote by ζ_{A} and η_{A} for Alice and by ζ_{B} and η_{B} for Bob. For example, when p = 0.5 + sqrt{5}/6 = 0.873 we will have ζ_{A} = 26.57, η_{A} = 116.57, ζ_{B} = 333.43, and η_{B} = 243.43. This is a special case in which the asymptotic probabilities are rational numbers (Mermin 2007, Appendix D). Another special case is p = 0.5 + sqrt{6{sqrt{5}-13}/2 = 0.823 when we will have ζ_{A} = 35.11, η_{A} = 111.46, ζ_{B} = 324.89, and η_{B} = 248.54. This p value gives the maximum value of pr_{−−}(ζ_{A}, ζ_{B}) which is 0.09. Allowing for minor changes in the form of the initial wavefunction, and noting that 111.46 = 180 - 68.54, both this p and these angles are in line with calculations by Hardy (1993).
Our instruction functions r_{A}(θ_{A}) and r_{B}(θ_{B}) are now functions of only two angles. Assigning a colour as above to each angle the four possible A-instructions can be written as ( + + )_{A}, ( + − )_{A}, ( − + )_{A}, ( − − )_{A}. The possible B-instructions can be written similarly, and there are sixteen possible instruction pairs, such as ( − + )_{A} ( + − )_{B}. For ease of reading, this pairing will be written as ( − + _{A} + − _{B}).
Set measurement direction for Alice:
ζ_{A} = 26.57
η_{A} = 116.57
Set parameters using this slider:
p = 0.873
Set a pair of directions:
26.57, 333.43Set measurement direction for Bob:
ζ_{B} = 333.43
η_{B} = 243.43
Alice
Bob
Alice | |
Bob |
p | θ_{A} | θ_{B} | n | f^{A}_{+} | f^{B}_{+} | f_{++} | f_{+−} | f_{−+} | f_{−−} |
---|---|---|---|---|---|---|---|---|---|
0.873 | 26.57 | 333.43 | 0 | ||||||
0.873 | 26.57 | 243.43 | 0 | ||||||
0.873 | 116.57 | 333.43 | 0 | ||||||
0.873 | 116.57 | 243.43 | 0 |
What instruction pairs can give rise to these results?
As already mentioned, we start with sixteen possibilities. However, after running “all four settings 10000 times”, there are (or, at least for p not ½ or 1 there should be) exactly three “nevers” in the above table and this rules out several of them.
The never in the second row of the table indicates that pr_{−−}(ζ_{A}, η_{B}) = 0 and this rules out four possibilities of the form ( − ± _{A} ± − _{B}).
From the third row, pr_{−−}(η_{A}, ζ_{B}) = 0 and this rules out three more of the form ( ± − _{A} − ± _{B}).
From the fourth row, pr_{++}(η_{A}, η_{B}) = 0 and this rules out four of the form ( ± + _{A} ± + _{B}).
The only instruction pairs which are left in the set of those which can have positive probability are
( + + _{A} + − _{B}),
( + + _{A} − − _{B}),
( + − _{A} + + _{B}),
( + − _{A} + − _{B}),
( − − _{A} + + _{B}).
This is a contradiction as we can see from the first row of the table that pr_{−−}(ζ_{A}, ζ_{B}) > 0 and this means that there needs to be at least one instruction pair of the form ( − ± _{A} − ± _{B}) which has positive probability.
It follows that it is not possible to set up a system of instruction pairs even just to explain the results of measurements for the sets of values of (p, θ_{A}, θ_{B}) considered by Hardy, let alone for any larger set.
It is fundamental to human existence to think about what might have happened:
Would I have tripped on the stairs if I had turned the light on?
Well maybe next time I should be more careful.
Physics can be expressed as a theory of counterfactuals to just the same extent as it can be expressed as a theory of prediction: If this had been the state of the world, then this would have be what would have been likely to happen and this is how likely it would have been. If we had changed the state, then we would have changed the outcome, or at least the outcome probabilities. Setting our face against fatalism by learning from our mistakes is a significant driver of science.
Hardy’s analysis can also be discussed in terms either of predictions or of counterfactuals. If Alice chooses ζ_{A} and Bob chooses η_{B} then, after a long run of tests, we see that r_{A} = − and r_{B} = − never happen at the same time and so we can predict that if we were to run the same test again, we would still not see that outcome. But we could also claim that, if the same choices had been made on one additional occasion, and if Bob’s result had been − then Alice’s result would have had to be +. The same holds if Alice chooses η_{A} and Bob chooses ζ_{B}. Finally, if Alice chooses η_{A} and Bob chooses η_{B} then the outcome (r_{A}, r_{B}) = (+ +) is ruled out.
Now adopt assumption two and consider a single particle-pair with Alice choosing ζ_{A} and Bob choosing ζ_{B}. Suppose that Alice gets the result r_{A} = −.
Suppose that counterfactuals make sense for this particle-pair so that we can ask whether there is anything interesting to be said about what would have happened if the directions chosen had been different.
According to assumption two, Alice’s result is unaffected by Bob’s choice. This means that Alice would still have got the result r_{A} = −, even if Bob had chosen η_{B}. But, in that case, with ζ_{A}, η_{B} and r_{A} = −, Bob would have had to get the result r_{B} = +.
And he would still have got that result even Alice had chosen η_{A} (to follow this argument, see the table below). But, in that case, with η_{A}, η_{B} and r_{B} = +, she would have had to get the result r_{A} = −. And she would still have got the result r_{A} = −, even if Bob had chosen ζ_{B}. But, in that case, with η_{A}, ζ_{B} and r_{A} = −, Bob would have had to get the result r_{B} = +. And then we know that Bob must have got the result +, as he has in fact chosen ζ_{B} and we know that his result is independent of whether Alice chooses ζ_{A} or η_{A}.
Once again we have ruled out the possibility, which quantum theory tells us can actually happen, that (θ_{A}, r_{A},θ_{B}, r_{B}) = (ζ_{A}, −, ζ_{B}, −). As well as summarizing the argument, the following table draws attention to the fact that, in its course, we consider three counterfactual situations.
θ_{A} | θ_{B} | r_{A} | r_{B} | argument |
---|---|---|---|---|
ζ_{A} | ζ_{B} | − | premise | |
ζ_{A} | η_{B} | − | r_{A} independent of Bob’s choice | |
ζ_{A} | η_{B} | − | + | never − − |
η_{A} | η_{B} | + | r_{B} independent of Alice’s choice | |
η_{A} | η_{B} | − | + | never + + |
η_{A} | ζ_{B} | − | r_{A} independent of Bob’s choice | |
η_{A} | ζ_{B} | − | + | never − − |
ζ_{A} | ζ_{B} | + | r_{B} independent of Alice’s choice | |
ζ_{A} | ζ_{B} | − | + | premise |
ζ_{A} | ζ_{B} | − | + | conclusion and contradiction: never − − |
This is essentially another version of the previous argument, with instruction functions being interpreted as telling us what would have happened had different angles been chosen. A not uncommon response to these problems (e.g. Mermin 2007, Appendix D) is simply to suggest that, by definition, counterfactuals refer to what does not happen, and so their analysis is irrelevant. This is a mistake, however, because what can be said about counterfactuals can be said to just the same extent about predictions. If we can’t say what might have happened, how can we say what might be going to happen? In other words, how are the actual results caused, if there is no way in which the results for differing measurements directions can be predicted when the particles separate, and yet Alice and Bob are free to choose those directions after that time?
According to the textbooks, a pair of spin-half particles has a spin state described by a four-component wavefunction of the form a_{++}|++> + a_{+−}|+−> + a_{−+}|−+> + a_{−−}|−−> where the a's are complex numbers satisfying |a_{++}|^{2} + |a_{+−}|^{2} + |a_{−+}|^{2} + |a_{−−}|^{2} = 1.
When Alice and Bob both set their measurement direction to 0, the probability of them both finding their spins to be in that direction and thus of them both getting the + result is
pr(r_{A} = + & r_{B} = +) = |a_{++}|^{2}.
Similarly, pr(r_{A} = + & r_{B} = −) = |a_{+−}|^{2}, pr(r_{A} = − & r_{B} = +) = |a_{−+}|^{2}, and pr(r_{A} = − & r_{B} = −) = |a_{−−}|^{2}.
A change in θ_{A} corresponds to a rotation of the direction in which Alice makes her measurement. This can be expressed by a coordinate change for the wavefunction calculated by applying a 4 × 4 unitary matrix to the a-coefficients. The result is to change (a_{++}, a_{+−}, a_{−+}, a_{−−}) to
(a_{++} cos θ_{A}/2 + a_{−+} sin θ_{A}/2, a_{+−} cos θ_{A}/2 + a_{−−} sin θ_{A}/2, -a_{++} sin θ_{A}/2 + a_{−+} cos θ_{A}/2, -a_{+−} sin θ_{A}/2 + a_{−−} cos θ_{A}/2).
A change in θ_{B} has a similar effect. Following these rotations, the four equations above are replaced by equations of the form
pr(r_{A} = ± & r_{B} = ±|θ_{A}, θ_{B}) = |a_{±±}(θ_{A}, θ_{B})|^{2}.
It is through this joint functional dependence of measurement result probabilities that quantum theory violates assumptions two and three. However,
pr(r_{A} = +|θ_{A}, θ_{B}) = pr(r_{A} = + & r_{B} = +|θ_{A}, θ_{B}) + pr(r_{A} = + & r_{B} = −|θ_{A}, θ_{B}) = |a_{++}(θ_{A}, θ_{B})|^{2} + |a_{+−}(θ_{A}, θ_{B})|^{2}
is independent of θ_{B} and similarly pr(r_{B} = +|θ_{A}, θ_{B}) = |a_{++}(θ_{A}, θ_{B})|^{2} + |a_{−+}(θ_{A}, θ_{B})|^{2} is independent of θ_{A}.
It is because of this independence that Alice cannot detect Bob’s measurement settings from her results. The independence can be proved, either by explicit calculation, or by using the fact that the quantum operators which define the rotations and the measurement projections can be considered to act locally on the wavefunction in the sense, for example, that Alice’s measurement operator commutes with Bob’s rotation operator.
In the process considered above, the first stage is the choice of the parameter p. This makes a choice of the values for the a-coeffients in the standard coordinates, by taking the spin wavefunction to be
sqrt{p} |++> + sqrt{1 − p} |−−>.
This choice makes p a measure of the entanglement between the two spins, with p = ½ corresponding to maximal entanglement and p = 1 corresponding to no entanglement. Given this choice, the a-coefficients in the coordinates determined by Alice and Bob’s angle choices (dependent on p as well as on the angles), will always remain real under the single-axis rotations we consider here.
This button will , the squared amplitudes (the |a_{± ±}(p, θ_{A}, θ_{B})|^{2}), and the quantum probabilities pr^{A}_{+}(θ_{A}) = pr(r_{A} = +|θ_{A}, θ_{B}) and pr^{B}_{+}(θ_{B}) = pr(r_{B} = +|θ_{A}, θ_{B}) at each of the processes considered above. In the tables, the frequencies f_{± ±} should tend, for large n, to these squared amplitudes (so that pr_{± ±} = |a_{±±}|^{2}) while f^{A}_{+} tends to pr^{A}_{+} and f^{B}_{+} to pr^{B}_{+}. Note the strange but well-known property of spin-half particles that a rotation by 360 degrees on either particle will have the effect of changing the sign of the wavefunction. In order to demonstrate Hardy’s argument, the wavefunctions considered here are all symmetric. Experimental tests of quantum non-locality on identical spin-half particles usually use anti-symmetric spin wavefunctions and Bell’s inequalities rather than Hardy’s theory. This would restrict us to p = ½, but otherwise the changes to the mathematics are minor. Discussions in the literature often involve photon polarization states with the result that angles are halved relative to the angles used here. Non-locality seems to be a ubiquitous problem in quantum theory, and not just in the systems where we can measure it.
This webpage runs this computer program on your browser. This simulates events with the squared amplitude quantum probabilities discussed above. At each simulated experiment, your browser takes a random number Rndm produced by your computer which is supposed to be uniformly distributed on the interval [0, 1]. The program divides the interval [0, 1] into four disjoint subintervals with lengths |a_{± ±}(p, θ_{A}, θ_{B})|^{2} and returns results for Alice and Bob according to which subinterval Rndm happens to be in. Although, at least with the minor changes required to replace symmetric with anti-symmetric wavefunctions and with the restriction to p = ½, this is an accurate simulation of the predicted and observed results of genuine experiments, it does not seem to be a particularly accurate model of the animation at the top of this webpage. Rather, what it seems to be modelling is this:
The information flow suggested here, starts with the setting of the measurement directions by Alice and Bob. Then, before the particles separate, the directions are communicated to them, and the outcomes of the pre-determined measurements are decided and then shared out to the places where the measurements are supposed to occur. If this is at all realistic as a model of reality, then our understanding of causality, of spacetime geometry, and even of the direction of time are seriously flawed.
For each value of p, Hardy’s directions are specially chosen to produce the three “nevers” in the corresponding table of results. These special choices are sufficient in themselves to show that quantum theory is not compatible with assumptions one, two, and three. Such incompatibility problems, however, are much more widespread. Indeed, if we restrict ourselves to the measurements considered above, for any p ≠ 1, there are open intervals of four angles, a pair for Alice and a pair for Bob, as in the Hardy scenario, such that, in each case, the quantum probabilities violate an inequality (a Bell inequality) which must hold for probabilities compatible with the assumptions.
To express this compatibility formally, we may suppose that there is a set Ω of possible particle-pairs together with the information attached to them, that there is a probability distribution q on Ω, and that each actual experiment corresponds to some ω ∈ Ω. Then suppose that, given ω, the results r_{A}(θ_{A}) and r_{B}(θ_{B}) for measurement directions θ_{A} and θ_{B} have probabilities pr(r_{A}(θ_{A})|ω) and pr(r_{B}(θ_{B})|ω) and that any possible correlation between Alice’s result and Bob’s is determined only by the “common cause” ω of the initial particle-pair and its information and not is affected by any later decision. It follows that
pr(r_{A}(θ_{A}) & r_{B}(θ_{B})|ω) = pr(r_{A}(θ_{A})|ω) pr(r_{B}(θ_{B})|ω).
In other words, it follows that, given ω (the “actual” particle-pair), the results of Alice’s measurements are independent of the results of Bob’s measurements.
Quantum theory does not give Alice and Bob any way to know about ω, so it is often referred to as a “hidden variable”. What they can measure is the long-term relative frequencies pr^{A}_{±}(θ_{A}), pr^{B}_{±}(θ_{B}), and pr_{± ±}(θ_{A}, θ_{B}). If the analysis just given is correct, then we should have a set of identities such as
pr_{+ +}(θ_{A}, θ_{B}) = ∫ pr(r_{A}(θ_{A}) = + & r_{B}(θ_{B}) = +|ω) q(ω) dω.
These identities together with the previous equation give rise to various inequalities (Bell’s inequalities and their generalizations). These inequalities do not in fact always hold. They fail both for the theoretical quantum probabilities, and for certain empirically measured relative frequencies. This failure is the most powerful way of analysing the non-locality of quantum theory. We shall not go into the derivation of Bell’s inequalities here. Instead, we shall use the scenarios developed above to show how often they fail, and and we shall also give an analysis of a situation in which a local hidden variable theory is possible.
The most widely used version of Bell’s inequality was developed by Clauser, Horne, Shimony, and Holt 1969. They showed, that if Alice and Bob each choose two angles (ζ_{A}, η_{A}, ζ_{B}, η_{B}), then the quantity
S_{CHSH}(ζ_{A}, η_{A}, ζ_{B}, η_{B}) = C(ζ_{A}, ζ_{B}) + C(η_{A}, ζ_{B}) + C(ζ_{A}, η_{B}) - C(η_{A}, η_{B})
where C(θ_{A}, θ_{B}) = pr_{+ +}(θ_{A}, θ_{B}) - pr_{+ −}(θ_{A}, θ_{B}) - pr_{− +}(θ_{A}, θ_{B}) + pr_{− −}(θ_{A}, θ_{B})
will always satisfy the inequality |S_{CHSH}| ≤ 2 if Bell’s common cause assumption holds. However, if we use the quantum probabilities pr_{± ±}(θ_{A}, θ_{B}) then we can only show that |S_{CHSH}| ≤ 2 √ 2 = 2.828.
S_{CHSH} may appear quite complicated, but in fact it is not difficult to calculate explicitly in the present case. Indeed, noting the dependence on the parameter p, the function C(θ_{A}, θ_{B}, p) is given by
C(θ_{A}, θ_{B}, p) = cos θ_{A} cos θ_{B} + 2 (p(1-p))^{1/2} sin θ_{A} sin θ_{B}.
In making our calculation, we shall actually replace S_{CHSH} by its maximum over interchanges between ζ_{A} and η_{A} and between ζ_{B} and η_{B} as this gives a better indication of the impossibility of a theory compatible with assumptions one, two, and three.
We can also use the program driving this webpage to simulate S_{CHSH} using the frequencies f_{± ±}(θ_{A}, θ_{B}) in place of the limiting probabilities. We shall use the notation F_{CHSH}(ζ_{A}, η_{A}, ζ_{B}, η_{B}, p, n) for this simulation. Here n denotes the number of times the experiment has been simulated with each of these angle-pairs, and so we expect that as n → ∞, this will converge to S_{CHSH}. Note that frequencies are not subject to the same constraints as their limits. F_{CHSH} has a maximum of 4 and because of the maximisation involved in its calculation, it may often be above S_{CHSH}.
For each value of p < 1, Gisin 1991 proved that there are choices of angles for which the Bell inequality |S_{CHSH}| ≤ 2 is violated. A slider below allows Gisin’s angles to be chosen for each value of p. The inequality violation is maximal for these angles, and for p = ½, S_{CHSH} reaches its absolute maximum of 2 √ 2. We start with these settings. For each p < 1, there are four distinct one-dimensional paths in the four-dimensional angle space on which the maximal Gisin value for |S_{CHSH}| is attained. For each setting of p, except p = 1 where the path degenerates, another slider allows one of these paths to be traced. |S_{CHSH}| is continuous, and so Bell’s inequality is violated throughout a neighbourhood of this path. For ½ < p < 1, the inequality is also violated with Hardy’s choice of settings. Example two above shows that Bell’s inequality must be obeyed for p = 1 while example three below will show that it must be obeyed for all p if we chose η_{A} = ζ_{A}. All this can be investigated by playing with the settings below.
Set measurement directions for Alice using these sliders:
ζ_{A} = 0
η_{A} = 90
Set p using this slider:
p = 0.5
Set p and Gisin angles.
Set p and Hardy angles.
Trace a maximal path.
Set measurement directions for Bob using these sliders:
ζ_{B} = 45
η_{B} = 315
Alice
Bob
Example Three: Suppose that Alice’s direction θ_{A} is fixed and public and that she is not allowed to make any other choice. Bob however is still free to choose any measurement direction. In this circumstance, it turns out that Bell’s inequalities are satisfied and that it is possible to construct a theory compatible with assumptions one, two, and three and with our initial animation.
The quantum probabilities are given as above in the form a_{± ±}(θ_{A}, θ_{B})^{2}. As the a's are real, we do not need to take absolute values. Write
x_{0}(θ_{B}) = a_{++}(θ_{A}, θ_{B})^{2}, noting that the dependence on θ_{A} is not significant because θ_{A} is assumed to be both fixed and known at the moment when the particles separate,
x_{1} = a_{++}(θ_{A}, θ_{B})^{2} + a_{+−}(θ_{A}, θ_{B})^{2} = pr^{A}_{+}, noting that, as discussed above, this sum does not depend on θ_{B},
x_{2}(θ_{B}) = a_{++}(θ_{A}, θ_{B})^{2} + a_{+−}(θ_{A}, θ_{B})^{2} + a_{−+}(θ_{A}, θ_{B})^{2}.
Finally, note that a_{+ +}(θ_{A}, θ_{B})^{2} + a_{+−}(θ_{A}, θ_{B})^{2} + a_{−+}(θ_{A}, θ_{B})^{2} + a_{−−}(θ_{A}, θ_{B})^{2} = 1.
Now we can set Ω = [0, 1] with q being the uniform distribution. Given ω ∈ Ω we can define local probabilities as follows:
pr(r_{A} = +|ω) = 1 if and only if ω ∈ [0, x_{1}],
pr(r_{A} = −|ω) = 1 if and only if ω ∈ [x_{1}, 1],
pr(r_{B}(θ_{B}) = +|ω) = 1 if and only if ω ∈ [0, x_{0}(θ_{B})] ∪ [x_{1}, x_{2}(θ_{B})],
pr(r_{B}(θ_{B}) = −|ω) = 1 if and only if ω ∈ [x_{0}(θ_{B}), x_{1}] ∪ [x_{2}(θ_{B}), 1].
We can now define pr(r_{A} & r_{B}(θ_{B})|ω) = pr(r_{A}|ω) pr(r_{B}(θ_{B})|ω) and then, for example,
pr(r_{A} = + & r_{B}(θ_{B}) = +|ω) = 1 if and only if ω ∈ [0, x_{1}] = x_{0}(θ_{B}).
This, and similar equations, shows that this provides a local theory in which, for each θ_{B}, the probabilities
pr_{± ±}(θ_{A}, θ_{B}) = ∫ pr(r_{A}(θ_{A}) & r_{B}(θ_{B})|ω) dω
agree with the quantum probabilities.
The intuition now is that each actual pair of particles carries a triple consisting of (p, θ_{A}, ω). Each particle will then have sufficient information to decide its response to the measuring device that it meets, as the A-particle will be able calculate x_{1}, and the B-particle will be able to calculate both x_{1} and x_{0}(θ_{B}).
This example may be useful as a demonstration of what might be meant by a local theory, but it depends on restricting the possible measurements available to Alice, and the premise that θ_{A} could somehow be available to the B-particle remains absurd.
As our final topic, we shall take the point of view of an individual observer, and very briefly consider where this might lead us.
At the beginning of each experiment, Alice sets or learns the value of p:
p = 0.875
Alice can then work out that, in the initial choice of coordinates, the particle-pair spin wavefunction is
The particles are now released and Alice sets her measurement direction: θ_{A} = 0
At this point, Alice would like to work out the particle-pair wavefunction with respect to the new angle coordinates. However, this is impossible as she does not yet know what direction Bob might choose, or might have chosen, for his measurement. She is limited instead to knowledge of the effective spin-state (or “density matrix”) of her own rapidly-approaching particle. This is
As she is going to measure the spin along her chosen direction, the relevant part of this for Alice's present purposes is the coefficient of |+><+|. This equals pr^{A}_{+} and is the probability of her getting the result +. The coefficient of |−><−| is of course pr^{A}_{−} = 1 - pr^{A}_{+}. The other coefficients enable her to calculate outcome probabilities for measurements which she might have made in other directions.
Her result, simulated with the correct probability, is ?
It seems intuitively clear that this result cannot be affected by a direction that Bob has not yet chosen, or which he might have chosen but not yet set on his measuring device. Indeed, this seems clear not only if these events lie in the future of Alice’s measurement, but even if they are merely spacelike separated from Alice’s measurement. If Alice and Bob are sufficiently far apart, there may be no answer to the question “which measurement came first?” and so there may be no way that a causal history can even be drawn, let alone investigated. And yet if we believe that Alice’s result is unaffected by Bob’s choice of direction and that Bob’s result is unaffected by Alice’s choice of direction, then we run smack into the contradiction with quantum probabilities which we have already discussed.
Now suppose that Alice learns what Bob’s measurement direction is going to be, before she learns the result of his measurement.
Bob’s measurement direction is going to be: θ_{B} = ?
Alice can now use textbook quantum mechanics to calculate the probability pr(r_{B} = +|r_{A}) with which she expects to see Bob tell her that the result of his measurement is +:
In coordinates with these angles, the wavefunction before either measurement will be
Alice’s result corresponds to measurement of the projection operator
pr(r_{B} = +|r_{A}) =
So far, this is entirely uncontroversial. Quite what it means, however, is not so clear. It might be that Bob has already made his measurement in the sense that his measurement lies in Alice’s past, but that Alice has yet find out what it is. In that case, it would be natural to interpret pr(r_{B} = +|r_{A}) as the credence, or degree of belief, that Alice should assign to one of the possible outcomes of an unrevealed event, as if she was betting on a number on a scratch card. On the other hand, if Bob’s measurement lies in Alice’s future, we might interpret pr(r_{B} = +|r_{A}) as the propensity for an outcome of a pending quantum event. However, at spacelike separation, the meaning would seem, somewhat absurdly, to be coordinate dependent.
I cannot myself see any way of avoiding this absurdity and the other problems of quantum non-locality, except by arguing that, for Alice, pr(r_{B} = +|r_{A}) is always a propensity for an outcome of a pending quantum event until she actually learns r_B. That however means that, for Alice, Bob’s result happens not when he learns it, but when she learns it. And similarly, of course, for Bob, Alice’s result happens not when she learns it, but when he learns it. This produces a many-minds interpretation. According to the simplest version of that scenario, there will be two Alices after her measurement, one seeing +_{A} and the other seeing −_{A}, and there will be four after the first two learn Bob’s result, seeing +_{A}+_{B}, +_{A}−_{B}, −_{A}+_{B}, and −_{A}−_{B}. In this scenario, it is true that the result r_{A} that any Alice herself sees is independent of Bob’s choice, but the result r_{A} that a Bob sees Alice seeing does depend on his choice. The fact that there are no instruction pairs then indicates that the space of posssibilities for quantum theory, on which the fundamental propensities are to be defined, should be a space of possible observers rather than a space of possible physical events.
I am grateful to Yael Loewenstein for making me realise that I needed to try and clarify my explanations of this subject. I am also grateful to the people responsible for WW3 schools from where I have learnt much of the Javascript used here.
Bell, J.S. (1987) Speakable and Unspeakable in Quantum Mechanics. (Cambridge)
Clauser, J.F., Horne, M.A., Shimony, A., and Holt, R.A. (1969) “Proposed experiment to test local hidden-variable theories.” Phys. Rev. Lett. 23, 880–884. DOI:10.1103/PhysRevLett.23.880
Gisin (1991) “Bell’s inequality holds for all non-product states.” Phys. Lett. A 154, 201–202. DOI:10.1016/0375-9601(91)90805-I
Hardy, L. (1998) “Spooky action at a distance in quantum mechanics.” Contemp. Physics 39, 419–429. DOI:10.1080/001075198181757
Hardy, L. (1993) “Nonlocality for two particles without inequalities for almost all entangled states.” Phys. Rev. Lett. 71, 1665–1668. DOI:10.1103/PhysRevLett.71.1665
Hardy, L. (1992) “Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories.” Phys. Rev. Lett. 68, 2981–2984. DOI:10.1103/PhysRevLett.68.2981
Maudlin, T. (2002) Quantum Non-Locality and Relativity. (Blackwell)
Mermin, N.D. (2007) Quantum Computer Science. (Cambridge)
Redhead, M. (1987) Incompleteness, Nonlocality, and Realism. (Oxford)
October 2019